ENGINEERING RESEARCH INSTITUTE THE UNIVERSITY OF MICHIGAN ANN ARBOR Final Report ELLIPTIC FUNCTIONS WITH COMPLEX ARGUMENTS F. M. Henderson Senio.r Leturer in Civril Engineering Uni-ersity- -of,TNew. Zeealand Project 2666 NATIONAL SCIENCE FOUNDATION WASHINGTON, D, C, October 1957

ACKNOWLEDGEMENT The work embodied in this repbort was done while the author was a visiting professor in the Engineering Mechanics Department, The University of Michigan, during the academic year 1956-570 The computing work was done on the University's IBM 650 digital computer. The author wishes to acknowledge not only the generous support given to the project by the National Science Foundation, but also the helpfulness of its advisors and officers in speeding up the processing of the grant so that the large amount of computing and drafting work could be completed in the limited time available. He wishes also to thank the officers of the University's Engineering Research Institute for their very helpful attitude, particularly when difficulties arose due to time limitations; similar thanks are due to the staff of the statistical laboratory for advice and assistance in programming the computing worko Most of all I should like to acknowledge the fine work of the five students who did all the drafting work in the extremely short time of three months, under the able leadership of Mro Gunay Aktay. It is a pleasure to acknowledge the help received from Professor G. W. Spenceley, of Miami University, Ohio, when this project was first conceived two years ago. His advice, encouragement, and support of the scheme in the appropriate quarters have helped greatly to bring the project to a successful conclusiono

ABSTRACT Values of the elliptic functions sn w, cn w, dn w, and E(w) are tabulated for complex values of the argument w = u + iv. Results are not only tabulated but are also plotted in the form of charts which illustrate the behavior of these functions in the w plane. Contours of x and y, the real and imaginary parts of the elliptic function, are drawn on the w plane, giving a representation of the conformal mapping problem to which elliptic functions provide the solution. Advantages claimed for this arrangement are compactness, convenience, and the simple illustration of the general behavior of the functions. OBJECTIVE The objective of this report is the computation of values of the elliptic functions sn w, cn w, dn w, and E(w) for a full range of complex values of the argument, w = u + iv. Values of the modulus k = sin G are such that G occurs at 5~ intervals, except for values of 1~ and 89~ at the extreme ends of the range.

1,0 INTRODUCTION The results contained in the Appendices to this report are intended to fill the need for complex values of the elliptic functions that arises in connection with plane harmonic, or "potential" problems, i.e., the mapping problems of hydrodynamics and electromagnetic theory. When such problems require the mapping of the interior of a rectangle, or of some other figures consisting of mutually perpendicular straight lines, on to the upper half of another plane, then use of the Schwarz-Cristoffel transformation almost always introduces one or another of the elliptic integrals, The form of the problem has suggested the method of presentation; as will be shown in more detail below, the charts in the Appendices are drawn up so as to display the solution of the typical mapping problem, and hence to show in compact and convenient form the (complex) values of the elliptic functions for the whole range of (complex) values of the argument. Conversely we may,. in the case of the sn-function, obtain the value of the elliptic integral of the first kind for any complex argument without the interpolation usually necessary in the use of tables of elliptic functions. Following this mode of presentation has of course meant a sacrifice of precision, However, the charts can be read directly to two significant figures and by estimation to three figures; this, after all, corresponds closely to the "slide-rule accuracy" which governs, in practice, the solution of many problems in physics and engineering. More precise tables could easily be drawn up by modern high-speed computing machines, but the resulting volume would be extremely bulky. Furthermore, the trend seems to be away from the formal publication of such tables since on the rare occasions when precise values are needed they can be easily obtained on high-speed computers. There is still, however, a strong case for the type of display embodied in this report, from which quick rough values, and a general picture of the function's behavior, can be obtained. Opposite each chart is a table giving values to four significant figures, with u and v taken at intervals of 1/10 K and 1/10 K', respectively. These tables may be useful to those who need rather more precision than the charts can offer, although the intervals of u and v are rather -too large for satisfactory linear interpolation. Space limitations, however, precluded taking u and v at smaller intervals. The functions dealt with are = sn w, z = cn w, z = dn w, and E = E(w)o This last is plotted as a function of w, since it is most easily dealt with as an explicit function of w; it would be possible, although more difficult, to treat it as a function of z, i.e,, as the elliptic integral of the second kind, 1

and in this form the results would be more directly usefulo This point will be amplified in the discussion below, in the meantime it may be remarked that the tabulation and plotting of E = E(z) suggests itself as a desirable addition to the work embodied in this report. The following paragraphs are intended to summarize enough of the theory of elliptic integrals and elliptic functions to make the tables and charts understandableo They are essentially a statement of certain equations and propositions without any pretension to completeness in the manner of treatment. There are many useful tests available on the subject. Many texts on advanced calculus, e go, Franklin's,1 have a chapter giving a brief basic treatment of the subject; Bowman2 goes further and gives a thorough treatment of complex values, MilneThomson3 gives a remarkably complete, although brief, discussion of the equations necessary for manipulating the elliptic functions and calculating all real and complex values, together with a basic set of tables; Neville has an attractive account of the underlying theory. Byrd and Friedman5. make a very complete job of the more modest task of showing how to reduce any integral basically elliptic in form (see Section 2o0), to the standard forms of elliptic integral. The principal authority for the advanced general theory is Whittaker and Watson9s6 comprehensive treatment of the subjecto Where the text goes beyond what is strictly necessary to introduce the tables and charts, it has been in the attempt to summarize information on how the elliptic integrals may be evaluated for all real and imaginary values of the argument. This is a necessary background to the problem of coping with complex arguments o 2o0 THE ELLIPTIC INTEGRALS The integrals xx dx F (k,x) =, O<k< (1) Jo (l-x) (l-k2x 2) E x X t (nkx) = —, O<k<l (5) Jo (l+nx2) (1-x2) (1-k2x2) are defined as elliptic integrals of the first, second, and third kinds, respec2

tivelyo The integral of the first kind may be transformed by the substitution x = sin 0 into: F (ko) = d dX- (4a) o J-l-k2sin2o and the integral of the second kind transforms under the same substitution into: E (k,0) = / l-k2sin20 do. (4b) These are known as Legendre's normal forms. 0 is termed the amplitude and k the modulus. We shall refer to x, whether real, imaginary, or complex, as the argument o The integral of the third kind will not be discussed further: since it is a function of three variables, it is usually regarded as being beyond the scope of any systematic tabulation. Since k may take any value between 0 and 1, it has become customary to regard k itself as the sine of an angle, usually termed a or G. In most tables the integral is tabulated as a function of a and 0, with values for both angles taken at 1~ or 5~ intervals from 0~ to 90~o It is shown in standard texts on the subject that if X is any quartic function of x, it is always possible to reduce the integral dx to an elliptic integral of the first kind. Depending on the form of X it is sometimes, but not always, possible to express the integral xndx J E in terms of elliptic integrals of the first and second kind. In many cases, however, this is more easily said than done, but Byrd and Friedman5 provide a very useful guide. 2o1 THE BEHAVIOR OF THE ELLIPTIC INTEGRALS Considering the form of the integral given in Eq. (1), it will be seen that for all values of x in the range 0 < x < 1 the integrand is real; hence the integral is realo For 1 < x < 1/k the integrand is imaginary; hence the integral is complex with a constant real component equal to the integral taken between the limits 0 and lo For x > 1/k the integrand is once more real; hence the integral is complex with constant imaginary component equal to the integral taken between the limits 1 and 1/k.'5

When x is an imaginary number equal to, say, i, where ~ is a real number, we have: dx F (k, i) = J (l-x2)(l-k2x2 The substitution x = iy leads to the integral dy io (l+y2)(l+k2y2)' and the further substitution y = tan 0 leads to tan-1 _ d ____ 1 Jo' - (l-k2)sin2' which can be written as i F'" )-k2a ) Hence we have, finally, F (k,i ) = i F (l-k2, J+ ) (5) The positive root N-k2 is usually written as k9, and is called the complementary modulus., The complete elliptic integral is the integral of Eq. (1) taken between the limits 0 and 1. It is thus a function of the modulus k alone and is given the symbol K. The complete integral corresponding to the complementary modulus k' is denoted by K' It is possible to evaluate the integral n:1Edx F (kil/k) = Jo F (k~l/k) fl /(l-x2)(1-k2x2) in terms of the complete integrals K and K', by means of suitable substitutions. We have F (kl/k) = K+ 1 J'1(_X2) (1k2X2) The substitution 1 - k2x2 = (1-k2) sin2 0 reduces the integral to the form 4I

i -t/2 d _ o.1.-(.1-k2) sin2. i e iK', and we have: o 4-(1-k2) sin2' (6) F (k,l/k) = K + iK' It is possible to generalize this result further to cover all real values of x. For x = xo, 1 < xo < 1/k the above substitution leads to the integral Ir /2 _ d sin-1 l-k2xo0 41' - (1-k2) sin20 l-k2 which is equal to i K' - F k 1 -k2XO21l-k2 Hence F (k,xo) =K + iK' - iF l, -k2 (7) 1-k2 For x = xo, > 1/k we observe that where there is no likelihood of confusion, the notation for the elliptic integral may be shortened bo the form F(x), and introduce the further notation F (X2) = F(x2) -F(xl) =. dx \~1 /~x1 Fi )(l-x )(-k2x2) Considering the integral (1)x dx F = (1-x2)(1-k2x2) the substitution x = l/k~ leads to the integral l/k Jl/kxo (1-2)(l-k2 2)' whence we have (F o = F 1/kx 5

Subtracting F (l/k) from both sides of the equation we obtain finally F ) = F((8) Equations (5) - (8) make it clear that working from tabulated values of elliptic integrals for 0 < x < 1, we may deduce values for all real and imaginary values of x, However when we consider complex values of x (or, more properly, of z = x + iy) the elliptic integral proves to be intractable whatever path of integration we select between the limits z = 0 and z = x + iy. The basis of the difficulty is that there is no true addition formula for elliptic integrals, so that there is no means of evaluating F(x+iy) in terms of F(x) and F(iy) The only way out of the difficulty is to invert the elliptic integral and to discuss the properties of the inverse function so obtained, called the elliptic function, With the elliptic integral of the second kind, it is better to postpone the kind of discussion given above until after consideration of the elliptic functionso However, it may be noted here that the complete integrals corresponding to K and K} are written as E and E, and that E(k,l/k) can be evaluated by exactly the same substitutions as are used above to obtain F(k,l/k), In this case, however, they lead to the result E(k,l/k) = E+ i(K-E'). (9) 3.0 JACOBI S ELLIPTIC FUNCTIONS We have seen that Eqs. (1) and (4) u = F (k, o) =,x ) - ) (1),(4) Jo N/`-kasin^ o 91(l-x2)(l-k2x2) -define u as a function of 0 or x, for a given fixed value of k. They also define 0 as a function of u, 0 = am u (ie.o, amplitude of u)o Since x = sin 0, we may also write x = sin (am u), or more briefly sn Uo We may in fact define three related functions thus: sn u = sin (am u) = sin g = x (lOa) cn u = cos (am u) = cos 0 = J/i-x2 (lOb) dn u = A (am u) = A 0 ='/l-k2sin2~ = /l-k;x2 (lOc) 6

These functions were introduced by Jacobi, Abel, and Gauss, who were the first to realize that the elliptic function, rather than the elliptic integral, should be regarded as fundamental in the general theory of functions, The elliptic function as defined above can be developed and discussed at some length (as a function of a complex variable) through its relationship with certain other functions known as Theta-functions. In particular the very fast convergence of the series which define the Theta functions provides a ready means of computing and tabulating the elliptic functions, whereas there is no direct method (other than numerical integration) of computing values of elliptic integrals. Most important of all, for the purposes of the present discussion, addition formulae can be derived for the elliptic functions, so that the values of the function cf a complex number may be readily computed. The addition formulae are: sn u cn v dn v ~ sn v cn u dn u sn (u + v) = (lla) 1 - k2sn2u sn2v n (u v) cn u cn v: sn u sn v dn u dn v (llb) cn (u ~ v) = 2 —------ (11b) 1 - k2sn2u sn2v d ( v) = dn u dn v t.k2 sn u sn v cn u cn v (i dn (u ~ v) (llc) 1 - k2sn2u sn2v The function sn u is odd, while cn u and dn u are even. That is to say sn(-u) = -sn u, cn(-u) = cn u, dn(-u) = dn u (12) Where it is necessary in the interests:!of clarity, the modulus k is included in the notation by writing sn(u,k) instead of sn u. In general, however, we shall simply write sn u for sn(u,k), and for sn(u,k' ) we shall write sn'u. We may note here the elementary results sn(O) = O, sn K = 1; cn 0 = 1, cn K = O; dn(O) = 1, dn K = k'? (15) sn'(O) = O, sn'K' = l; cn'(O) = 1, cn'K'=O; dn'(O) = 1, dn'K' = k Equations (5) - (8) can now be rewritten sn i u = i snu (5a) cn'u sn (K + iK') = 1/k (6a) sn (K + iv) = cn' + k2sn'2v (7a) sn (K + iK'-u) = dn u (8a) k cn u 7

Equation (5a) can readily be deduced from Eqo (5), and Eqo (6a) is a trivial restatement of Eq. (6). Equations (7a), and (8a) are most readily obtained by working from Eqs. (11) and (13), but with some manipulation they could be derived directly from Eqs. (7) and (8). A further condensation of the notation is possible by writing sn u cn u --- = sc u = cd u, etc. cn u dn u There will, however, be very little occasion to use this notation in this report, 3.1 THE PERIODICITY OF THE ELLIPTIC FUNCTIONS From Eqs. (lla) and (13) it is easy to derive the equations sn (u + 2K) = - sn u (14a) sn (u + 4K) = sn u (14b) sn (2K - u) = sn u (14c) These equations make. it clear that the function sn u has a period 4K and in general if u is a real number sn u behaves in similar fashion to the sine function, with the period 4K corresponding to the period 2it, as shown in Fig. 1. snu a-_. \ i.. / _ ~ U.-...., K 3 /4 K 5KE Fig. 1 From Eqs. (lla) and (13) we can further derive the equation sn (u + 2iKg) = sn u (15a) and from (5) and (13) the further equation 8

sn i K' = 0 (15b) These equations, together with the knowledge that sn u is an odd function, show that the function sn u has a second period of 2iK', and in general if u is an imaginary number equal to, say, iv, where v is real, sn iv, as a function of v, behaves in similar fashion to the tangent function, with the period 2iK' corresponding to the period c, as shown in Fig. 2. sn iv I yQo / K 2K/ 3K' -3K' /2K -K1 / Fig. 2 Considering now the elliptic function of a complex number, sn w = sn (u + iv), we see that, just as the variation of the sine and the tangent of an angle ~ can be compactly shown by describing their behavior in the range 0 < G < t/2; so sn w can be completely defined by describing its behavior inside, and on, the rectangle bounded by the lines u = O, u = K, v = O, v = iK'. Consideration of the exact mode of its behavior within the rectangle can conveniently be deferred until the Schwarz-Cristoffel transformation is discussed in Section 3.3. The doubly periodic property deduced above is peculiar to elliptic functions; in fact, from the viewpoint of the theory of functions it is the defining property of the generalized elliptic function. The rectangle having a side of 4K parallel to the real axis and a side of 2K' parallel to the imaginary axis is called a period-parallelogram. 3.2 THE ELLIPTIC FUNCTIONS AND THE INTEGRAL OF THE SECOND KIND To obtain an addition formula for this integral, it is necessary to transpose it from a function of x into a function of u. Since x = sn u, we have: l-k2x _ dn u 1-x ecn u and 9

dx du = I(lk-k2)(l-k2) dx cn u dn u Hence E = lk2 dx = dn2u du (16) E(u) is not periodic in either real or imaginary values of u, although it is possible to define a related function, the Zeta function, which is singly periodic. However, the main interest in this report attaches to E as a function of x: we consider it as a function of u simply to obtain an addition formula to work witho By using Zeta functions it is possible to derive the addition formula E (u+v) = E(u) + E(v) - k sn u sn v sn (u+v) o (17) An imaginary transformation can also be obtained: E(iu) = iu + i dncu sc'u - i E'(u) (18) It would have been possible, by an argument like those in Section 21o, to obtain an imaginary transformation for E considered as a function of xo However, there is little point in doing so, as Eqo (18) is already in the correct form for application to the addition formula (17)o 353 ELLIPTIC FUNCTIONS IN PLANE-POTENTIAL PROBLEMS Many plane-potential (also called plane-harmonic) problems can be reduced to the problem of mapping the interior of a polygon in one plane onto the upper half of another plane. When this polygon is a rectangle, the Schwarz-Cristoffel transformation leads immediately to an elliptic integral of the first kind. Consider, as shown in Figo 3, the mapping of the interior of the rectangle ABCD on the w plane on to the upper half of the z plane. The Schwarz-Cristoffel theorem states that such a mapping may be effected my means of the transformation dw - Go= (z - k) dw(z - ) (19) where Qk is the abscissa of the point on the real axis of the z-plane corresponding to the vertex of the polygon having an internal angle Bk, and G is an arbitrary constant: 10

b -c -I +1 +c _ _ _ _ _ _ _ _ __-t- u - - - 1- - -.. - B o a _~C A B O C D w plane z plane Fig. 3 In our case all the internal angles of the polygon are equal to n/2 and hence Pk/T - 1 = -1/2. If the lengths a, b, and c are as defined in Fig. 3, we have: dx w = G (z2l)z22) + C G | ds c J 4(l-z2)[l(z2/c2)] + C I dz = J Gk (-(l)(1k2z2) + C, where k = l/c = Gk F (k,z) + C. (20) Considering the boundary values, we have: Pt. 0 0: = 0 + C.C = 0 Pt. C: a =GkK Pt, D: a+ib = GkF (k,l/k) = Gk (K + iK') b = GkK'. We may without loss of generality select G so that Gk = 1, whence a = K, b = K' (21) 11

and EqO (20) becomes simply w = F (k,z) (22) or r (22) z = sn w These results confirm those obtained in the preceding paragraph, while at the same time making it clear how the function sn w varies within the rectangle OCDE, which corresponds to the whole of the first quadrant on the z plane. The point E corresponds to the point at infinity, and the boundary OCDE corresponds of course to the real axis, confirming in general the points made at the beginning of Section 2.1 about the behavior of the elliptic integral, and the results given in Eqs. (7) and (8). The only modification to this equation which now appears to be necessary is that when z is real and greater than 1/k, the negative sign must be taken for the integrand, so that the real part of F(z) returns to zero as z goes to (real) infinity. The flow situation represented is that of flow from a source at E into the boundary of the rectangle, or alternatively (interchanging the roles of the two sets of lines in the flow net) flow about a doublet, or dipole, at E, with the boundary of the rectangle becoming the bounding streamline, If the rectangle is now cut at E, and the sides AE and DE folded back so that E goes to infinity on both sides, as in Fig. 4, we have the case of flow across a horizontal floor containing a rectangular slot. In this case two of the angles are equal to i/2, giving Pk/t - 1 = -1/2, and the other two are equal to 35/2, giving Pk/A - 1 = +1/2. Hence we obtain w = Gk d-k z + C V Y E ~- _ _E rd i H A DE b aI I~-c -I | + c "B5u 0 UA B~c o C D w plane x plane Figo 4 12

We can again set Gk = 1, and the boundary values lead this time to: a = E, b = K'-E' (23) from Eq. (9). 4.0 THE CONSTRUCTION OF THE CHARTS It should now be clear that the charts presented in the Appendices contain all the information needed to solve the type of problems outlined in the preceding section. Considering the first problem, the first quadrant of the z. plane is to be folded up inside the rectangle OCDE on the w plane, and this is precisely what is shown in the charts in Appendix I (x = sn w) where contours of x and y are drawn in on this rectangle on the w plane. Appendices II and III display in similar fashion the behavior of the functions z = cn w and z = dn w; the corresponding problems are simple extensions of the first problem. Again we may think of the point E as being a source; if the x-contours are considered as streamlines, then flow is from E to the boundaries OE, OC, for z = cn w, and to the boundaries OE, OC, CD, for z = dn w. Scales on the w plane have been selected so as to make the rectangle OCDE a unit square; in other words, the quantities measured along the axes of the w plane are not u and v, but u/K and v/K'. Although this arrangement distorts the x-y net so as to conceal its orthogonal property, it has a generality and a convenience that appear to outweigh this disadvantage. Considering the second problem, the difficulty is that the E-function has to be computed as a function of w, and not as a function of z, where z = sn w. It has been plotted on the w plane (Appendix IV) because this was the simplest course to follow in the limited time available, but it would of course have been better to plot contours of x and y on the E plane, and hence to show directly the solution to the mapping problem. However, values of E can be obtained for a given value of z by first obtaining the appropriate value of w from Appendix I, and then entering this value into the charts in Appendix IV. The mapping problem has still been kept in mind by dividing ER and EI, the components of E, by E and K'-E', respectively, i.e., the lengths OC and CD in Fig. 4. Contours of the resulting normalized values, ER' = ER E R E and EI EI' = _I K' -E' are plotted on the w plane in the charts in Appendix IV. 15

The range of values of k selected is as follows: sin-1 1, sin1 50, sin-l 100... sin-1 80~, sin- 85, sin- 890, making 19 values in all. This should be sufficient for the purposes of most physical problems It remains only to give the equations from which the tables and charts have been computed. From Eq. (5) we have sn v sn iv = i --- cn'v whence /sn v\2 cn iv = + cn cnv' and //2 dn iv = k2 dnvv cn5 v Hence from Eqso (11) we have: sn u dn'v + i cn u dn u sngv cnv sn (univ) = ~ 2.. (24) cna2v + k2 sn2u sn2 v cn (u+iv) cn u cnvv - i sn u dn u sniv dnvv (25) Cnl ( UtiV_) = r,,...~ —.:.-:.-.5 cn Y2v + k2 sn2u sn 2. dn u cn'v dn'v - i k2 sn u cnusn v dn (u+iv) = (26) cn 2v + k2 sn2U sn'2v For the E function, we use Eqs. (17), (18), and (24) to obtain: k2 sn u cn u dn u sn 2v E(u+iv) = ER + iEi =_ E(u) + cn.v + kZ snusn2 c n -vcnk2v + kn sn u sn (v [v - E_(v) + sn'v cn'v dntv dn2u (27) l4

The denominator in Eqs. (24) - (27) could be expressed in other ways, e.g., as 1 - dn2u sn2v, but the form given was selected as being better for computing purposes since it is a sum rather than a difference of two terms. The computing was done on the University's IBM 650 digital computer, which handles numbers having a maximum of ten digits. Fixed-point arithmetic was used, the decimal point being kept fixed between the fourth and fifth digits in both input and output data. Since all the constants fed in (sn u, cn u, etc.) lay between 0 and 1, this meant that these numbers had at least five, and usually six, significant figures. However, for the lower values of k some of the results had only two significant figures because there were four zeros after the decimal point. This was not a serious source of error in plotting the charts, but for the four-figure tables such results, and any others of doubtful precision, were recomputed manually to at least three significant figures. In most cases, however, the results had six nonzero digits, which were rounded off to four before printing. The reader can take it, therefore, that whether three or four digits are given, the last digit is reliable. The values of sn u, cn u, etc., fed into the machine were taken straight from Spenceley's7 twelve-figure tables of elliptic functions, 15

REFERENCES 1o Franklin, Po, Methods of Advanced Calculus, McGraw-Hill Book Coo, New York, 1944. 2o Bowman, Fo, Introduction to Elliptic Functions, Wiley and Sons, Inc., New York, 1953. 35 Milne-Thomson, L. Mo, Jacobian Elliptic Function Tables, Dover Publications, Inco, New York, 1950. 4. Neville, Eo H., Jacobian Elliptic Functions, Oxford Univ. Press, New York, 1944. 5o Byrd, P. F., and, Friedman, Mo Do, Handbook of Elliptic Integrals for Engineers and Physicists, Lange, Maxwell and Springer, Berlin, 1954. 60 Whittaker, E, T., and, Watson, G. No, A Course of Modern Analysis, Cambridge Univo Press, New York, 1952. 7. Spenceley, G. Wo, and Spenceley, Ro M., Smithsonian Elliptic Functions Tables, Smithsonian Institution, Washington, 1947 (Vol. 109, Smithsonian Miscellaneous Collections)o 16

APPENDIX I THE sn-FUNCTION

THE sn-FUNCTION, x + iy = sn(u + iv) k = sin 1* = 0.0174524; K = 1.570916, K' = 5.434910 u 0 0.1 0.2 0.3 04 0.5 0.6 0.7 0.8 0.9 1.0 K - K' x 0 0.1564 0.3090 0.4540 0.5878 0.7071 0.8090 0.8910 0.9511 0.9877 1.000 y 0 0 0 0 0 0 0 0 0 0 0 x 0 0.1801 0.3558 0.5228 0.6768 0.8142 0.9315 1.026 1.095 1.137 1.151 0.l y 0.5707 0.5636 0.5427 0.5084 0.4616 0.4034 0.3354 0.2590 0.1763 0.08924 0 x 0 0.2584 0.5104 0.7499 0.9708 1.168 1.336 1.471 1.570 1.631 1.651 0'2 y 1.314 1.298 1.250 1.171 1.063 0.9289 0.7721 0.5963 0.4058 0.2054 0 x 0 0.4153 0.8202 1.205 1.559 1.875 2.145 2.362 2.520 2.617 2.650 0.3 y 2.456 2.426 2.336 2.188 1.986 1.735 1.442 1.113 0.7576 0.3835 0 x 0 0.6995 1.381 2.028 2.623 3.153 3.604 3.967 4.231 4.392 4.446 y 4.346 4.291 4.131 3.867 3.508 3.064 2.544 1.964 1.336 0.6758 0 x 0 1.204 2.376 3.484 4.500 5.399 6.161 6.769 7.211 7.480 7.570 ~05 y 7.570 7.473 7.187 6.720 6.087 5.306 4.399 3.389 2.302 1.164 0 x 0 2.120 4.172 6.094 7.833 9.348 10.61 11.60 12.32 12.74 12.89 y 13.19 13.01 12.48 11.62 10.48 9.082 7.490 5.744 3.888 1.961 0 x 0 3.921 7.670 11.07 14.02 16.46 18.40 19.85 20.85 21.43 21.63 0.7 y 23.33 22.90 21.84 20.10 17.85 15.23 12.37 9.358 6.268 3.141 0 x 0 8.475 16.05 22.23 26.84 30.05 32.15 33.45 34.21 34.59 34.71 y 43.60 42.56 39.29 34.70 29.39 23.91 18.58 13.56 8.840 4.358 0 x 0 29.45 48.41 56.96 57.80 56.52 54.45 52.51 51.01 50.08 49.77 0.9 y 100.4 92.24 73.84 54.81 39.42 28.01 19.61 15.26 8.212 3.931 0 x 0 366.3 185.4 126.2 97.48 81.03 70.82 64.31 60.24 58.01 57.30 1.0 y 0 0 y oo oo0 0 0 0 0 0 0 0 0

o,~~~~~~~~~~~ O0 cN o ii o N CII OR 1~ 0 0 6 oq 0 0 0 0 0i0 c'I _j ir ~ ~ ~ ~ ~ ~ -C ( O O~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~q AA~~~~~~1 IN I' K I \1 11N JI I A N ( IIII — I I I X i M/~ ~ ~ ~ ~e it~ o ~~~~~~~~~~~~~~~~~o I — A IN I I \ I I o o, ao re ct, In ~~~~~~~~~~~~~~~~~~~~~1, L LIIA o I X I I X I0I N Ij 1 IL Il)

THE sn-FUNCTION, x + iy = sn(u + iv) k = sin 5~ = 0.0871557; K = 1.575792, K' = 3.851742 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 K v K' x 0 0.1567 0.5096 0.4547 0.5885 0.7078 o.8095 0.8914 0.9512 0.9877 1.000 y 0 0 0 0 0 0 0 0 0 0 0 x 0 0.1685 0.3327 0.4886 0.6524 0.7604 o.8696 0.9573 1.021 1.061 1.074 0'1 y 0.3927 0.3878 0.3732 0.3494 0.3169 0.2767 0.2298 0.1773 0.1206 0.06102 0 x 0 0.2057 0.4060 0.5960 0.7709 0.9264 1.059 1.165 1.242 1.289 1.305 2 y 0.8444 0.8338 0.8022 0.7507 0.6805 0.5938 0.4927 0.3800 0.2583 0.1307 0 x 0 0.2747 0.5420 0.7948 1.027 1.232 1.405 1.544 1.645 1.706 1.727 0, 3 y 1.424 1.4 1.1.352 1.264 1.144 0.9966 0.8257 0.6359 0.4318 0.2183 0 x 0 0.3891 0.7664 1.121 1.444 1.726 1.963 2.151 2.286 2.368 2.395 y 2.225 2.195 2.107 1.965 1.773 1.540 1.272 0.9764 0.6615 0.3339 0 x 0 0.5759 1.131 1.645 2.104 2.497 2.820 3.070 3.247 3.352 3.387 ~5 y 3.387 3.338 3.193 2.961 2.655 2.288 1.876 1.432 0.9652 0.4857 0 x 0 0.8981 1.749 2.514 3.168 3.702 4.117 4.424 4.632 4.752 4.791 0.6 y 5.157 5.067 4.809 4.405 3.892 3.302 2.667 2.008 1.340 0.6702 0 x 0 1.536 2.932 4.093 4.987 5.630 6.070 6.354 6.525 6.616 6.644 0.7 y 8.056 7.860 7.312 6.506 5.556 4.555 3.566 2.616 1.713 0.8463 0 x 0 3.199 5.762 7.443 8.365 8.779 8.908 8.902 8.853 8.808 8.790 y 13.59 12.97 11.39 9.374 7.384 5.627 4.146 2.904 1.841 0.8925 0 x o 10.81 15.27 15.53 14.50 13.33 12.34 11.59 11.08 10.78 10.69 0.9 y 29.22 24.89 17.13 11.11 7.265 4.850 3.260 2.147 1.308 0.6205 0 x 0 73.21 37.09 25.23 19.50 16.21 14.17 12.87 12.06 11.62 11.47 1.0 y 0 0 y 0 o0 0 0 0 0 0 0 0 0

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THE sn-FUNCTION, x + iy = sn(u + iv) k = sin 10~ = 0.173648; K = 1.582843, K' = 5.153385 u 0 O.1 0.2 0.3 0,4 0.5 0.6 0.7 0.8 0.9 1.0 K v K' x 0 0.1576 0.3112 0.4568 0.5907 0.7098 0.8112 0.8924 0.9517 0.9879 1.000 Y 0 0 0 0 0 0 0 0 0 0 x 0 0.1658 0.5272 0.4801 0.6207 0.7454 0.8514 0.9563 0.9985 1.036 1.049 0 y 0.3208 0.3166 0.3045 0.2845 0.2572 0.2239 0.1855 0.1426 0.09681 0.04892 0 x 0 0.1914 0.5775 0.5532 0.7142 0.8565 0.9765 1.072 1.142 1.184 1.198 y 0.6750 o.6661 0.6597 0.5968 0.5592 0.4686 0.3875 0.2976 0.2018 0.1019 0 x o 0.2585 0.4696 0.6865 0.8855 1.055 1.200 1.515 1.595 1.444 1.461 053 y 1.101 1.086 1.041 0.9691 0.8727 0.7558 0.6226 0.4770 0.3226 0.1626 0 x 0 0.3157 0.6196 0.9015 1.155 1.368 1.545 1.681 1.778 1.836 1.855 0.4 y 1.650 1.626 1.554 1.459 1.288 1.108 0.9065 0.6905 0.4649 0.23557 0 x 0 0.4420 0.8619 1.241 1.569 1.858 2.050 2.208 2.516 2.379 2.400 0.5 y 2.400 2.559 2.240 2.054 1.816 1.545 1.247 0.9398 0.6276 0.3159 0 x 0 0.6650 1.276 1.801 2.225 2.545 2.775 2.951 3.032 5.087 5.105.6 3.490 5.414 5.198 2.875 2.485 2.059 1.627 1.204 0.7926 0.5950 0 x 0 1.111 2.075 2.805 5.299 5.607 5.782 5.874 5.918 5.957 5.942 0.7 y 5.230 5.059 4.596 5.957 5.260 2.585 1.963 1.407 0.9060, 0.4453 0 x 0 2.295 3.940 4.810 5.156 5.176 5.096 4.987 4.891 4.828 4.806 o'8 y 8.552 7.987 6.677 5.190 5.878 2.852 2.021 1.584 0.8642 0.4154 0 x 0 7.474 9.437 8.881 7.919 7.088 6.458 6.012 5.716 5.547 5.492 0.9 y 17.95 14.28 8.777 5.258 3.281 2.129 1.406 0.9158 0.5543 0.2619 0 x 356.54 18.51 12.61 9.749 8.115 7.100 6.455 6.051 5.830 5.759 1.0 y 0 0 0 0 0 0 0 0 0 0

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THE sn-FUNCTION, x + iy = sn(u + iv) k = sin 15~ = 0.258819; K = 1.598142, K' = 2.768063 0 0.1 0.2 0.3 0,4 0.5 0.6 0.7 0.8 0.9 1.0 K v K' x 0 0.1591 0.5159 0.4602 0.5944 0.7132 0.8138 0.8942 0.9526 0.9881 1.000 y o o 00 0o 0 0 0 0 0 x 0 0.1656 0.3267 0.4787 0.6179 0.7407 0.8445 0.9272 0.9873 1.024 1.036 0.1 Y 0.2806 0.2768 0.2654 0.2471 0.2225 0.1928 0.1588 0.1217 0.08233- 0.04151 0 x 0 0.1863 0.3669 0.5566 0.6908 0.8260 0.9392 1.029 1.093 1.12 1.145 Y 0.5847 0.5764 0.5520 0.5129 0.4608 0.5981 0.5272 0.2501 0.1688 o0.8503 0 x 0 0.2243 0.4406 0.6418 0.8221 0.9776 1.106 1.206 1.277 1.319 1.333 y 0.93935 0.9252 0.8837 0.8177 0.7311 0.6282 0.5135 0.3907 0.2628 0.1520 0. x 0 0.2871 0.5612 0.8113 1.030 1.212 1.559 1.469 1.546 1.591 1.605 1.381 1.358 1.290 1.185 1.050 0.8932 0.7232 0.5457 0.3648 0.1825 0 x 0 0.3911 0.7573 1.080 1.348 1.560 1.719 1.833 1.909 1.952 1.966 Y 1.966 1.926 1.814 1.643 1.431 1.197 0.9533 0.7095 6.4693 0.2333 0. x 0 0.5761 1.095 1.520 1.840 2.066 2.216 2.312 2.368 2.398 2.407 Y 2.798 2.724 2.518 2.220 1.W876 1.522 1.180 0.8588 0.5589 0.2752 0 x 0 0.9563 1.746 2.295 2.625 2.798 2.875 2.901 2.904 2.901 2.899 y 4.114 3.945 3.502 2.924 2.334 1.798 1.335 0.9401 0.5979 0.2904 0 x 0 1.962 3.227 3.763 3.871 3.797 3.669 3.547 3.452 3.394 3.374 y 6.609 6.070 4.855 3.597 2.582 1.830 1.278 0.8623 0.5332 0.2549 0 x 0 6.154 7.127 6.375 5.537 4.887 4.420 4.097 3.888 3.769 3.731 Y 13.77 10.28 5.790 3.290 1.994 1.272 0.8313 0.5379 0.3242 0.1528 0 x g 24.29 12.31 8.395 6.500 5.417 4.748 4.321 4.056 3.910 3.864 1.0 y 0 Y 0 0 0 0 0 0 0 0 0 0

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THE sn-FUNCTION, x + iy = sn(u + iv) k = sin 20~ = 0.342020; K = 1.620026, K' = 2.504550 u 0 0.1 0.2 0.3 o.4 0.5 0.6 0.7 0.8 0.9 1.0 K V K' x o 0.1612 0.3178 0.4652 0.5997 0.7180 0.8176 0.8966 0.9538 0.9884 1.000 y 0 0 0 0 0 0 0 0 0 0 0 x 0 0.1669 0.3288 0.4809 0.6193 0.7406 0.8424 0.9229 0.9810 1.016 1.028 y 0.2534 0.2497 0.2387 0.2211 0.1979 0.1703 0.1394 0.1062 0.07146 0.03592 0 x o 0.1848 0.3634 0.5301 0.6803 0.8105 0.9184 1.003 63 1.100 1.112 0.2 y 0.5250 0.5169 0.4933 0.4557 0.4065 0.3485 0.2841 0.2157 0.1448 0.07268 0 x o 0.2181 0.4273 0.6198 0.7899 0,9341 1.051 1.141 1.204 1.241 1.253 03 y 0.8362 0.8222" 0.7817 0.7180 0.6362 0.5414 0.4383 0.3307 0.2210 0.1106 0 x o 0.2736 0.5324 0.7645 o.9626 1.124 1.249 1.342 1.405 1.442 1.454 y 1.216 1.193 1.126 1.024 0.8965 0.7532 o.6025 0.4499 0.2983 0.1485 0 x o 0.3667 0.7048 o.9940 1.225 1.401 1.527 1.614 1.669 1.700 1.710 y 1.710 1.670 1.558 1.391 1.193 0.8908 0.7694 0.5652 0.3703 0.1830 0 x 0 0.5343 1.003 1.369 1.627 1.795 1.899 1.959 1.991 2.007 2.012 y 2.405 2.329 2.122 1.833 1.515 1.203 0.9155 0.6565 0.4227 0.2067 0 x 0 0.8813 1.574 2.014 2.245 2.343 2.370 2.365 2.350 2.338 2.333 7 y 3.497 3.322 2.88 2.334 1.808 1.358 0.9882 0.6855 0.4314 0.2083 0 x 0 1.793 2.831 3.172 3.167 3.045 2.906 2.787 2.700 2.647 2.630 y 5.569 5.015 3.842 2.726 1.892 1.309 0.8989 0.5993 0.3677 0.1750 0 x 0 5.411 5.825 5.019 4.284 3.750 3.379 3.127 2.965 2.874 2.845 0.9 y 11.54 8.095 4.228 2.307 1.369 0.8623 0.5590 0.3597 0.2160 0.1016 0 x ~ 18.14 9.202 6.285 4.875 4.072 3.576 3.261 3.065 2.958 2.924 1.0 y 0000 F O 0 0 0 0 0 0 0 0 0 0

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THE sn-FUNCTION, x + iy = sn(u + iv) k = sin 25~ = 0,422618; K = 1.648995, K' = 2.508787 0 0.1 0.2 0.5 0,4 0.5 0.6 o.7 0.8 0.9 1.0 K K' x 0 0.1640 0.3229 0.4717 o.6o66 0.7243 0.8225 0.8998 0.9554 0.9888 1.000 Y 0 0 0 0 0 0 0 0 0 0 0 x 0 0.1692 0.3328 0.4857 0.6237 0.7435 0.8431 0.9212 0.9771 1.011 1.022 1 y 0.2333 0.2295 0.2185 0.2012 0.1786 0.1524 0.1236 0.09341 0.06249 0.03129 0 x 0 0.1856 0.3642 0.5295 0.6768 0.8030 0.9063 0.9862 1.043 1.076 1.088 y 0.4817 0.4735 0.4497 0.4124 0.3646 0.3095 0.2499 0.1881 0.1254 0.06268 0 x 0 0.2162 0.4222 o.6094 0.7722 0.9077 1.016 197 1.153 1.186 1.197 y 0.7631 0.7489 0.7078 0.6444 0.5648 0.4750 0.3802 0.2840 0.1883 0.09377 0 x o 0.2677 0.5183 0.7387 0.9222 1.067 1.177 1.256 1.309 1.339 1.349 y 1.102 1.078 1.010 0.9083 0.7842 0.6495 0.5125 0.3783 0.2486 0.1231 0 x o 0.3549 0.6767 0.9436 1.149 1.297 1.400 1.467 1.509 1.531 1.538 0.5 y 1.538 1.497 1.381 1.215 1.023 0.8265 0.6380 0.4625 0.3000 0.1474 0 x 0 0.5133 0.9511 1.275 1.489 1.618 1.690 1.727 1.745 1.752 1.754 y 2.147 2.067 1.854 1.568 1.266 0.9843 0.7354 0.5199 0.3313 0.1610 0 x 0 0.8420 1.471 1.855 1.996 2.046 2.043 2.022 1.998 1.982 1.976 0.7 y 3.101 2.917 2.466 1.938 1.460 1.071 0.7650 0.5235 0.3264 0.1567 0 x 0 1.698 2.573 2.781 2.710 2.566 2.426 2.315 2.237 2.191 2.175 0.8 y 4.912 4.332 3.178 2.166 1.460 0.9889 0.6691 0.4416 0.2691 0.1275 0 x 0 4.924 4.968 4.158 3.506 3.054 2.747 2.543 2.412 2.339 2.315 0'9 y 10.14 6.680 3.262 1.722 1.004 0.6259 0.4028 0.2578 0.1542 0.07240 0 x " 14.43 7.329 5.016 3.901 3.267 2.877 2.630 2.477 2.393 2.366 Y 10 0 0 0 0 0 0 0 0

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THE sn-FUNCTION, x + iy = sn(u + iv) k = sin 30~ = 0.500000; K = 1.685750, K' = 2.156516 0 0.1 0.2 0.3.4 0.5 0.6 0.7 0.8 0.9 1.0 K V K' 0 x 0 0.1676 0.3293 0.4800 0.6152 0.7321 0.8285 0.9037 0573 0.9893 1.000 y 0 0 0 0o o o o o o o0 0 01 x o 0.1725 0.3386 0.4928 0.6305 0.7489 0.8461 0.9214 0.9749 1.007 1.017 y 0.2178 0.2139 0.2025 0.1849 0.1626 0.1372 0.1101 0.08238 0.05469 0.02725 0 0.2 x 0 0.1880 0.3679 0.5329 0.6781 0.8007 0.8996 0.9750 1.028 1.059 1.069 y 0.4486 0.4401 0.4155 0.3777 0.3302 0.2769 0.2210 0.1646 0.1089 0.05412 0 0 x 0 0.2171 0.4223 0.6064 0.7635 0.8917 0.9916 1.066 1.116 1.145 1.155 y 0.7081 0.6932 0.6506 0.5861 0.5071 0.4207 0.3323 0.2454 0.1613 0.07984 0 4 x 0 0.2664 0.5130 0.7253 0.8972 1.029 1.126 1.194 1.239 1.264 1.272 y 1.018 0.9930 0.9223 0.8182 0.6955 0.5668 0.4405 0.3210 0.2089 0.1028 0 x 0 0.3506 0.6627 0.9130 1.097 1.225 1.308 1.361 1.393 1.409 1.414 05 1.414 1.370 1.249 1.080 0.8920 0.7071 0.5366 0.3836 0.2462 0.1202 0 0.6 x 0 0.5041 0.9212 1.213 1.392 1.491 1.540 1.562 1.570 1.572 1.573 y 1.964 1.879 1.656 1.369 1.079 0.8210 0.6023 0.4197 0.2647 0.1279 0 x 0 0.8228 1.404 1.705 1.818 1.835 1.813 1.782 1.755 1.738 1.732 0.7 y 2.824 2.627 2.163 1.648 1.207 0.8656 0.6076 o.4o14 0.2536 0.1211 o x 0 1.642 2.389 2.498 2.384 2.231 2.097 1.994 1.924 1.883 1.870 y 4.458 3.844 2.698 1.770 1.161 0.7716 0.5151 0.3367 0.2038 0.09624 o 9 x 0 4.571 4.350 3.558 2.974 2.584 2.325 2.153 2.045 1.985 1.966 9 y 9.185 5.667 2.602 1.335 0.7670 0.4733 0.3025 0.1925 0.1147 0.05372 0 10 x 11.94 6.073 4.167 3.251 2.732 2.414 2.213 2.089 2.022 2.000 Y1oo.0 0 0 0 0 0 0 0 0

ooo'o =oO2 u!s = W (z ~Z -)(z -I)/ / zp =()=A+n (A! +n)us = A!+x=z'uoipun-us a@L n 0'1 6'0 8'0 Z'O 9'0 9'0 O ~0' ~'0'0'0 0 010 0O 0 i~ ~i ~ ~n 8i' L'O. 0IN AoO $i i0 j Wj jt g~~~g j i!'k AI 8' 0 -f -I- /f I I I I zl u-l I I I.: I I I'0 0'1 0'1..i~~~~i,. ~ g~~~o 9,0~~~~~

THE sn-FUNCTION, x + iy = sn(u + iv) k = sin 35~ = 0.573576; K = 1.731245, K' = 2.034715 K 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 v K' x 0 0.1720 0.3373 0.4900 0.6257 0.7414 0.8357 0.9083 0.9595 0.9899 1.000 Y 0 0 0 0 0 0 0 0 0 0 o 01 x 0 0.1767 0.3462 0.5020 0.6397 0.7564 0.8508 0.9231 0.9739 1.004 1.014 y 0.2054 0.2012 0.1894 0.1712 0.1487 0.1238 0.09802 0.07249 0.04767 0.02361 0 0.2 x 0 0.1918 0.3742 0.5397 o.6832 0.8023 0.8967 0.9677 1.017 1.046 1.055 y 0.4224 0.4133 0.3876 0.3485 0.3007 0.2485 0.1956 0.1438 0.09423 0.04655 o 0 x 0.2201 0.4265 0.6087 0.7610 0.8825 0.9752 1.042 1.088 1.114 1.122 y 0.6652 o.6493 0.6044 0.5377 0.4582 0.3741 0.2910 0.2120 0.1379 0.06784 o 0.4 x 0 0.2685 0.5138 0.7201 0.8822 1.003 1.089 1.147 1.185 1.206 1.212 y 0.9537 0.9266 0.8518 0.7443 0.6215 0.4974 0.3802 0.2731 0.1758 0.08594 0 05 x 0 0.3514 0.6579 0.8949 1.062 1.171 1.240 1.281 1.305 1.317 1.320 y 1.320 1.273 1.145 0.9709 0.7851 0.6097 0.4543 0.3200 0.2032 0.09851 o 0.6 x 0 0.5029 0.9052 1.171 1.321 1.395 1.427 1.438 1.440 1.439 1.438 y 1.828 1.736 1.501 1.210 0.9305 0.6920 0.4983 0.3422 0.2136 0.1025 0 x 0 o.8164 1.359 1.609 1.681 1.675 1.642 1.606 1.578 1.560 1.554 0 y 2.621 2.408 1.926 1.421 1.012 0.7099 0.4899 0.3267 0.2000 0.09502 o 0.8 x 0 1.610 2.248 2.280 2.138 1.983 1.856 1.763 1.700 1.664 1.653 y 4.127 3.470 2.328 1.472 0.9408 0.6143 o.4048 0.2620 0.1575 0.07409 0 x 0 4.296 3.877 3.111 2.586 2.245 2.022 1.877 1.786 1.736 1.720 09 y 8.490 4.892 2.121 1.061 0.6011 0.3674 0.2330 0.1474 0.08743 0.04082 0 0 x c 10.14 5.169 3.558 2.786 2.352 2.086 1.919 1.817 1.761 1.743 y1. 0 0 0 0 0 0 0 O

U K 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.-r \ I _e 0.9 0 _ A 0.9 0 0.1 02 03 04 05 06 07 08 0.9.0 ~~~~~~k= sin 5~~3~~5~0~0.=075736 VIl A N 0.6 0.6 0.3 0.1 i0.5 I i' 1~ i Tt~~~~ Th nfncin =x y sn( V zt dz- u + IV F W -_-2 2 2

THE sn-FUNCTION, x + iy = sn(u + iv) k = sin 40~ = 0.642788; K = 1.786769, K' = 1.935581 u 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.o K v K' x 0 0.1773 0.5469 0.5021 0.6382 0.7525 0.8442 0.9157 0.9621 0.9906 1.000 y 0 0 0 0 0 0 0 0 0 0 0 x 0 0.1820 0.3556 0.5136 0.6512 0.7659 0.8572 0.9261 0.9738 1.002 1.011 1 y 0.1953 0.1908 0.1782 0.1592 0.1362 0.1116 0.08696 0.06340 0.04124 0.02028 0 x 0 0.1969 0.3830 0.5495 0.6914 0.8070 0.8969 0.9632 1.009 1.055 1.043 y O.4013 0.3916 0.364o 0.3231 0.2743 0.2229 0.1725 0.1251 0.08099 0.05972 0 x 0 0.2251 0.4341 o.6152 0.7632 0.8784 0.9641 1.025 1.065 1.088 1.096 0~3 y 0.6310 0.6138 0.5657 0.4959 0.4152 0.3327 0.2543 0.1825 0.1173 0.05729 0 x 0 0.2733 0.5194 0.7209 0.8742 0.9844 1.060 1.111 1.143 1.160 1.165 4 y 0.9028 0.8735 0.7935 0.6811 0.5574 0.4371 0.3278 0.2318 0.1475 0.07155 0 x 0 0.3562 0.6598 o.8854 1.036 1.130 1.186 1.218 1.236 1.245 1.247 5 y 1.247 1.195 1.058 0.8785 o.6940 0.5271 0.3852 0.2670 0.1675 o. 8065 0 x 0 0.5077 0.8988 1.140 1.265 1.320 1.339 1.342 1.339 1.336 1.335 6 y 1.72 1.622 1.373 1.077 0.8067 0.5861 0.4140 0.2800 0.1729 0.08244 0 x 0 0.8195 1.328 1.533 1.573 1.549 1.509 1.471 1.443 1.426 1.420 0.7 y 2.465 2.234 1.731 1.236 0.8557 0.5868 0.3979 0.2619 0.1589 0.07503 0 x o 1.595 2.134 2.104 1.944 1.791 1.673 1.588 1.533 1.501 1.491 Y 3.877 5.171 2.028 1.237 0.7713 0.4948 0.3218 0.2062 0.1231 0.05763 0 x 0 4.072 2.497 2.764 2.289 1.989 1.796 1.672 1.595 1.552 1.539 y 7.967 4.269 1.753 0.8567 0.4789 0.2898 0.1822 0.1145 0.06753 0.03142 0 x oo 8.773 4.484 3.098 2.438 2.067 1.843 1.703 1.617 1.570 1.556 1.0 y oo y a 0 0 0 0 0 0 0 0 0 0

8Z9'0 =oO U!S = 1 (ZZ Z - I (ZZ -1) (Z l-|i)(z-i I: (Z) = Al+ n zp J (A +n) us = A! +x =z' uo!iunj-us a9 n 0'1 6'0 8'0 L'O 9'0' O _' 0' Vo2' 0 o0 1'0 0.~~~~~~~~~0~~ ~ n I1~~~~~~~~~~~~~~~~~~Z'0 I;FI'0 I 4, C 1 v I I J. 7 l + \ I i I i \ \ 1 N i I 1 1 I I I/ 1 1 1 1 1 1 1 I I I I I I. I.I.I.I.I A k A S 1 1 111 6'0 II I I I 91 _O_ 7 __ 7 TF _ _ T_ w 7 v D a~rwn M 1R1 NAT11N 11 1 ZI' 1\1 _ 1.- I - 11I I I 11 0 1 b'OK I i t IS; I SIj W! 1\N11 X IO O' I OI

THE sn-FUNCTION, x + iy = sn(u + iv) k = sin 45~ = 0.707107; K = K' = 1.854075 0 0.1 0.2 0.3 0,4 0.5 0.6 0.7 0.8 0.9 1.0 K V K' x 0 0.1838 0.3585 0.5166 0.6529 0.7654 0.8539 0.9198 0.9650 0.9914 1.000 y 0 0 0 00 0 0 0 0 0 0 x 0 0.1885 0.3671 0.5276 0.6651 0.7775 0.8652 0.9301 0.9745 1.000 1.009 y 0.1870 0.1822 0.1685 0.1484 0.1247 0.1002 0.07661 0.05493 0.03525 0.01719 0 x 0 0.2035 0.3942 0.5623 0.7027 0.8144 0.8995 0.9610 1.002 1.026 1.034 0.2 y 0.3841 0.3734 0.3436 0.3002 0.2502 0.1992 0.1512 0.1077 0.06885 0.03348 0 x 0 0.2320 0.4448 0.6255 0.7693 0.8782 0.9572 1.012 1.048 1.068 1.074 0~3 y 0.6033 0.5844 0.5323 0.4586 0.3761 0.2950 0.2210 0.1558 0.09883 0.03785 0 x 0 0.2807 0.5291 0.7266 0.8715 0.9719 1.039 1.082 1.109 1.123 1.127 y o.8620 0.8297 0.7429 0.6251 0.5000 0.3831 0.2814 0.1955 0.1227 0.05904 0 x 0 0.3645 0.6672 0.8822 1.018 1.099 1.144 1.168 1.181 1.187 1.189 0.5 y 1.189 1.132 0.9844 0.7974 0.6139 0.4551 0.3256 0.2218 0.1374 0.06560 o x 0 0.5174 0.8995 1.118 1.221 1.259 1.268 1.266 1.260 1.256 1.254 y 1.641 1.529 1.263 0.9622 0.7004 o.4965 0.3436 0.2286 0.1395 0.o6604 0 x 0 0.8299 1.307 1.471 1.484 1.447 1.403 1.365 1.338 1.322 1.317 0.7 y 2.344 2.091 1.564 1.078 0.7254 0.4862 0.3238 0.2101 0.1262 0.05923 0 0.8 x 0 1.592 2.039 1.957 1.786 1.638 1.529 1.453 1.404 1.377 1.368 y 3.682 2.920 1.777 1.045 0.6360 0.4009 0.2571 0.1629 0.09645 0.04493 0 x 0 3.880 3.182 2.484 2.054 1.789 1.622 1.515 1.449 1.414 1.402 9 y 7.562 3.749 1.461 0.6985 0.3852 0.2306 0.1436 0.08947 0.05244 0.02430 0 x o 7.693 3.944 2.738 2.166 1.848 1.656 1.538 1.465 1.427 1.414 1.0 y y 0 0 0 0 0 0 0 0 0 0 0

u K 0 o.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.0 Io0 o.8:1- 5ll 0V8i-.1.0 2_ _ IK l i" @'1 4L' ) "r ^^^^^^^?ntltfjo.,5 o. 0.3 4 5 0 K The sn-function, z = x + iy = sn (u + iv) / dz uiv=FZ- NJI-z) (I-kz2) k0.= sin 45~=0.7071

THE sn-FUNCTION, x + iy = sn(u + iv) k = sin 50~ = 0.766044; K = 1.935581, K' = 1.786769 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 K V K' x 0 0.1917 0.3724 0.5336 0.6701 0.7802 0.8649 0.9267 0.9683 0.9922 1.000 y 0 0 0 0 0 0 0 0 0 0 0 x 0 0.1965 0.3810 0.5444 0.6815 0.7911 0.8747 0.9352 0.9758 0.9990 1.007 y 0.1802 0.1748 0.1599 0.1383 0.1138 0.08933 0.06679 0.04694 0.02966 0.01432 0 x 0 0.2118 0.4082 0.5782 0.7168 0.8243 0.9042 0.9607 0.9979 1.019 1.026 y 0.3699 0.3581 0.3254 0.2791 0.2274 0.1769 0.1312 0.09161 0.05763 0.027.74 0 x 0 0.2408 0.4588 0.6393 0.7789 0.8813 0.9535 1.002 1.034 1.051 1.057 y 0.5806 0.5596 0.5027 0.4242 0.3397 0.2599 0.1901 0.1314 0.08205 0.03933 0 x 0 0.2906 0.5427 0.7365 0.8730 0.9639 1.022 1.059 1.081 1.093 1.097 y 0.8290 0.7930 0.6983 0.5739 0.4471 0.3338 0.2394 0.1630 0.1008 0.04803 0 x 0 0.3761 0.6793 0.8839 1.006 1.074 1.109 1.128 1.137 1.141 1.143 y 1.143 1.079 0.9187 0.7239 0.5415 0.3908 0.2731 0.1824 0.1114 0.05272 0 x 0 0.5318 0.9058 1.103 1.185 1.209 1.210 1.204 1.197 1.192 1.191 y 1.575 1.451 1.165 0.8593 0.6067 0.4188 0.2835 0.1853 0.1116 0.05239 0 x 0 0.8469 1.293 1.418 1.408 1.363 1.317 1.280 1.255 1.240 1.236 0.7 y 2.248 1.968 1.417 0.9407 0.6141 0.4019 0.2626 0.1678 0.09961 0.04642 0 x 0 1.598 1.955 1.831 1.655 1.514 1.414 1.347 1.304 1.280 1.273 y 3.529 2.701 1.559 0.8840 0.5250 0.3248 0.2051 0.1284 0.07529 0.03485 0 x 0 3.708 2.913 2,252 1.863 1.629 1.484 1.392 1.337 1.306 1.297 9 y 7.244 3.300 1.222 0.5723 0.3112 0.1840 0.1133 0.06988 0.04062 0.01872 0 x o 6.811 3.505 2.446 1.948 1.673 1.509 1.409 1.348 1.316 1.305 1.0 y m 0 0 0 0 0 0 0 0 0 0

U K 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 O1.0...~lk..lF. 1_0001.0 V V 0.6,4 mm: —g4 # S E;W H ~ e M X \ ] | I 10,4 0. - 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9.0 The sn-function z x+iy sn(u+iv) u+iv =F(z)= - dz) 0o~ ~~~~~~k s 00076 0. 0.2 0,3 0.4 0.5 0.6 0.7 0.8 0,9 I.6 u-'v = Fz2)=(I-_k2z2 k= sin 50~=0.7660

THE sn-FUNCTION, x + iy = sn(u + iv) k = sin 55~ = 0.819152; K = 2.034715, K' = 1.731245 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 K v K' x 0 0.2012 0.3891 0.5539 0.6901 0.7972 0.8773 0.9343 0.9719 0.9931 1.000 y 0 0 0 0 0 0 0 0 0 0 0 x 0 0.2061 0.3979 0.5645 0.7010 0.8071 0.8859 0.9414 0.9779 0.9985 1.005 0 y 0.1746 0.1686 0.1520 0.1287 0.1032 0.07881 0.05736 0.03936 0.02441 0.01164 0 0. x 0 0.2219 0.4252 0.5974 0.7339 0.8367 0.9109 O.9620 0.9950 1.013 1.019 2 y 0.3583 0.3450 0.3088 0.2590 0.2054 0.1553 0.1121 0.07646 0.04721 0.02246 0 x 0 0.2519 0.4762 0.6568 0.7917 0.8872 0.9526 0.9957 1.023 1.037 1.042 ~03 y 0.5621 0.5385 0.4756 0.3916 0.3048 0.2265 0.1613 0.1088 0.06674 0.03163 0 x 0 0.3033 0.5603 0.7502 0.8781 0.9596 1.010 1.041 1.059 1.068 1.071 y 0.8021 0.7617 0.6574 0.5255 0.3972 0.2878 0.2009 0.1337 0.08119 0.03827 0 x 0 0.3913 0.6958 0.8897 0.9978 1.054 1.082 1.095 1.101 1.104 1.105 y 1.105 1.033 0.8583 0.6652 0.4744 0.3322 0.2261 0.1478 0.08876 0.04157 0 x 0 0.5508 0.9168 1.092 1.154 1.167 1.163 1.154 1.146 1.141 1.139 y 1.522 1.383 1.076 0.7646 0.5220 0.3501 0.2312 0.1482 0.08787 o.o4087- 0 x 0 0.8700 1.284 1.371 1.343 1.292 1.246 1.212 1.189 1.176 1.172 0.7 y 2.172 1.860 1.282 0.8176 0.5170 0.3298 0.2109 0.1324 0.07758 0.03585 0 x 0 1.610 1.880 1.720 1.543 1.410 1.320 1.261 1.224 1.204 1.198 y 3.407 2.503 1.365 0.7458 0.4318 0.2618 0.1625 0.1002 0.05809 0.02669 0 x 0 3.550 2.678 2.056 1.705 1.498 1.373 1.295 1.248 1.223 1.215 0'9 y 6.993 2.903 1.023 0.4689 0.2510 0.1463 o.08888 0.05413 0.03115 0.01426 0 x 0 6.069 3.137 2.204 1.769 1. 531 1.391 1.307 1.256 1.229 1.221 1. v * 0 0 y 0 0 0 0 0 0 0

u K 0 0.,1,0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.0 _ I I I _I I I I I I I _ _ _ _ _ I I _I -I I -I I - I I. I I I _I I I - I - I I I I I I I I I,. _ _ I _. - UJ1 0.9 t | t t t: 0/t Z" XA0.9 —. 05 0.5 d K7 II ly t~K ~I~tt ~ IV 0.8I0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 K The sn-function, z= x+ iy = sn(u+iv) u+iv=F(z)== -.dz-k k = sin 55~-08192

THE sn-FUNCTION, x + iy = en(u + iv) k = sin 60~ * 0.866025; K = 2.156516, K' - 1.685750 0 0.1 0.2 0. 0.4 0.5 0.6 0.7 0.8 0.9 1.0 K K' x 0 0.2128 0.4093 0.5779 0.7154 0.8165 0.8912 0.9427 0.9758 0.9941 1.000 Y O O O 0 O O O O O O O x 0 0.2i79 0.4181 0.5885 0.7235 0.8252 0.8982 0.9482 0.9802 0.9979 1.004 Y 0.1700 0.1631 0.1445 0.1192 0.09264 0.0684821 0.00.0215 0.01949 0.009165 0 X o 0.2344 0.4460 0.6205 0.7543 0.8515 0.9193 o.9647 0.9952 1.009 1.014 y 0.3488 0.3356 0.2951 0.2391 0.1836 0.1542 0.09582 0.06219 0.03756 0.01762 0 x 0 0.2657 0.4976 0.6781 0.8076 0.8958 0.9540 0.9913 1.014 1.026 1.050 Y 0.5471 0.5201 0.4498 0.35596 0.2705 0.1943 0.1339 0.08792 0.05275 0.02465 0. 0 0.3192 0.5821 0.7677 0.8862 0.9583 1.001 1.027 1.041 1.049 1.051 0.7804 0.7543 0.6185 0.4784 0.5489 0.2442 0.1652 0.1070 0.06365 0.02961 0 x 0 0.4106 0.7167 0.8988 0.9929 1.038 1.059 1.068 1.072 1.074 1.075 y 1.075 0.9931 0.8006 0.5889 0.4109 0.2781 0.1837 0.1171 0.06893 0.03188 0 x 0 0.5749 0.9318 1.083 1.128 1.132 1.123 1.113 1.105 1.100 1.099 y 1.480 1.523 0.9900 0.6752 0.4441 0.2884 0.1855 0.1160 0.06755 0.03106 0 x 0 0.8993 1.277 1.329 1.286 1.231 1.187 1.156 1.136 1.125 1.121 ~07 y 2.111 1,761 1.154 0.7048 0.4506 0.2670 0.1667 0.1025 0.05908 0.02702 0 x 0 1.628 1.808 1.620 1.445 1.523 1.245 1.192 1.161 1.144 1.159 0.8 y 5.511 2.517 1.188 0.6245 0.5519 0.2086 0.1269 0.07684 0.04590 0.01998 0 x 0 3.396 2.467 1.885 1.570 1.390 1.282 1.216 1.178 1.157 1.151 Y 6.793 2.542 0.8525 0.5822 0.2011 0.1152 0.06879 0.04124 0.02542 0.01065 0 x 5.427 2.821 1.998 1.619 1.414 1.296 1.225 1.183 1.162 1.155 1.000 Y w 0 0 0 0 0 0 0 0 0 0

u K I O * _ _ _ _ _ 1 O l I _ i I I I _ lop _ I 1~t3Ir I I I/ 11 1 10F41 I I I I I 1 1, N51 I I I II I. I I I I I 1 9171 1 1 1 1 1 1 IPS81 1 1 1 1 I 1/ 1 I I Iv I L I L 0.2 85: 4t: 0.5 0 O0.i 0.47 0. U The sn-function, z=Nxiy= sn(u+iv) 0,t \ I-RhK z )(J I I-X - kz)V k= sin 60~=08660 1T67_811h1~~1\ I T ~t~~i'fiTl~~,~\iJ=;CS~ —~tr ~ ~ n-turtt-rrrIr r II1 i i~un ~ l I~-tt- i \T - r~-tM LIII IU I1\ rl i I1 LLL~cA Al N 070.1 0.2 0.3 0.4 0.5 0.6 0.7 0, 8 0.9 1.0

THE sn-FUNCTION, x + iy = sn(u + iv) k = sin 65~ " 0.906308; K 2.508787, K' - 1.648995 0 0.i 0. 0.2 0.4 0.5 0.6 0.7 0.8 0.9 1.0 V K' x 0 0.2272 0.4340 0.6066 0.7405 0.8384 0.9065 0.9517 0.9799 0.9952 1.000 y 0 0 0 0 0 0 0 0 0 0 0 x 0 0.2327 0.4430 0.6168 0.7499 0.8460 0.9123 0.9560 0.9831 0.9978 1.002 y 0.1663 0.1583 0.1371 0.1095 0.08181 0.05799 0.03929 0.02550 0.01492 0.006891 0 x 0 0.2500 0.4715 0.6481 0.7782 0.8688 0.9295 0.9687 0.9926 1.005 1.009 y 0.3411. 0.5235 0.2776 0.2188 0.1615 0.1133 0.07613 0.04875 0.02865 0.01521 0 x 0 0.2830 0.5238 0.7037 0.8268 0.9069 0.9576 0.9890 1.008 1.017 1.020 y 0.5350 0.5037 0.4244 0.3270 0.2361 0.1627 0.1079 0.06850 0.04003 0.01839 0 x 0 0.3391 0.6089 0.7889 0.8972 0.9596 0.9952 1.016 1.027 1.033 1.035 y 0.7630 0.7096 0.5798 0.4309 0.3011 0.2023 0.1318 0.08268 0.04795 0.02194 0 x 0 0.4346 0.7423 0.9109 0.9906 1.026 1.040 1.047 1.049 1.050 1.050 5 y 1.050 0.9563 0.7432 0.5231 0.3495 0.2274 0.1449 0.08958 0.05151 0.02346 0 x 0 0.6049 0.9504 1.077 1.105 1.101 1.090 1.079 1.072 1.068 1.067 y 1.446 1.266 0.9050 0.5883 0.3710 0.2321 0.1443 0.08786 0.05006 0.02269 0 x 0 0.9352 1.272 1.290 1.234 1.179 1.138 1.110 1.093 1.084 1.081 y 2.062 1.665 1.030 0.5992 0.3523 0.2115 0.1282 0.07691 0.04343 0.01960 0 x 0 1.650 1.738 1.528 1.359 1.249 1.179 1.136 1.111 1.097 1.093 y 3.234 2.135 1.023 0.5160 0.2821 0.1628 0.09657 0.05718 0.03206 o 0.01441 0 x 0 3.242 2.273 1.734 1.454 1,298 1.207 1.153 1.122 1.106 1.101 y 6.636 2.206 0.7035 0.3079 0.1587 0.08898 0.05199 0.03052 0.01703 0.007637 0 x o 4.856 2.542 1.819 1.490 1.316 1.217 1.159 1.126 1.109 1.103 1. Y 00 0 0 0 0 0 0 0 0 0 0

U 1.0........... I I I. I. I... I I.. I I I I... I I ~~~ ~1.0 ~K' I~~~~K K o O.l 00.3 0.4,g!|u|- - - - F= zL d "AL I1+-.1 I F i I o O.i 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 7.(? VV' S g gSU+i f o. 5 k= sin 65~=0.9063

THE sn-FUNCTION, x + iy = sn(u + iv) k = sin 70~ = 0.939695; K = 2.504550, K' = 1.620026 u 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 K V K' x 0. 2456 0. 4,49 0.6415 0.7723 0.8632 0.9234 0.9615 0.9843 0.9963 1.000 ~ y 0 0 0 0 O 0 0 0 0 0 x 0 0.2515 0.4741 0.6513 0.7807 0.8696 0.9279 0.9646 0.9865 0. 9980 1.002 0.1 y 0.1634 0.1539 0.1294 0.09903 0.07040 0.04739 0.03057 0.01884 0.01073 o.on4844 0 x 0 0.2700 0.5032 0.6815 0.8063 0.8888 0.9414 0.9738 0.9929 1.003 1.006 0.2 y 0.351 0.142.2613 0.1971.138 0.0921 0.05898 0.03618 0.02055 0.009259 0 x 0 0.3050 0.5564 0.7345 0.8495 0.9203 0.9631 0.9885 1.003 1.011 1.013 0'3 y 0.5255 0.4885 0.3976 0.2924 0.2005 0.1313. 08305 0.05055,0.02857 0.01284 0 x 0 0.3645 0.6417 0.8143 0.9110 0.9634 0.9919 1.008 1.016 1.020 1.022 o04 y 0.7494 0.6863 0.5391 0.3810 0.2528 0.1616 0.1005 0.06056 0.0340? 0.01524 0 x 0 0.4651 0.7732 0.9256 0.9903 1.016 1.026 1.030 1.031 1.031 1.032 10.5 y 1.032 0.9208 0.6829 0.4554 0.2889 0.1791 0.1093 0.06507 0,03629 0.01620 0 x 0 0.6424 0.9722 1.072 1.085 1.074 1.062 1.052 1.046 1.043 1.042 o.6 y 1.420 1.209 0.8167 0.5017 0.3010 0.1802 0.1076 0.06325 0.03502 0.01557 0 x 0 0.9787 1.266 1.251 1.187 1.133 1.097 1.074 1.060 1.053 1.051 0.7 y 2.025 1.567 0.9048 0.4978 0.2801 0.1617 0.09457 0.05490 0.03019 0.01338 0 x 0 1.674 1.666 1.441 1.282 1.185 1.126 1.091 1.071 1.061 1.058 Y 3.176 1.948 0.8649 0.4168 0.2199 0.1228 0.07055 0.04055 0.02217 0.009796 0 x 0 3.079 2.089 1.597 1.352 1.220 1.146 1.103 1.079 1.066 1.063 0.9 y 6.515 1.884 0.5703 0.2428 0.1219 0.06649 0.03774 0.02154 0.01174 0.005176 0 x oo 4.332 2.289 1.659 1.378 1.233 1.153 1.107 1.081 1.068 1.064 y O.0 y 0 0 0 0 0 0 0 0 0 0

U K 10, 1' i' i,, i,, p, __,3,..P PI4 _92P 0_____ 0,61, 1,,,,, 0, J O 4 XI I~ 0.9 1 IAAI I I I X I i I 11 0.9 0.7 0 07 0.8 0..0 V v 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 U K The sn-function, z=x+iy= sn(u+ iv) u+i v=F(z)= dz k= sin 70 z I 0 -k z k- sin 70~=0.9397

THE sn-FUNCTION, x + iy = sn(u + iv) k = sin 75~ = 0.965926; K = 2.768065, K' = 1.598142 0 0.1 0.2 0.3 0.4 05.6 0.7 0.8 0.9 1.0 K' 0 x 0 0.2702 0.5047 0.6846 0.8099 0.8913 0.9417 0.9717 0.9887 0.9974 1.ooo y o o o 0 0 o o o o o 0 01 x 0 0.2765 0.5142 0.6939 0.8172 0.8964 0.9450 0.9738 0.9901 0.9983 1.001 y 0.1611 0.1495 0.1207 0.08712 0.05796 0.03647 0.02206 0.01284 0.006981 0.003055 0 2 x 0 0.2965 0.5459 0.7224 0.8393 0.9116 0.9548 0.9800 0.9941 1.001 1.003 y 0.3306 0.3050 0.2429 0.1725 0.1132 0.07055 0.04239 0.02458 0~01533 0.005828 0 0.35 x 0 0.3342 0.5976 0.7717 0.8761 0.9363 0.9705 0.9897 1.000 1.006.007 y o.5184 0.4732 0.3673 0.2536 0.1627 0.09972 0.05950 0.03416 0.01847 0.008057 0 0o4 x 0 0.3980 0.6826 0.8444 0.9274. 9693 0.9909 1.002 1.008 1.011 1.012 y 0.7392 0.6624 0.4931 0.3262 0.2024 0.1214 0.07118 0.04067 0.02188 0.009524 0 05 x 0 0.5048 o.8102 O.9427 0.9917 1.009 1.015 1.017 1.017 1.017 1.017 y 1.017 0.8834 0.6155 0.3829 0.2276 0.1328 0.07663 0.04337 0.02321 0.01008 0 0.6 x 0 0.6900 o.9966 1.067 1.066 1.052 1.039 1.031 1.026 1.024 1.023 y 1.400 1.148 0.7200 0.4122 0.2326 0.1317 0.07466 0.04185 0.02227 0.009643 0 x 0 1,031 1.257 1.211 1.142 1.093 1.063 1.045 1.035 1.029 1.028 0.7 y 1.997 1.460 0.7727 0.3979 0.2121 0.1165 0.06495 0.03606 0.01910 0.008249 0 x 0 1.697 1.587 1.356 1.211 1.129 1.082 1.056 1.041 1.034 1.032 y 3.132 1.745 0.7086 0.3237 0.1634 0.08736 0.04804 0.02648 0.01397 0.006020 0 x 0 2.897 1.908 1.469 1.260 1.153 1.095 1.063 1.046 1.037 1.034 0.9 y 6.425 1.567 0.4479 0.1844 0.08939 0.04691 0.02556 0.01402 0.007372 0.003174 0 1.0 x 0 3.832 2.051 1.512 1.278 1.162 1.099 1.065 1.047 1.038 1.035 y 1 0 0 0 0 0 0 0 0 0 0

u K 0 0.1 0.2 _ 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.0 6k= sinE 8 75 00.9659 0.9 X I IINt I II/ I 0 X i. t K O 5 2 t4 u 0. 0.iE 0.42 i 4 y x:II> I A: y0.4 021I I Fl I 111' tM \/ 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 K The sn-function, z= x+iy = sn(u+iv) u+iv=F(ZL)=o:l)I-kXz ) k = sin 75~=09659

THE sn-FUNCTION, x + iy = sn(u + iv) k = sin 80~ = 0.984808; K = 3.153385, K' = 1.582843 0.1 0.2 0.5 0.4 0.5 0.6 0.7 0.8 0.9 1.0 K V K' x 0 0.3054 0.5595 0.7403 0.8552 0.9251 0.9613 0.9822 0.9952 0.9985 1.000 y o0 0 0 0 0 0 0 0 0 0 x 0 0.3124 0.5691 0.7487 0.8611 0.9267 0.9634 0.9834 0.9939 0.9989 1.000 0.1 y 0.1596 0.1446 0.1095 0.07246 0.04380 0.02505 0.01384 0.007427 0.003773 0.001577 0 x 0 0.3344 0.5990 0.7743 0.8787 0.9375 0.9696 09869 0.9959 1.000 1.001 0.2 y 0.5274 0.2945 0.2193 0.1425 0.08502 0.04821 0.02650 0.01418 0.007192 0.005004 0 x 0 0.3758 0.6525 0.8179 0.9076 0.9547 0.9795 0.9924 0.9991 1.002 1.003 0 y 0.5134 0.4554 0.3288 0.2072 0.1209 0.06758 0.03685 0.01962 0.009931 0.004143 0 x 0 0.4451 0.7352 0.8803 0.9467 0.9774 0.9922 0.9995 1.003 1.005 1.005 y 0.7321 0.6341 0.4356 0.2622 0.1482 0.08133 0.04388 0.02323 0.01172 0.004882 0 x 0 0.5595 0.8553 0.9616 0.9943 1.004 1.007 1.007 1.008 1.008 1.008 0.5 y 1.008 0.8381 0.5328 0.3012 0.1637 0.08782 0.04681 0.02462 0.01238 0.005148 0 x 0 0.7530 1.022 1.060 1.047 1.032 1.021 1.015 1.012 1.011 1.010 y 1.387 1.073 0.6057 0.3156 0.1639 0.08585 0.04517 0.02560 0.01183 0.004910 0 x 0 1.095 1.241 1.167 1.099 1.058 1.055 1.023 1.016 1.013 1.012 0.7 y 1.978 1.327 0.6254 0.2956 0.1464 0.07490 0.03895 0.02022 0.01010 0.004188 o x 0 1.710 1.495 1.268 1.145 1.080 1.046 1.029 1.020 1.015 1.014 y 3.101 1.506 0.5473 0.23355 0.1108 0.05555 0.02860 0.01478 0.007565 0.003049 0 x 0 2.677 1.721 1.344 1.176 1.095 1.054 1.033 1.022 1.017 1.015 09 y 6.362 1.259 0.3512 0.1 00 0.05983 0.02960 0.01514 0.007797 0.003879 0.001605 0 x ~ 3.325 1.815 1.372 1.187 1.100 1.056 1,034 1.022 1.017 1.015 y c O O O O O O O O O O

U K O.9'0.9 0.8 O~f~~~t~,-tt~t~t~e 1 I. 8 o 00.05. 0.6 \.7 0.8 0.N.0 The sn-function,z z=x+ y = sn(u+iv) 0u+iv=F(z)= f d I 0.3. I A s\ A0.3 I (- z)(I-kAz2) k si 80 = 0.9848 0.1 0.2 0.3 0. 4 0.5 0.6 0.7 0.8 0.9I 0.

THE sn-FUNCTION, x + iy = sn(u + iv) k = sin 85~' 0.9961951 K - 3.831742, K' = 1.573792 U 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 K' v K' x 0 0.3655 0.6452 0.8184 0.9121 0.9591 0.9817 0.9924 0.9973 0.9994 1.000 y o 0 0 0 0 0 0 0 0 0 0 x 0 0.3734 0.6546 o.8251 o.916 o0.9610 0.9827 0.9928 o.9975 0.9995 1.000 1 y 0.1587 0.1371 0.09193 0.05193 0.02661 0.01297 0.006157 0.002872 0.001293. 0s4956 0 x 0 0.3985 0.6834 0.453 0.9273 0.9668 0.9855 0.9942 0.9982 0.9999 1.000 2 y 0.3255 0.2783 0.1826 0.1012 0.05124 0.02481 0.01174 0.005470 o0.02461 o.o3943o 0 x 0 0.4450 0.7337 0.8787 o.9455 0.9759 0.9899 0.9963 0.9992 1.000 1.001 0.3,y 0.5105 0.4278 0.2698 0.1448 0.07191 0.03447 0.01624 0.007544 0.003391 0.001299 0 x 0 0.5218 0.8087 0.9247 0.9695 0.9876 0.9954 0.9989 1.000 1.001 1.001 0.4 y ).7279 0.5896 0.3496. 1791 o.08668 0.04101 0.01920 0.008893 0.003991 0.001528 0 x 0 0.6452 0.9121 0.9818 0.9975 1.001 1.002 1.002 1.002 1.002 1.002.5 y 1.002 0.7665 0.4145 0.1999 0.09378 0.04370 0.02031 0.009378 0.004202 0.001608 0 x 0 0.8449 1.046 1.046 1.027 1.015 1.008 1.005 1.003 1.003 1.002 6 y 1.379 0.9547 0.4521 0.2027 0.09185 0.04213 0.01944 0.008945 0.004002 0.001530 0 x 0 1.172 1.205 1.112 1.056 1.027 1.014 1.008 1.005 1.003 1.003 0.7 y 1.966 1.127 0.4432 0.1832 0.08029 0.03629 0.01663 0.007630 0.003409 0.001302 0 x 0 1.693 1.371 1.171 1.079 1.038 i.o18 1.010 1.006 1.004 1.003 o.8 y 3.084 1.182 0.3663 0.1401 0.05964 0.02663 0.01214 0.005555 0.002479 o.039468 0 x 0 2.369 1.504 1.212 1.095 1.044 1.021 1.011 1.006 1,004 1.004 0.9 y 6.326 0.8697 0.2112 0.07626 0.03181 0.01409 0.006400 0.002924. 0.001304 0.034979 0 x 1o 2.746 1.556 1.227 1.101 1.047 1.023 1.012 1.007 1.004 1.004 y1.Go 0 0 0 0 0 0 0 0 0 0

u K 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.9 s- 90.9 0.8 d U 00 S I I0.1 8 I / i V 1 L 11" 1 Fl Wli a=:=^,1Yt ^::fj - 3l:: 1 E _ _ _- I r^It _|.:I II _ _ I1 A \ I I I K' 0.5 oi FtiX( tii~i~ 3 0 540N 0.4 S tS'' 0' 0.4 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0,9 1.0 If \ I I IIIAIN If o N NI-Z)(-k z2) 0 O.I 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9962

THE sn-FUNCTION, x + ty = sn(u + iv) k = sin 89~ 0.999848; K - 5.434910, K' = 1.570916 0 0.1 0.2 0.3 o04 0.5 0.6 0.7 0.8 0.9 1.0 K K' x 0 0.4956 0.7958 0.9262 0.9745 0.9914 0.9971 0.9991 0.9997 1.000 1.000 y 0 0 0 0 0 0 0 0 0 0 0 x 0 0.5050 0.8030 0.9294 0.9757 0.9918 0.9973 0.9991 0.9998 1.000 1.000 0.1 y 0.1584 0.1188 0.05719 0.02207 0.007803 0.002674 0.039066 0.03oo60 O.091021 0.043092 0 x 0 0.5341 0.8247 0.9389 0.9792 0.9930 0.9977 0.9993 0.9998 1.000 1.000 0.2 y 0.3249 0.2389 0.1117 0.04241 0.01490 0.005093 0.001726 0.0 5822 0.031942 0.045882 0 x 0 0.5869 0.8609. 9542 0.9848 0.9949 0.9983 0.9995 0.9999 1.000 1.000 0.3 y 0.5096 0.3614 0.1605 0.05931 0.02062 0.007024 0.002377 0.038015 0.032673 0.048096 0 X 0 0.6703 0.9113 0.9741 0.9918 0.9973 0.9992 0.9998 1.000 1.000 1.000 0.4 y 0.7266 0.4852 0.1997 0.07118 0.02441 0.008277 0.002796 0.039424 0.033143 0.049517 0 x 0 0.7958 0.9745 0.9971 0.9997 1.000 1.000 1.000 1.000 1.000 1.000 0.5 y 1.000 0.6057 0.2246 0.07662 0.02587 0.008727 0.002943 0.039912 0.033305 0.031001 0 x 0 0.9790 1.047 1.021 1.008 1.003 1.001 1.000 1.000 1.000 1.000 6 y 1.376 0.7087 0.2295 0.07463 0.02480 0.008322 0.002801 0.039430 0.033143 0.049518 0 x 0 1.236 1.123 1.044 1.015 1.005 1.002 1.001 1.000 1.000 1.000 0.7 y 1.963 0.7609 0.2093 0.06491 0.02125 0.007096 0.002385 0.038024 0.032674 0.048097 0 x 0 1.560 1.191 1.063 1.021 1.007 1.002 1.001 1.000 1.000 1.000 y 3.078 0.6980 0.1613 0.04801 0.01553 0.005166 0.001734 0.035831 0.031945 0.045883 0 x 0 1.877 1.239 1.076 1.025 1.008 1.003 1.001 1.000 1.000 1.000 y 6.314 0.4414 0.08826 0.02554 0.008196 0.002719 0.039120 0.033066 0.031022 0.043093 0 x w 2.018 1.257 1.080 1.026 1.009 1.003 1.001 1.000 1.000 1.000 1,0 Y ~ 0 0 0 0 0 0 0 0 0 0

u K 0 0.1 0.2 0 04 05 06 07 0.8 0.9 10 1.0 4.... 1.0 IA~~~~~~~~~ Vd- x-kz2 0.9.99 I~~~~~~~~1 0. 5 I 1.I IV _' __Ttt l _i._ bl, /1+ 1 1ltl |1ltltl N I - Ik I I I/ - I1 - - 111 Ilo 0. 1Lf1 A 44 I I1 0.4 M AtS X{E V~IiI / 0f tAt~0U \t I4 /W IX~~ \!W II 1i dt ffi ffiS g X; g H l~~~~~~~~f I W\f I I -- A, 11( t1 E FW \ / \1; I 11 1 I VI I NMI 0.6 W 0gX | | i iX 2 2 2 W~~~~~~~~~~~~~~~~~~~g l 4 0 > 0; g 0. 6~~~~~~~0. H- Z X L \ _ f W~vII A\ I I/ i it; r/= -f i 0.72 I 1e 0.2 Y1 1 I I 0.3CI I IF- I - I I 1\ 0.7 DE tS;E t50001 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 P 0.9 1.0 U The sn-function, z = x iy = sn (u + iv) u~v =F~ z ) =o l-z)(I-kzN) k =sin89~=.9999

APPENDIX II THE cn-FUNCTION

THE cn-FUNCTION, x + iy = cn(u + iv) k = sin 1~ = 0.0174524; K = 1.570916, K' = 5.454910 0 0.1 0.2 0.3 0.4 0.5 0.6 0 7 0.8 0.9 1.0 K V K' x 1.000 0.9877 0.9511 0.8910 0.8090 0.7071 0.;878 0.4540 0.3090 o.1564 0 y 0 0 0 0 0 0 0 0 0 0 0 0 x 1.151 1.137 1.095 1.026 0.9314 o.8141 0.6767 0.5226 0.5557 0.1801 0 0.1 y 0 0.08928 0.1764 0.2591 0.5554 0.4055 0.4616 o. io84 0.5 426 0.5655 0.5705 x 1.651 1.631 1.570 1.471 1.336 1.167 0.9705 0.7494 0.5100 0.2582 0 0.2 y 0 0.2056 0.4062 0.5967 0.7725 0.9292 1.063 1.171 1.249 1.297 1.514 x 2.652 2.619 2.522 2.362 2.144 1.873 1.557 1.202 0.8181 0.4141 0 0.3 y 0 0.3846 0.7596 1.116 1.444 1.757 1.986 2.187 2.554 2.424 2.454 x 4.459 4.404 4.259 3.968 3.600 3.144 2.611 2.015 1.571 0.6956 0 0.4 y 0 0.6817 1.346 1.976 2.557 3.073 3.512 3.865 4.125 4.280 4.552 x 7.635 7.538 7.250 6.779 6.140 5.552 4.457 5.419 2.523 1.174 0 0.5 y 0 1.194 2.356 3.454 4.461 5.352 6.107 6.710 7.148 7.414 7.505 x 13.22 1304 12.51 11.65 10.51 9.109 7.512 5.761 5.899 1.967 0 6 y 0 2.114 4.160 6.077 7.810 9.20 10.58 11.57 12.28 12.71 12.85 x 25.35 22.93 21.86 20.12 17.87 15.25 12.58 9.367 6.274 3.144 0 0.7 y 0 3.917 7.663 11.06 14.00 16.45 18.58 19.85 20.85 21.41 21.61 x 43.61 42.58 39.28 54.72 29.40 23.91 18.59 15.56 8.844 4.359 0 0.8 356 8844 4 y 0 8.472 16.04 22.27 26.84 50.04 32.14 55.44 54.19 54.58 54.70 x 100.4 92.24 73.84 54.81 59.45 28.01 19.60 13.26 8.211 3.950 0 0~9 y 0 29.48 48.41 56.35 57.79 56.51 54.45 52.50 51.00 50.07 49.76 x oo 0 0 0 0 0 0 0 0 0 0 1.0 66. 185.4 126.2 97.47 81. 7081 640 6.23 58.01 57.29 y 5 566.5 185.4 126.2 97.47 81.02 70.81 64.50 60.25

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THE cn-FUNCTION, x + iy = cn(u + iv) k = sin 5~ = 0.0871557; K = 1.573792, K' = 3.831742 u 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 v K' x 1.000 0.9876 0.9509 0.8907 0.8085 0.7064 0.5871 0.4533 0.3085 0.1561 0 y 0 0 00 0 o o o o 0 0 x 1.074 1.061 1.021 0.9566 o.8682 0.7585 0.6302 0.4866 0.3311 0.1676 0 O- 1 y o 0.06157 0.1216 0.1785 0.2308 0.2774 0.3171 0.3489 0.3721 0.3862 0.3910 x 1.309 1.292 1.244 1.164 -1.056 0.9221 0.7656 0.5908 0.4018 0,2033 0 ~'2 y 0 0.1327 0.2619 0.3843 0.4967 0.5965 0.6813 0.7492 0.7987 0.8288 o.8390 x 1.740 1.718 1.652 1.545 1.400 1.220 1.011 0o.7793 0.5295 0.2677 0 0~3 y 0 0.2248 0.4435 0.6500 0.8391 1.006 1.147 1.260 1.342 1.391 1.408 x 2.439 2.407 2. 311 2.156 1.947 1.691 1.398 1.074 0.7278 0.3674 0 ~4 y 0 0. 549 0o6988 1.022 1.315 1.572 1.786 1.956 2.078 2.151 2.176 x 3.532 3.481 3.531 3.090 2.772 2.391 1.961 1.497 1.010 0.5083 0 0.5 Y 0 0.5523 1.084 1.576 2.015 2.391 2.698 2.935 3.103 3.203 3.236 x 5.253 5.162 4.9o0 4.491 3.969 3.370 2.723 2.052 1.370 0.6852 0 y 0 o.8816 1.716 2.466 3,106 3.627 4.032 4.329 4.531 4.648 4.686 x 8.118 7.921 7.371 6,561 5.606 4.599 3.602 2.644 1.732 0.8560 0 Y 0 1.524 2.908 4.058 4.942 5.577 6.008 6.286 6.453 6.542 6.568 x 13.61 13.01 11.42 9.408 7.414 5.653 4.168 2.921 1.852 0.8983 0 Y 0 3.190 5.745 7.417 8.331 8.738 8.862 8.851 8.799 8.751 8.733 x 29.24 24.90 17.14 11.12 7.279 4.862 3.270 2.154 1.313 0.6232 0 9 0 10.80 15.25 15.51 14.47 13.29 12.30 11.55 11.04 10.74 10.64 x 0 0 0 0 0 0 0 0 0 0 1 7.02 11.57 11.43 Y = 73.20 37.05 25.21 19.47 16.18 14.14 12.84 12.02 11.57 11.43

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THE cn-FUNCTION, x + iy = cn(u + iv) k = sin 10~ = 0.173648; K = 1.582843, K' = 3.153385 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 K V K' x 1.000 0.9875 o.9504 0.8896 o.8069 0.7044 0.5848 0.4512 0.3069 0.1553 0 0 y o o o o o 0 o o o o 0 x 1.050 1.037 0.9978 0.9336 0.8464 0.7386 0.6129 0.4727 0.5214 0.1626 0 0.1 y 0 0.05061 0.09979 0.1462 0.1886 0.2259 0.2575 0.2825 0.3007 0.3117 0.3154 x 1.206 1.191 1.145 1.070 0.9688 0.8440 0.6993 0.5385 0.3657 0.1848 0 2 y 0 0.1070 0.2109 0.3085 0.3974 0.4754 0.5409 0.5927 0.6501 0.6527 0.6602 x 1.487 1.468 1.409 1.313 1.185 1.029 0.8495 0.6522 0.4418 0,2250 0 0 y 0.1765 0.3471 0.5066 0.6505 o.7754 0.8792 0.9604 1.019 1.053 1.065 x 1.930 1.902 1.819 1.688 1,514 1.305 1.071 0.8172 0.5511 0.2774 0 y 0 0.2699 0.5291 0.7684 0.9806 1.161 1.308 1.420 1.500 1.547 1.562 x 2.600 2.556 2.430 2.252 1.978 1.684 1.365 1.031 0.6894 0.3452 0.5 y 0 0.4078 0.7944 1.142 1.440 1.684 1.874 2.014 2.109 2.164 2.181 x 3.630 3.553 3:332 3.000 2.596 2.158 1,710 1.267 0.8360 0.4150 0 0.6 y 0 0.6571 1.224 1.726 2.125 2.426 2.639 2.784 2.874 2.924 2.939 x 5.325 5.155 4.686 4.042 3.336 2.650 2.019 1.450 0.9353 0.4581 0 7 y 0 1.091 2.033 2.745 3.224 3.516 3.678 3.759 3.796 3.810 3.813 x 8.590 8.044 6.733 5.242 3.925 2.874 2.056 1.411 0.8822 0.4245 0 y 0 2.278 3.908 4.763 5.074 5.101 5.011 4.893 4.791 4.724 4.701 x 17.98 14.30 8.804 5.283 3.304 2.148 1.422 0.9284 0,5629 0.2663 0 y 0 7.460 9.409 8.840 7.865 7.023 6.384 5.930 5.628 5.457 5.401 x o0 0 0 0 0 0 0 0 0 0 l. y 36.53 18.48 12.57 9.697 8.051 7.029 6,375 5.968 5.743 5.671

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THE cn-FUNCTION, x + iy = cn(u + iv) k - sin 15~ = 0.258819; K = 1.598142, K' = 2.768065 u 0 0.1 0.2 0.5 o.4 0.5 0.6 o 7 0.8 0.9 1.0 K' x 1.000 0.9873 0.9495 0.8878 o.8041 0.7010 0.5811 0.4477 0.3042 0.158 0 y 0 0 o o o o0 0 0 o 0 x 1.039 1.025 0.9856 0.9211 o.8336 0.7261 0.6015 0.4631 0.3144 0.158 o0 0.1 y 0 0.04472 0.08797 0.1284 0.1649 0.1967 0.2230 0.2437 0.2585 0.2674 0.2705 x 1.158 1.143 1.097 1.023 0.9240 0.8026 0.6631 0.5093 0.3452 0.1743 0 0.2 y 0 0.09394 0.1846 0.2689 0.3445 0.4097 0.4634 0.5051 0.5548 0.5525 0.5584 x 1.372 1.352 1.295 1.203 1.081 0.9336 0.7672 0.5866 O.3961 0.1995 0 y 0 0.1534 0.3007 0.4363 0.5562 0.6578 0.7401 0.8029 0.8470 0.8730 0.8816 0.. 1.705.1.678 1.599 1.474 1.312 1.122 0.9135 0.6926 0.4647 0.2331 0 y 0 0.2323 0.4530 0.6524 0.8241 0.9651 1.076 1.157 1.213 1.245 1.256 x 2.205 2.163 2.042 1.856 1.625 1.366 1.094 0.8182 0.5433 0.2708 0 5 y 0 0.3483 0.6728 0.9554 1.187 1.366 1.498 1.590 1.649 1.682 1.692 x 2.972 2.895 2.683 2.374 2.016 1.644 1.282 0.9375 0.6125 0.3023 0 y 0 0.5420 1.028' 1.421 1.712 1.912 2.040 2.118 2.161 2.185 2.189 x 4.233 4.063 3.616 3.031 2.431 1.882 1.405 0.9945 0.6349 0.3091 0 y 0 0.9285 1.691 2.214 2.520 2.672 2.731 2.742 2.735 2.725 2.721 08 x 6.684 6.144 4.927 3.664 2.643 1.883 1.322 0.8965 0.5564 0.2666 0 y 0 1.938 3.180 3.695 3.782 3.689 3.547 3.412 3.308 3.244 3.223 x 31.81 10.32 5.824 3.322 2.024 -1.298 0.8527 0.5543 0.3354 0.1585 0 y 0 6.133 7.085 6.515 5.457 4.790 4.309 5.976 3.758 3.634 5.594 x10 0 0 0 0 0 0 0 0 0 y 24.27 12.27 8.335 6.422 5.324 4.641 4.204 3.931 3.780 3.732

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THE cn-FUNCTION, x + iy = cn(u + iv) k = sin 20~ = 0.342020; K = 1.620026, K' = 2.504550 U K 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 v K' x 1.000 0.9869 o.9482 0.8852 0.8002 0.6960 0.5758 0.4428 o.3004 0.1517 0 y 0 0 0 0 0 o o o o o o x 1.032 1.018 0.9774 0.9117 0.8233 0.7153 0.5910 0.4541 0.3078 0.1554 o y 0 0.04093 0.08028 0.1166 0.1489 0.1764 0.1987 0.2158 0.2278 0.2349 0.2372 x 1.129 1.114 1.067 0.9929 o.8934 0.7733 0.6366 0.4875 0.3296 0.1661 o 0.2 y 0 0.08579 0.1679 0.2433 0.3095 0.3652 0.4099 0.4437 0.4673 0.4811 o.4856 x 1.304 1.284 1.226 1.134 1.013 0.8706 0.7116 0.5416 0.3644 0.1831 0 y 0 0.1397 0.2725 0.3925 0.4959 0.5808 0.6473 0.6964 0.7298 0.7492 0.7555 4 x 1.574 1.547 1.467 1.343 1.186 1.006 0.8125 0.6120 0.4086 0.2043 0 y 0 0.2110 0.4088 0.5829 0.7276 0.8415 0.9265 0.9866 1.026 1.048 1.055 x 1.981 1.938 1.816 1.633 1.411 1.172 0.9283 0.6880 0.4538 0.2253 0 0.5 y 0 0.3160 0.6048 0.8471 1.035 1.172 1.266 1.326 1.362 1.381 1.587 x 2.605 2.526 2.312 2.011 1.676 1.344 1.033 0.7470 0.4843 0.2379 o y 0 0.4926 0.9207 1.248 1.470 1.607 1.683 1.721 1.738 1.744 1.745 x 3.637 3.461 3.013 2.458 1.922 1.457 1.071 0.7491 0.4747 0.2302 o 0.7 y 0 0.8460 1.505 1.912 2.113 2.184 2.188 2.164 2.136 2.115 2.108 x 5.658 5.103 3.927 2.806 1.965 1.372 0.9512 0.6397 0.3952 0.1889 0 y 0 1.762 2.770 3.082 3.051 2.905 2.746 2.611 2.512 2.452 2.433 x 11.58 8.137 4.269 2.346 1.404 0.8928 o.5844 0.3793 0.2293 0.1084 0 0.9 y 0 5.383 5.769 4.937 4.177 3.621 3.232 2.965 2.792 2.695 2.663 x o 0 0 0 0 0 0 0 0 0 y o 18.11 9.148 6.205 4.772 3.947 3.433 3.104 2.898 2.784 2.747

U K 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.0 1.0 0.97 I hX\ RI O. 9 I A I I I Ii I 1i L t K'0.8 o;~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~i.8 Iclill \+ Iv) =V AL SS W)F fsWSN E I ~~~~~~_Z' VA -7 0. <i X \ O.5 axSW m / U C0.490. 0.1 O| 0.2 0.3 0.4 0. 5 0.6 0.7 0.8 0.9 1.00 U. —[\\,. K The cn - function, z = x + iy = cn (u+ iv) =\/1I- sn2 ( u +I v) k = sin 20~=0.3420

THE cn-FUNCTION, x + iy = cn(u + iv) k = sin 25~ = 0.422618; K = 1.648995, K' = 2.508787 u K 0 0.1 0.2 0.5 0.4 0.0.6 0 7 0.8.9 1.0 v K' x 1.000 0.9865 0.9464 0.8817 0.7950 o.689, o0.5688 0.4565 0.2954 0.1490 0 0 y 0 0 0 0 0 0 x 1.027 1.013 0.9709 0.9055 0.8134 0.7044 0.5802 0.4445 0.5006 0.1516 0 0.1 y 0 0.03835 0.07491 0.1081 0.1370 0.1608 0.1796 0.1936 0.2051 0,2086 0.2104 x 1.110 1.094 1.046 0.9698 0.8692 0.7491 0.6141 0.4686 0.3159 0.1590 0 0 y 0 0.08035 0.1566 0.2252 0.2839 0.3318 o.3688 0.3958 0.4140 0.4244 0.4278 0.3 x 1.258 1.237 1.178 1.084 0.9652 0.8225 0.i6684 0.5062 0.5595 0.1701 0 Y 0 0.1508 0.2537 0.3625 0.4528 0.5242 0.5777 0.6154 0.6400 o.6558 0.6582 0.4 x 1.488 1.459 1.377 1.252 1.096 0.9212 0.7381 0.5523 0.3669 0.1829 0 Y 0 0.1978 0.3802 0.5361 0.6601 0.7526 0.8176 0.8605 0.8869 0.9010 0.9054 x 1.835 1.790 1.663 1.479 1.262 1.056 0.8115 0.5964 0.3911 0.1955 0 Y 0 0.2968 0.5621 0.7752 0.9310 1.036 1.100 1.138 1.157 1.166 1.169 0.6 x 2.369 2.286 2.065 1.765 1.445 1.141 0.8653 0.6201 0.3994 0.1955 0 y 0 0.4642 0.8541 1.133 1.305 1.396 1.436 1.448 1.447 1.443 1.441 x 3.258 3.072 2.615 2.078 1.587 1.182 0.8573 0.5947 0.3748 0.1812 0 Y 0 0.7994 1.387 1.710 1.857 1.854 1.824 1.780 1.740 1.714 1.705 0.8 x 5.013 4.431 3.274 2.256 1.541 1.060 0.7283 0.4872 0.3001 0.1433 0 y 0 1.659 2.498 2.671 2.566 2.393 2.229 2.098 2.005 1.950 1.932 x 10.19 6.728 3.309 1.766 1.044.0.6606 0.4317 0.2801 0.1694 o. 8008 0 y 0 4.888 4.898 4.055 3.373 2.894 2.564 2.340 2.196 2.115 2.088 x o 0 0 0 0 0 0 0 0 0 0 1.0149 7.260 4.915.77.110 2.698 2.42 2.266 2.174 2.145 Y 14.59 7.260 4.915 3.770 3.110 2.698 2.432 2.266 2.174 2.145

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THE cn-FUNCTION, x + iy = cn(u + iv) k = sin 30~ = 0.500000; K = 1.685750, K' = 2.156516 U 0 0.1 0.2 o.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 K' - K' x 1.000 0.9859 0.9442 0.8773 0.7883 0.6813 0.5600 0.4282 0.2892 0.1456 0 y 0 0 0 0 0 0 0 0 0 0 0, x 1.023 1.009 0.9651 0.8954 0.8032 0.6928 0.5685 0.4340 0.2927 0.1473 0 y 0 0.03657 0.07106 0.1018 0.1277 0.1483 0.1638 0.1749 0.1821 0.1862 0.1875 x 1.096 1.079 1.029 0.9505 0.8479 0.7271 0.5932 0.4508 0.3029 0.1521 0 y 0 0.07667 0.1485 0.2118 0.2641 0.3050 0.3351 0.3560 0,3694 0.3768 0.3791 x 1.225 1.204 1.141 1.045 0.922 0.7822 0.6318 0.4759 0.3178 0.1590 0 y 0 0.1250 0.2407 0.3402 0.4198 0.4795 0.5216 0.5494 0.5663 0.5753 0.5781 0.4 x 1.427 1.397 1.311 1.181 1.025 0,8537 0.6784 0.5043 0.3335 0.1658 0 y 0 0.1894 0.3610 0.5023 0.6090 0.6834 0.7314 0.7601 0.7760 0.7835 0.7858 x 1.732 1.684 1.551 1.363 1.148 o.9306 0.7221 0.5266 0.3435 0.1694 0 0.5 y 0 0.2852 0.5337 0.7237 0.8526 0.9306 0.9724 0.9915 0.9984 0.9999 1.000 0.6 x 2.204 2.116 1.884 1.582 1.273 0.9899 0.7426 0.5279 Q.3383 0.1651 0 y 0 0.4477 0.8097 1.050 1.181 1.237 1.249 1.242 1.228 1.218 1.214 x 2.996 2.797 2.326 1.801 1.346 0.9867 0.7082 0.4880 0.3064 0,1478 0. y 0 0.7729 1.305 1.560 1.630 1.609 1.556 1.499 1.453 1.424 1.414 x 4.569 3.953 2.803 1.869 1.250 0.8498 0.5800 0.3868 0.2379 0.1135 0 y 0 0.1596 2.299 2.366 2.213 2.026 1.862 1.736 1.648 1.597 1.580 x 9.239, 5.721 2.654 1.384 0.8109 o 0.5117 0.3343 2.171 0.1314 0.06218 0 y 0 4.528 4.266 3.433 2.813 2.391 2.103 1.910 1.785 1.715 1.692 x m 0 0 0 0 0 0 0 0 0 0 y 1 11.89 5.990 4.045 3.093 2.542 2.197 1.974 1.834 1.757 1.732

U K 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~1.0. 0.70. U~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ i I I ~i~uy I 1I;r Ii i l~ r iI III i Ir irT I A I — ~~~~~~~~~~~~ -ifY0 A I.. 0.5 0. 5.... K~~~~~~~~~ The cn-function,z~x+iycnuv) IS n(uIV) k=- si 3Q0=Q5VQN 0.9i_ 0.3000 _~~~~~~~~~~~~~~~~N NII AN Xi 0 0.2 N I r 0.2 0~~~~~.1 0. A X2 0.3 A. I. 0.6 0.7 0.8. 8. I —.... K;- - - ~~~Thcn-fntoz=~y nui)kI n(~v'Ne/~~~ =- s i n 30=0.00

THE cn-FUNCTION, x + iy = cn(u + iv) k = sin 35~ = 0.573576; K = 1.731245, K' = 2.034715 U 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 V K' x 1.000 0.9850 0.9414 0.8717 0.7801 0.6710 0.5491 0.4183 0.2816 0.1416 0 y 0 0 0 0 0 0 0 0 0 0 0 x 1.021 1.005 0.9595 0.8869 0.7920 0.6799 0.5552 0.4222 0.2839 0.1426 0 y 0 0.03537 0.06832 0.09692 0.1201 0.1377 0.1502 0.1585 0.1656 0.1662 0.1671 x 1.086 1.067 1.015 0.9331 0.8278 0.7057 0.5726 0.4331 0.2900 0.1453 0 0.2 y 0 0.07426 o.1429 0.2016 0.2481 0.2825 0.5062 0.3214 O. 3304 0.3349 0.3363 x 1.201 1.178 1.112 1.012 0.8864 0.7462 0.5987 0.4485 0.2982 0.1488 0 0.3 y 0 0.1213 0.2318 0.3235 0.3934 0.4424 0.4740 0.4928 0.5030 0.5078 0.5091 x 1.382 1.349 1.258 1.124 0.9649 0.7963 0.6276 0.4635 0.3051 0.1513 0 y 0 0.1844 0.3479 0.4769 0.5683 0.6264 0.6595 0.6759 0.6828 0.6849 0.6854 x 1.656 1.604 1.464 1.269 1.055 0.8450 0.6494 0.4703 0.3052 0.1501 0'5 0 0.2787 0.5145 0.6846 0.7899 0.8450 0.8674 0.8717 0.8684 0.8640 0.8622 x 2.084 1.988 1.744 1.437 1.136 0.8715 0.6472 0.4570 0.2916 0.1420 0 y 0 0.4391 0.7791 0.9855 1.081 1.108 1.099 1.077 1.055 1.039 1.034 x 2.805 2.590 2.101 1.585 1.161 0.8395 0.5974 0.4096 0.2564 0.1235 0 0.7 y 0 0.7591 1.246 1.442 1.466 1.416 1.346 1.281 1.231 1.200 1.189 x 4.247 3.588 2.441 1.578 1.037 0.6984 0.4745 0.3159 0.1942 0.09261 0 y 0 1.557 2.143 2.126 1.939 1.744 1.583 1.462 1.379 1.331 1.316 x 8.549 4.950 2.177 1.113 0.6484.0.4087 0.2673 0.1739 0.1054 0.04993 0 0.9 y 0 4.246 3.777 2.965 2.397 2.018 1.763 1.591 1.481 1.419 1.399 x o0 0 0 0. 0 0 0 0 y0 5.072.414 21.517 450428 O 10.09 5.072 3.414 2.601 2.128 1.831 1.638 1.517 1.45o 1.428

It IN O o.oJ~~~~~~~~~~~~~~~~~~~~~~~~~~J 0 " + 1 C + II O 0 0 0o 0o 0 0 - 0) 0 0iIIIi 1 ll ll ~ 1il i] iii Iii IIll I II X A A Y1 1 I., l I - I 11 I I -- i I iI III I I Ir I I Ii UI i I I Il, ~ III i ll rT 1 1 I II i i I 71 I r I II L1-r m rrr i \1"I, N IN A I 1 I OI /I I I I lilt I L~l 14 I x., N _~~~~~~~~~~~~~~~~~~~O

THE cn-FUNCTION, x + iy = cn(u + iv) k = sin 40~ = 0.642788; K = 1.786769; K' = 1.935581 u 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 v K' x 1.000 0.9841 0.9579 0.8648 0.7699 o.6586 0.5561 o.4064 0.2726 0.1567 o y 0 0 0 0 0 0 0 0 0 0 0 y o o o o o o o o o o o x 1.019 1.002 0.9558 0.8776 0.7794 0.6651 0.5401 0.4087 0.2758 0.1572 0 0.1 y o o.03466 0.06644 0.09517 0.1138 0.1285 0.1380 0.1457 0.1467 0.1480 0.1484 x 1.078 1.058 1.003 0.9165 0.8077 0.6859 0.5515 0.4149 0.2767 0.1585 0 0 0.07287 0.1590 0.1957 0.2549 0.2650 0.2806 0.2904 0.2952 0.2972 0.2977 x 1.182 1.158 t.o87 0.9818 0.8532 0.7124 0.5674 0.4225 0.2798 0.1392 0 y 0 0.1194 6.2258 0.3107 0.3714 0.4103 0.4322 0.4427 o.4468 0.4478 0.4479 x 1.347 1.512 11214 1.074 0.9121 0.7452 0.5824 0.4274 0.2800 0.1585 0 y 0 0.1820 0335953 0.4575 0.5343 0.5774 0.5969 0.6025 9.6016 0.5993 0.5982 x 1.599 1.542 1. 92 1.190 0.9754 0.7719 0.5878 0.4228 0.27553 0.1341 0 0'5 y 0.2761 0.5018 o.6538 0.7374 0.7719 0.7775 0.7694 0.7578 0.7488 0.7455 x 1.992 1.888 1.668 1.316 1.023 0.7743 0.5698 0.4000 0.2543 0.1236 0 y 0 0.4562 0.7578 0.9334 0.9980 0.9993 0.9728 0.9394 0.9103 0.8912 0.8846 x 2.661 2.427 1.916 1.409 1.013 0.7254 0.5112 0.3492 0.2182 0.1050 0 0~* 7 y 0. 7544 1.200 1.544 1.529 1.256 1.174 1.103 1.051 1.019 1.008 x 4.004 5.296 2.149 1.349 0.8734 0.5839 0.3956 0.2651 0.1618 0.07718 0 y 0 1.534 2.015 1.928 1.717 1.518 1.361 1.245 1.166 1.121 1.106 x 8.030 4.551 1.812 0.9122 0.5292 0.5557 0.2186 0.1425 0.08661 0.04107 0 o0'9 y 0 4.014 5.582 2.595 2.071 1.727 1.497 1.343 1.243 1.188 1.170 x o0 0 0 0 0 0 0 0 0 0 1.0 y c 8.716 4.571 2.952 2.225 1.810 1.548 1.578 1.271 1.211 1.192

U K 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 08 09.0 1.0 I.0 0.9 a X O.9 _0. 0.7.7 The en-function, z=x+iy= cn(u+iv)=\/ - sn-u+iv) = sin 40~ =0. 70. 0.;

THE cn-FUNCTION, x + iy = cn(u + iv) k = sin 45~ = 0.707107; K = K' = 1.854075 0 0.1 0.2 o0. 0.4 0.5 0.6 0.7 0.8 0.9 1.0 v K' x 1.000 0.9830 0.9335 0.8563 0.7574 0.6436 0.5205 0.3924 0.2621 0.1311 0 y O O O 0 0 O O O O O O x 1.017 0.9994 0.9476 0.8671 0.7649 0.6481 0.5228 0.3934 0.2624 0.1311 0. y O 0.03436 0.06528 0.09029 0.1084 0.1202 0.1268 0.1299 0.1309 0.1311 0.1511 x 1.071 1.050 0.9906 0.8995 0.7866 0.6609 0.5291 0.3956 0.2627 0.1309 0 0.2 y 0 0.07236 0.1367 0.1877 0.2235 0.2455 0.2571 0.2617 0.2627 0.2624 0.2621 x 1.168 1.141 1.065 0.9537 0.8209 0.6792 0.5366 0.3971 0.2617 0.1299 0 0.3 y 0 0.1188 0.2223 0.5008 0.3525 0.3815 0.3941 0.3971 0.3956 0.3934 0.3924 x 1.320 1.282 1.176 1.029 0.8633 0.6978 0.5407 0.3941 0.2571 0.1268 0 y 0 0.1817 0.3342 0.4417 0.5047 0.5336 0.5407 0.5366 0.5291 0.5228 0.5205 x 1.554 1.492 1.330 1.119 0.9043 0.7071 0.5336 0.3815 0.2455 0.1202 0 y 0 0.2766 0.4940 0.6286 0.6915 0.7071 0.6978 0.6792 0.6609 0.6481 0.6436 x 1.921 1.806 1.529 1.210 0.9248 0.6915 0.5047 0.3525 0.2235 0.1084 0' y 0 0.4381 0.7429 0.8890 0.9248 0.9043 0.8633 0.8209 0.7866 0.7649 0.7574 x 2.549 2.292 1.758 1.259 0.8890 0.6286 0.4417 0.3008 0.1877 0.09029 0 y 0 0.7569 1.163 1.259 1.210 1.119 1.029 0.9537 0.8995 0.8671 0.8563 x 3.816 3.052 1.903 1.163 0.7429 O.4940 0.3342 0.2223 0.1367 0.06528 0 y 0 1.523 1.903 1.758 1.529 1.330 1.176 1.065 0.9906 0.9476 0.9335 x 7.628 3.814 1.523 0.7569 0.4381,0.2766 0.1817 0.1188 0.07236 0.03436 0. y 0 3.814 3.052 2.292 1.806 1.492 1.282 1.141 1.050 0.9994 0.9830 x a 0 0 0 0 0 0 0 0 0 0 7.628.816 549 921.20168 1.071 1.017000 Y m 7.628 3.816 2.549 1.921 1.554 1.320 1.168 1.071 1.017 1.000

U K 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 l.o 1.0 0. i.0 I.O 0.770.7 0.6 o.A I l i t 1' I:,1. I 1 - I FI TT I0.I 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 K The cn-function, z=x+iy= c n (u+iv) =N/- sn (u+iv) k = sin 45~ = 0 7071

THE cn-FUNCTION, x + iy = cn(u + iv) k = sin 50~ = 0.766044; K = 1.955581, K' = 1.786769 0 0.1 0.2 0.5 0.4 0.5 0.6 0.7 0.8 0.9 1.0 K K' x 1.000 0.9815 0.9281 0.8457 0.7423 0.6255 0.5019 0.3759 0.2498 0.1245 0 y O 0 0 0 0 0 0 0 0 0 0 x 1.016 0.9966 0.9405 0.8547 0.7478 0.6283 0.5028 0.5758 0.2493 0.1242 0 0.1 y 0 0.03446 0.06476 O.08811 0.1057 0.1125 0.1162 0.1168. 1161 0.1151 0.1147 x 1.066 1.043 0 9786. 8816 0.7659 0. 659. 5048. 5749 0.2477 0.1251 0 0.2 y 0 0.07268 0.1357 0.1851 0.2134 0.2293 0.2350 0.2348 0.2522 0.2297 0.2288 x 1.156 1.127 1.044 0.9258 0.7885 0.6456 0.5056 0.5715 0.24536 0.120) 0 0.3 y 0 o.1196 0.2208 0.2930 0.3356 o.5548 0.5586 0.3545 0.3481 0.3429 0.5410 x 1.299 1.256 1.142 0.9854 0.8163 0.6524 0.5009 0.5626 0.2554 0.1158 0 y 0 0.1855 0.5320 0.4290 0.4781 0.4952 0.4887 0.4761 0.4627 0.4532 0.4498 x 1.518 1.449 1.274 1.054 0.8386 0.6477 0.4845 0.5442 0.2207 0.1078 0 0.5 y 0 0.2800 0.4899 0.6070 0.6497 0.6477 0.6253 0.5978 0.5759 0.5581 0.5526 x 1.865 1.758 1.440 1.115 0.8374 0.6188 0.4483 0.3117 0.1971 0.09551 0 y 0 0.4441 0.7329 0.8496 0.8582 0.8185 076553 0.7159 0.6777 0.6540 0.6460 x 2.461 2.178 1.618 1.128 0.7852 0.5486 0.3857 0.2607 0.1625 0.07818 0 y 0 0.7654 1.152 1.182 1.104 0.9983 0.9011 0.8239 0.7690 0.7364 0.7256 8 x 3.668 2.858 1.690 1.007 0.6358 0.4211 0.2847 0.1895 0.1167 0.05575 0 y 0 1.520 1.805 1.608 1.566 1.168 1.019 0.9125 o.8413 0.8005 0.7870 x'7.515 3.368 1.287 0.6330 0.3660 0.2317 0.1527 0.1002 0.06114 0.02908 0 y 0 3.634 2.766 2.056 1.584 1.294 1.101 0.9714 o.8880 0.8410 0.8258 x o0 0 0 0 0 0 0 0 0 0 1. y 6.757 5.559 2.232 1.672 1.341 1.130 0.9922 0.9041 0.8550 0.8591

u K 00.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.0 1.0 zm-~~~~-C -77~~~~~~~~~~~~~~~~~~~~~~~i 0. 9 tth*)t tttt'ttftM+-?P 0.9 0.8 0.8~~~$~cPM ~ - -al ~ 0.7 0.7 0.6~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 0.9 ~~~~~~~~~~~~~~~~~~~~~~ v 0.I... 0.5 0. 5.... K I,~ F V tr V~~~~~~~~~~~~~1 e 0'.4gttt~t;'-~ftt~ 0.4 0.k 0.7 N N I/ I; i I 0.7 -0.7 ci _i i~~~1 I~-lo i i~~~~~~1~ 1 ~, i -YJ'Ir 1 C.o o~~~~~~~~~7 o ~~~~~~~~~~~~~~~~~~~~~~~~~0. 0.1 0,2 0.3 0,4 0.5 0.6 0.7 0.8 0.9 1.0 K The cn-funcfion, z~x+iy= cn(u+iv):~ I - sn (u+iv)I k = sin 50~ - 0 7660

THE cn-FUNQTION, x + iy a cn(u + iv) k sin 55~ 0 0.819152; K = 2.034715, K' = 1.731245 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 v K' x 1.000 0.9796 0.9212 0.8326 0.7237 0.6037 0.4799 0.3565 0.2355 0.1170 0 y o o 0o 0 o o 0 0 o o x 1.015 0.9936 0.9523 0,8400 0.7276 0.6051 0.4796 0.3556 0.2345 0.1164 0 0.1 y 0 0.03497 0.06486 0.08652 0.09945 0.1051 0.1059 0.1042 0.1018 0.09986 0.09915 x.62 1.037 0.9659 0.8617 0.7384 0.6080 0.4781 b.3522 0.2313 0.1145 0 0.2 y 0 0.07385 0.1359 0.1796 0.2042 0.2137 0.2136 0.2088 0.2031 0.1988 0.1972 x 1.147 1.114 1.024 0.8968 o.754o 0.6104 0.4733 0,3451 0.2251 0.1110 0 0'. y 0.1217 9.2212 0.2868 0.3200 0.3293 0.3246 0.3140 0.3032 0.2956 0.2929 x 1.282 1.234 1.108 0.9425 0.7695 0.6073 0.4618 0.3319 0.2144 0.1052 0 0.4 y 0 0.1872 d.3323 0.4183 0.4532 0.4547 0.4395 0.4192 0.4010 o.3888 0.3845 x 1.490 1.413 1.221 0.9916 0.7758 0.5917 0.4386 0.3098 0.1978 0.09642 0 0'5 y 0 0.2862 0.4890 0.5878 0.6103 0.5917 0.5576 0.5225 0.4941 0.4760 0.4699 x 1.821 1.678 1.358 1.027 0.7573 0.5531 0.3979 0.2755 0.1738 0.08410 0.6 y o 0.4540 0.7262 0.8129 0.7957 0.7588 0.6756 0.6206 0.5796 0.5546 0.5462 x 2.391 2.076 1.489 1.010 0.6901 o.4794 0.3339 0.2265 0.1411 0.06786 0 0.7 y 0 0.7795 1.105 1.110 1.006 0.8886 0.7869 0.7083 o.6534 0.6210 0.6104 x 3.551 2.645 1.501 0.8722 0.5455 0.3603 0.2436 0.1623 0.1000 0.04784 0 8 y 0 1.524 1.709 1.471 1.221 1.025 o.8809 0.7788 0.7108 0.6719. 6592 x 7.064 2.973 1.090 0.5315 0.3074 0.1952 0.1291 0.08496 0.05200 0.02477 0 9 0 3.466 2.513 1.814 1.392 1.123 0.9448 0.8249 0.7475 0.7039 0.6898 x. 0 0 0 0 0 0 0 0 0 0 10 y 5.986 2.974 1,964 1.459 1.160 0.9676 0.8410 0.7601 0.7148 0.7002

u K 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.,8 0.9 1.0 0,9 0. il \ i \ J^ 0.6 0,1 0.2 0 k- sin 55~-0.8192:L [/ N I.1\ 0.k 0.2 x P I J~~~~~~~~~~S u~v k = sin 55" L: C>.8192~L"I I

THE cn-FUNCTION, x + iy = cn(u + iv) k = sin 60~ = 0.866025; K = 2.156516, K' = 1.685750 0 0.1 0.2 0.5 0.4 0.5 0.6 0.7 0.8 0.9 1.0 v K' x 1.000 0.9771 0.9124 0.8161 0.7007 0.5774 0.4557 0.3338 0.2189 0.1082 o y 0 0 0 0 0 0 0 0 0 0 0 y O O O O O O O O o o o x 1.014 0.9901 0.9221 0.8219 0.7030 0.5775 0.4524 0.55322 0.2175 0.1075 0 0.1 y 0 0.05590 0.06552 0.08535 0.09535 0.09779 0.09572 0.09177 0.08785 0.08507 o.o8409 x 1.059 1.051 0.9518 0.8388 0.7092 0.5764 0.4480 0.5270 0.2155 0.1052 0 2 y 0 0.07588 0.1575 0.1769 0.1953 0.1985 0.1925 0.1855 0.1749 0.1690 0.1669 x 1.140 1.103 1.002 O. 854 0.7168 0.5724 0.4389 0.5172 0.2055 0.1010 0 0.3 y 0 0.1255 0.2235 0.2818 0.5048 o.3040 0.2912 0.2748 0.2603 0.2506 0.2472 x 1.268 1.214 1.075 0.8982 0.7212 0.5614 0.4225 0.5011 0.1935. 09460 0 y 0 0.1930 0.3349 o.4089 0.4287 0.4169 0.3917 0.3648 0.3425 0.3282 0.5233 x 1.468 1.580 1.170 0.9292 0.7159 0.5373 0.3946 0.2769 0.1761 0.08561 0 0.5 y 0.2954 0.4906 o.5697 0.5715 0. 575 0.4929 0.4516 0.4198 o.4000 0.3955 x 1.786 1.624 1.278 0.9415 0.6817 0.4922 0.5516 0.2424 0.1525 0.07570 0 y 0 0.4681 0.7219 0.7771 0.7349 0.6629 0.5915 0.5324 0.4895 0.4657 0.4552 x 2.355 1.982 1.566 0.9009 0.6060 0.4178 0.2900 0.1964 0.1225 0.05877 0 0'7 y 0 0.7988 1.079 1.040 0.9155 0.7867 0.6821 0.6052 0.5488 0.5170 0.5065 x 3.458 2.465 1.327 0.7556 0.4675 0.5081 0.2084 0.1590 0.08576 0.04105 0 y 0 1.552 1.619 1.543 1.088 0.8956 0.7567 0.6590 0.5942 0.5572 0.5451 x 6.866 2.614 0.9216 0.4462 0.2585,0.1647 0.1094 0.07217 0.04426 0.02111 0 ~09 y 0 3.305 2.282 1.615 1.222 0.9722 0.8064 0.6950 0.6230 0.5823 0.5692 x o0 0 0 0 0 0 0 0 0 0 10 y 5.334 2.658 1.750 1.275 1.000 0.8259 0.7074 0.6528 0.5909 0.5774

U K 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 F_ *00K'~~~ 0 00 0 0 t:; 0. 1 0.2 0.I.9 The cn - function, =X+iy= cn(u+iv)=I I k= sin 60~=0.8660 - j V~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Cr~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ U~~~~~~~~~~~~_ T I...~ ~ ~ ~ ~~QIi. /N t-T — toK 0.2~ ~ ~~~h nfucin ~y=c (~v=/-s2u v k = sin 60~= 0.8660I dI II

THE cn-FUNCTION, x + iy = cn(u + iv) k - sin 65~ - 0.906308; K 2.308787, K'. 1.648995 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 v K' x 1.000 0.9738 0.9009 0.7950 0.6720 0,5450 0.4222 0.3069 0.1995 0.09813 0 ~ 0y0 0 0 0 0 0 0 0 0 0 x 1.014 0.9861 0.9094 0.7992 0.6729 0.5438 0.4201 0.3049 0.1979 0.09729.1 y o0 o.03734 0.06681 0.08450 0.09117, 0.09021 0.08531 0.07934 0.07412 0.07068 0.6949 x 1.057 1.024 0.9351 0.8114 0.6747 0,5396 0.4136 0.2985 0.1931 0.09471 0 0.2 y o 0.07900 0.1400 0.1748 0.1863 0.1824 0.1711 0.1582 0.1473 0.1402 0.1377 x 1.154 1.091 0.9784 0.8298 0.6751 0.5305 0.4013 0.2870 o.1846 0.09027 0 0.3 y 0 0.1306 0.2272 0.2773 0.2891 0.2781 0.2575 0.2360 0.2185 0.2073 0.2035 x 1.258 1.195 1.040 0.8503 0.6697 0.5131 0.3812 0.2694 0.1720 0.08376 0 0.4 y 0 0.2013 0.3396 0.3997 0.4035 0.5784 0.3442 0.3117 02864.0.2706 02652 x 1.450 1.349 1.116 0.8646 0.6510 0.4829 0.3509 0.2445 0.1547 0.07500 o 0.5 y 0 0.3081 0.4943 0.5511 0.5318 0.4829 0.4296 0.3835 0.3493 0.3285 0.3215 x 1.758 1.573 1.197 0.8564 o.6084 0.4341 0.3078 0.2111 0.1324 0.06387 0 y 0 0.4869 0.7187 0.7401 0.6739 0.5886 0.5109 0.4491 0.4053 0.3793 0.3707 x 2.292 1.891 1.245 0.7972 0.5282 0.3615 0.2500 0.1689 0.1050 0.05043 0 0'7 0'0.8235 1.052 0.9692 0.8231 0.6895 0.5837 0.5055 0.4522 0.4213 0.4112 x 3.385 2.284 1.165 0.6466 0.3983 0.2621 0.1773 0.1183 0.07301 0.03493 0 0.8 y 0 1.543 1.527 1.220 0.9627 o.7756 0.6422 0.5492 0.4878 0.4527 0.4413 x 6.711 2.279 0.7738 0.3728 0.2164 0.1384 0.09222 0.06101 0.03747 0.01788 0 0.9 y 0 3.137 2.066 1.433 1.066 0.8545 0.6805 0.5770 0.5100 0.4722 0.4600 x 0 0 0 0 0 0 0 0 0 0 1.0 o o o o o o o 00.4788 o.466 1 r0 r.7S2 2a.337 1.519 1,105 0,8555 0.6939 0.5866 0.5176 O.4788 o,4665

u K 10,,., QI ____02_____f____i _____,_____L____~____.?.0 0O.I 8. 0 I 1 0 0 0 I 0, 8 K' ~~~~~~0.7 O.II R 1184 Tnn~2~-,-Kl NI1 I I!rn ~ 1 1 S Ii ll r i i t 1 / Pl i >i i 1 l4 I \ I I II SlI l l o o K The cn-function, z=x+iy= cn (u+iv) = k = sin 65~ =0.9063

THE cn-FUNCTION, x + iy = cn(u + iv) k - sin 70~ - 0.939693; K - 2.504550, K' = 1.620026 o 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 v Kx 1.000 0.9694 0.8854 0.7671 0.6352 0.5048 0.3840 0.2750 0.1767 0.08635 0 0 0 o 0 0 o o o o o o o o x 1.013 0.9808 0.8926 0.7699 0.6547 0.5027 0.3814 0.2727 0.1751 0.08549 0. y 0 0.03944 0.06876 0.08377 0,08659 0.08199 0.07437 0.06666 0.06047 o0.5655 0.05522 x 1.055 1.016 0.9142 0.7774 0.6325 0.4958 0.3754 0.2656 0.1701 0.08290 0 0.2 y 0 0.08547 0.1438 0.1728 0.1763 0.1652 0.1487 0.1326 0.1200 0.1120 0.1093 x 1.130 1.079 0.9501 0.7876 0.6265 0.4826 0.3591 0.2535 0.1615 0.07853 0 0.3 y 0 0.1381 0.2328 0.2726 0.2719 0.2504 0.2227 0.1971 0.1774 0.1653 0.1612 x 1.250 1.176 0.9993 0.7962 0.6126 0.4606 0.3372 0.2356 0.1492 0.07231 0 0.4 y 0 0.2127 0.3462 0.3896 0.3758 o.3379 0.2957 0.2590 0.2317 0.2151 0.2096 x 1.437 1.318 1.057 0.7948 0.5849 0.4266 0.3063 0.2114 0.1329 0.06419 0:05 y 0 0.3250 0.4994 0.5303 o.4892 0.4266 0.3662 0.3169 0.2814 0.2603 0.2533 x 1.737 1.520 1.110 0.7690 0.5350 o0.3768 0.2648 0.1805 0.1127 0.05419 0 0.6 6 0 0.5110 0.7150 0.6995 0.6101 0.5137 0.4317 0.3688 0.3252 0.2997 0.2913 x 2.258 1.797 1.122 0.6958 0.4541 0.3083 0.2122 0.1429 0.08855 0.04245 0 0.7 y 0 0.8538 1.021 0.8948 0.7320 0.5942 0.4889 0.4127 0.3614 0.3318 0.3222 x 3.329 2.099 1.008. 0.5476 0.3351 0.2201 o.1488 0.09917 0.06113 0.02922 0 o.8 y o 1.554 1.429 1.097 0.8414 0.6608 0.5339 0.4462 o.3885 0.3557 0.5451 x 6,591 1.959 0.6414 0.3079 0.1793 0.1150 0.07680 0.05085 0.03123 0.01490 o 0.9 y 0 2.962 1.857 1.260 0.9194 0.7053 0.5629 0.4672 0.4054 0.3705 0.3592 x 0 0 0 0 0 0 0 0 0 1.0 y, 4.215 2.059 1.324 0.9480 0.7210 0.5730 0.4744 0.4111 o.3755 0.3640

u K 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.0 1.0 0.9 0.9 0.8 0.8 0.7 0.7 0.6I- ~ ~i 0.6:- II ]]l\\~l\Si~~~cg~lx~l/iN,~X~[ l'",/. i [/1 I If F71-FT-li C[11] ~I 0.6 I I4 0 N0 00- IX gSC;0 iN k 1.m u~ IX SS t.W X~tW -kX 1 -4 I I v V ~~~~~~~~~~~~~~~~~~~~~~~~~~~0.5' K' 0.5gH 3~~~~ W\At I > -0-0-;-5 F + 1dfA10S /LFt 1 1 W 1 1111 0.4 0.4 0.3 - \1 - - 0.3 0.2 0.2 0.1 0.1 02; 0 /gl >;tX 400j~XX02'" X Y w/reil:Afoyll~~~~~~'N Y 1V'N/NII I 1 N1 D1\- V A W__ 1 11111 o _ 0 O.I 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 u K The cn-function, z =x+iy= cn(uiv)=/ - sn2 (u+iv) k= sin 70~ =0.9397

THE cn-FUNCTION, x + iy = cn(u + iv) k sin 75~ 0.965926; K = 2.768063, K' = 1.598142 - 0 0.1 0.2 0.- 0.4 0.5 o.6 0.7 0.8 0.9 1.0 v K' x 1.000 0.9628 0.8633 0.7289 0.5866 0.4534 0.3365 0.2362 0.1496 0.07244 0 0 y.o 0 o o o o o o o o o 1.013 0.9735 0.8691 0.7300 0.5848 o.4506 0.3337 0.2339 0.1480 0.07164 0 y -0 0.04246 0.07143 0.08281 0.08099 0.07255 0.06247 0.05345 0.04669 0.04258 0.04121 x 1.053 1.007 0.8862 0.7327 0.5791 0.4418 0.3250 0.2269 0.1433 0.06927 0.s2 y 0..08984 0.1491 0.1701 0.1641 o. 1456 0.1245 0.1061 0.09249 0.08425 0.08151 x 1.126 1.065 0.9140 0.7347 0.5674 0.4259 0.3101 0.2151 0.1354 0.06530 0 0 53 y 0. 1485 0.2402 0.2664 0.2512 0.2192. 1856 0.1572 0.1365 0.1241 0.1200 x 1.244 1.154 0.9501 0.7316 0.5466 0.4013 0.2882 0.1983 0.1242 0.05977 0 0.4 y 0 0.2284 0.3543 0.3765 0.5434 0.2931 0.2448 0.2056 0.1777 0.1612 0.1557 x 1.427 1.283 0.9884 0.7158 0.5123 0.3660 0.2586 0.1763 0.1098 0.05270 0 0.5 y 0 o.3476 0.5045 O.5043 0.4406 0.3660 o.3007 0.2502 0.2151 0.1946 0.1878 x 1.721 1.462 1.013 0.6752 0.4587 0.3180 0.2208 0.1490 0.09232 0.04420 0 0.6 y 0 0.5420 0.7081 0.6513 0.5402 0.4354 0.3515 0.2896 0.2476 0.2254 0.2155 X 2.233 1.691 0.9898 0.5932 0.3807 0.2560 0.1748 0.1169 0.07205 0.03441 0 y 0 0.8900 0.9816 0.8121 0.6362 0.4974 0.3949 0.3223 0.2742 0.2468 0.2379.8 x 3.287 1.898 0.8520 0.4531 0.2754 0.1803 0.1214 0.08056 0.04946 0.02357 0 y 0 1.560 1.320 o.9688 0.7189 o.5471 0.4283 o.3470 0.2940 0.2641 0.2544 x 6.502 1.643 0.5192 0.2488 0.1452 0.09332 0.06225 0.04111 0.02517 0.01198 0 y 0 2.764 1.646 1.089 0.7758 0.5796 0.4495 0.3624 0.3063 0.2747 0.2645 x = 0 0 0o 0 0 0 0 0 0 y. 3.699 1.791 1.134 0.7962 0.5909 0.4568 0.3676 0.3104 0.2783 0.2679

u K 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 V I 06II\ II\\1\\Nn~11 l n i 1, I/ I - III I 1 r 1 rI I II 1 I I I IL 0l N. OK' ~"t' I IaTmt-te~,'~~~s i1 0. O.9 The n-function,z=x+iy= cnuiv)I - IV k= sin 75~ = 0.9659 0 0.5 O 4 0.5 0 k = sin 75~ =0.9659

THE cn-FUNCTION, x + iy = cn(u + iv) k = sin 80~ = 0.984808; K = 3.153385, K' = 1.582843 u 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 K' x 1.000 0.9522 0.8289 o.6725 0.5183 0.3847 0.2755 0.1878 0.1164 0.05561 o y 0 0 0 0 0 0 0 0 0 0 0 x 1.013 0.9620 0.8329 0.6717 0.5155 0.3815 0.2727 0.1858 0.1151 o.o05496 0.1 y 0 o.04695 0.07485 o.08077 0.07317 o. 6086 0.04891 0.03932 0.03259 0.02867 0.02738 x 1.052 0.9923 0.8447 0.6693 0.5068 0.3718 0.2644 0.1796 o.1111 0.05302 0 0.2 y 0 0.9924 o.1556 0.1649 0.1474 0.1216 0.09718 0.07791 0.06448 0.05667 0.05411 x 1.124 1.045 0.8627 0.6629 o.4908 0.3550 0. 250 o0.1693 o.104 0. 04982 o 0o.3 y 0 o.1637 0.2487 0.2557 0.2235 0.1817 O.1441 0.1150 o.09495 o.o8335 0.07956 x 1.239 1.126 0.8835 o.6485 0.4654. 3304 0.2306 0.1550 0.09537 0.04540 o y 0 0.2507 0.3624 0.3559 0.3015 0.2406 0.1888 0.1498 0.1233 0.1081 0.1031 x 1.420 1.238 0.8995 0.6199 0.4277 0.2969 0.2047 0.1367 o.08383 0.03984 o 0'5 y 0 0.3787 0.5066 0.4672 0.3805 0.2969 0.2301 0.1814 o.1488 0.1302 0.1242 x 1.710 1.387 0.8948 0.5684 0.3748 0.2540 0.1729 0.1147 0.07008 0.03325 o y o 0.5825 o.6921 o.5882 0.4578 0.3487 0.2669 0.2090 0.1708 0.1493 0.1423 x 2.216 1.559 0.8398 0.4839 0.3043 0.2014 0.1 55 o.08932 0.05440 0.02578 o 0.7 y 0 0.9316 0.9242 0.7126 0.5289 0.3935 0.2976 0.2316 0.1887 0.1646 0.1569 x 3.259 1.659 0.5891 0.3584 0.2159 0.1401 0.09350 0.06120 0.03718 0.01759 0 y 0 1.552 1.187 0.8263 0.5875 0.4284 0.3208 0.2484 0.2019 0.1759 0.1676 x 6.441 1.315 0.4017 0.1923 0.1123,0.07191 0.04759 0.03111 0.01887 0.008923 0 0.9 y 0 2.522 1.418 0.9085 0.6264 0.4507 0.3353 0.2588 0.2100 0.1828 0.1741 x 0 0 0 0 0 0 0 0 0 0 1.0 y 0. 3.171 1.515 0.9389 0.6401 o.4584 0.3402 0.2623 0.2127 0.1852 0.1763 Y ~ 3.171 1.515 0 89

U K 0 O, I 0,2 0,3 0.4 0.5 71 1.0.................I 16 9 1 0 0.9 o.9 I_, 0.8 0.8 0.7.0.7 0~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~. 0.6 I\ iM A N I|[1X X X 7Z n \ 1,/H\III #IW /|/ Xi N, I 1k 1 NJ I 111\II V o {1alil@|\@~~I l l1XI1 XI N I f Xns N IA 11 f6I[s111 L W E$Zsta \ 5 V v 0.5~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 0.4 t01.gtO0 Ill' 0.3.0,3 v~~w I Ill 15 _ i V IS) N \I 1 1\1S \ I I N - k X:: I 1 -!/V1 I 111 1 f1. ILL 1 111. 0 111\1\ \, AIIIN4A \1\1 N IM' I I11 lr\f( I 1 4 > 1 - ~ /!> ^ VI /il Nl M 4~~~~ ~ ~ ~~ VA X XIt ~, \r II P 0.2 N X 0.2 I 0. ], I1 I I0.1 I I I 1 I O O~~~~~~~~~~~~~~~~~~~~ A~~~~~~~~~~~~~~~~~~. I 1 \1 zll /',% F\ti I —T ~:11l il + 111 1 1111111I|I11.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 U K The cn-function, z=x+iy= cn(u+ iv)=- sn2(u+iv) k = sin 80~ 0.9848

THE cn-FUNCTIQN, x + iy = cn(u + iv) k = sin 85~ = 0.996195; K = 3.831742, K' = 1.573792 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 K' V K' x 1.000 0.9308 0.7640 0.5746 0.4099 0.2831 0.1904 0.1232 0.07340 0.03421 0 y 0 0 0 0 0 0 0 0 0 0 0 x 1.013 0.9393 0.7656 0.5722 0.4066 0.2802 0.1882 0.1217 0.07252 0.03379 0 0.1 1466 0.01336 1 y O 0.05451 0.07859 0.07488 o.05944 0.04447 0.03215 O.02342 0.01779 0.01466 0.01336 x 1.052 0.9653 0.7698 0.5645 0.3964 0.2715 O.1818 0.1174 o.o69 8 0.03255 0 0.2 2 y 0 0.1149 0.1621 0.1515 0.1199 0.08835 0.06367 0.04632 0.03515 0.02896 0.02698 x 1.123 1.010 0.7745 0.5499 0.3788 0.2568 0.1711 0.1102 0.06555 0.03052 0 0.3 y 0 0.1885 0.2556 0.2313 0.1795 0.1310 0.09394 O.06818 0.05168 0.04257 0.03966 x 1.237 1.076 0.7753 0.5256 0.3527 0.2360 0.1563 0.1004 0.05961 0.02774 0 ~.*4 y 0 0.2860 0.3646 0.3151 0.2382 0.1716 0.1223 0.08849 0.06698 0.05514 0.05136 x 1.416 1.163 0.7645 0.4879 0.3173 0.2092 0.1375 0.08802 0.05219 0.02427 0 0.5 y 0 0.4253 0.4946 0.4023 0.2948 0.2092 0.1480 0.1067 0.08067 0.06636 0.06180 x 1.703 1.266 0.7289 0.4323 0.2717 0.1763 0.1150 0.07341 0,04346 0.02020 0 o.6 y 0 0.6371 o.6486 0.4904 0.3472 0.2425 0.1703 0.1224 0.09240 0.07596 0.07073 x 2.206 1.357 o.6490 0.3551 0.2157 0.1379 0.08939 0.05687 0.03362 0.01561 0 ~07 y 0 0.9733 0.8230 0.5740 0.3929 0.2703 0.1886 0.1352 0.1019 0.08369 0.07792 x 3.242 1.335 0.5026 0.2545 0.1501 0.09482 o.06114 0.03880 0.02291 o.01064 0 o.8 y 0 1.500 0.9992 0.6448 0.4287 0.2914 0.2022 0.1446 0.1088 0.08937 0.08176 x 6.404 0.9454 0.2791 0.1333 0.07714,0.04832 0.03105 0.01967 0.01161 0.005387 0 ~'9 y 0 2.179 1.138 0.6930 0.4516 0.3045 0.2106 0.1503 0.1131 0.09283 0.08641 x 0 0 0 0 0 0 0 0 0 0 1.0 y 2.558 1.192 0.7102 0.4596 0.3090 0.2134 0.1523 0.1145 0.09399 0.08749

u K 0 O.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.0 Io0 A.9 10.9:;|| =| l=^=::i= = h ^ H 6 O. I 0.2 0. 0.4 0.5 i i X 1 0.6.5 ~ 3~ i 1 I S \ I I N 1 I I I I I I I I TM' /\ I W s ii / 1 I /1 -I I 1X 1 I I 1I 1 1 111 1 \11 1 1 N IIA IIL I 3I II 0, 1 The cn-function, z = x+iy= cn (u+ iv) =/I - sn2 (u + iv) k = sin 85~ =0.9962

THE cn-FUNCTION, x + iy = cn(u + iv) k = sin 89~ = 0.999848; K = 5.434910, K' = 1.570916 0 0.1 0.2 0.3 0.4 6.5 0.6 0.7 0.8 0.9 1.0 _v K' V K' x 1.000 o.8685 0.6056 0.3771 0.2243 0.1310 0.07562. 04283 0.02293 0.009959 0 y o o o 0 0 o o o o o o x 1.012 0.8740 0.6035 0.3737 0.2218 0.1294 0.07470 0.04250 0.02265 0.009836 0 0.1 y 0 0.06862 0.07609 0.05488 0.03433 0.02050 0.01211 0.007227 0.004506 0.003143 0.002730 x 1.051 0.8902 0.5968 0.3636 0.2143 0.1248 0.07196 0.04074 0.02181 o.009471 o y 0 O.1434 0.1544 0.1095 0.06806 o.04054 0.02393. 0.1428 0.008902 0.006209 0.005393 x 1.122 0.9164 0.5837 0.3461 0.2019 0.1171 0.06746 0o03818 0.02043 0.008874 0 3 y 0 0.2314 0.2367 0.1635 0.1006 0.05968 0.03517 0.02098 0.01508 0.009122 0.007923 x 1.236 o.9504 0.5610 0.3208 0.1847 0.1066 0.06131 0.03467 0.01855 0.008057 0 y 0 0.3423 0.3245 0.2161 0.1311 o.07745 0.04558 0.02717 o.01694 0.01181 0.01026 x 1.414 0.9861 0.5244 0.2871 0.1627 0.09342 0.05363 0.03031 0.01622 0.007043 0'5 y 0 o. 4888 0.4174 0.2661 0. 1590 0.09342 0.05488 0.03270 0.02038 0.01421 0. 01234 x 1.701 1.008 o.4684 0.2444 0.1363 0.07787 0.04462 0.02521 0.01348 0.005855 0 6 y 0 0.6880 0.5132 0.3119 0.1834 0.1072 0.06284 0.03742 0.02331 0.01626 0.01412 x 2.203 0.9831 0.3879 0.1930 0.1061 0.06029 0.03449 o 01947 0.01042 0.004522 0 0.7 y 0.9563 0.6059 0.3511 0.2034 0.1183 o.06927 0.04123 0.02568 0.01791 0.01555 x 3.236 o0.8450 0.2801 0.1337. 07262 0.04112 0.02349 0.01326 0.007090 0.003078 0 y 0 1.289 0.6860 0.3816 0.2184 0.1265 0.07398 0.04402 0.02741 0.01911 0.01660 x 6.393 0.5145 0.1475 o.06850 0.036go 0.02084 0.01190 0.006713 0.003589 0.001558 0 0.9 y 0 1.610 0.7414 0.4009 0.2277 0.1316 0.07686 0.04572 0.02847 0.01985 0.01724 x oO 0 0 0 0 0 0 0 0 0 1.7612 4076 08 778 04629.02882 02010 0174 y 00 1.755 0.7612 0.4076 0.2308 0.1333 0.07783 O.04629 0.02882 0.02010 0.01745

U K 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.0 1.0 0,9 IIII I I I La T I 11 I IIl II I -JL I III I I L I I I 1 I - - IIII I|Ni ll1 7HA11E11 1111WtflIIIW1W 0.9 8 N 1 1 41 0.8 0.6 Iv A l l r I I vI I E\ ii I E\ _n E r I Z I _ r _ \1 a _:E rT I IY,1 T I 1 111 \1 I 1X 1 1 11\1 1 uX Ii I 1/\ 1 I R I 111 1 1 i > I' 11 1 1 1A 1i 1 1 1/1 Vl\ S{11II 1 K'0 0.5 0,41 1 II U ++I ll~l 1 I JII I V I 1 A1 1 18 1/ i I P.. 1NAITW 1XI 1 I oi l n \ T T 1 1 AM I 1 1 1 1 FI~~~~~~~~~~~~~~~~~ V 1. 1In 1 Ilium I I IN I "''' / "''"''L''1Y''\'' V'/N' "1;1 I I I'\`T 1 L''' 103tA I 1111 1; 114 TWU 1 1 1 7W 54 F~\1\ I\ 11 1ru!11 1 N I II l. I - I \ I I I 1 1 I IV1 vIM /A S 1~A;. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 U K -I~~~ ~~ k1 _ f\L I1I.1he cn-functionz=x+iy cn(u+iv) = - sn u 0.1 AR 0- 0 / 0.k = sin 89 =0. 19999 U/ I VU The~~~~~~~~~~~~~~~ cI- funin COI = xiy c(uiv)~i sn(z v 0.~~~~~~~~~~ i 9 0.699

APPENDIX III THE dn-FUNCTION

THE dn-FUNCTION, x + iy = dn(u + iv) k = sin 1~ = 0.0174524; K = 1.570916, K' = 5.434910 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 K v K' x 1.000 1.000 1.000 1.000 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9998 y 00 0 0 0 0 0 0 0 0 0 0 x 1.000 1.000 1.000 1.000 1.000 0.9999 0.9999 0.9999 0.9998 0.9998 0.9998.1 y 0 0.043092 0.045882 0.048095 0.049399 0.C41001 0.C49516 0.C48094 0.045881 0.043092 0 x 1.000 1.000 1.000 1.000 1.000 0.9999 0.9998 0.9997 0.9996 0.9996 0.9996 y 0 0.031021 0.031943 0.032674 0.033104 0.033304 0.033142 0.032673 0.031942 0.031021 0 x 1.001 1.001 1.001 1.001 1.000 0.9999 0.9996 0.9993 0.9991 0.9990 0.9989 -0. y 0 0.033066 0.035830 0.038023 0.039313 0.039911 0.039423 0.038014 0.05821 0.033060 0 x 1.003 1.003 1.002 1.002 1.001 0.9999 0.9990 0.9982 0.9975 0.9972 0.9970 0.4 y 0 o.o39118 0,001734 0.002385 0.002767 0.002943 0.002796 0.002376 0.001725 0.0o0o67 0 x 1.009 1.008 1.007 1.005 1.003 0.9999 0.9972 0.9949 0.9929 0.991-7.9913 0.5 y 0 0.002719 0.005165 0.007096 0.008219 0.008726 0.008276 0.007023 0.005093 0.002674 0 x 1.026 1.025 1.621 1.015 1.008 0.9996 0.9917 0.9846 0.9791 0.9756 0.9744 6 y 0 0.008196 0.01553 0.02125 0.02450 0.02587 0.02441 0.02062 0.01489 0.007802 0 x 1.080 1.073 1.063 1.044 1.021 0.9970 0.9739 0.9540 0.9388 0.9293 0.9260 ~0.7 y 0 0.02548 0.04800 0.06490 0.07370 0.07661 0.07117 0.05931 0.04240 0.02206 0 x 1.25 1.242 1.191 1.123 1.047 0.9744 0.9111 0.8607 0.8246 0.8029 0.7957 0.8 y o0 0.08846 o.1613 0.2093 0.2267 0.2246 0.1997 0.1605 0.1117 0.05718 0 x\ 2.018 1.877 1.560 1.236 0.9792 0.7959 0.6703 0.5868 0.5340 0.5049 0.4956 o0.9 y 0 0.4413 0.6979 0.7610 0.7000 0.6054 0.4851 0.3613 0.2389 0.1187 0 x o 0 0 0 0 0 0 0 0 0 0 1.0. y 6.313 3.076 1.963 1.376 0.9999 0.7265 0.5095 0.3249 0.1584 0 ~..

<1< 0 0 -I D (D (1) o o i I w\0 l:2 ~l1=4 W WI o0 o p N __ X II xic o 00 000000000 O + o o CDC IL It IF x J i MV x j I I;~.T

THE dn-FUNCTION, x + iy = dn(u + iv) k = sin 5~ = 0.0871557; K = 1.573792, K' = 3.831742 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 K V K' x 1.000 0.9999 0.9996 0.9992 0.9987 0.9981 0.9975 0.9970 0.9966 0.9963 0.9962 y 0 0 0 0 0 0 0 0 0 0 0 x 1.001 1.000 1.000 0.9996 0.9989 0.9981 0.9973 0.9966 0.9961 0.9957 0.9956 0.1 y 0 0.034960 0.039432 0.001297 0.001524 0.001601 0.001522 0.001294 0.039394 0.094937 0 x 1.003 1.002 1.002 1.001 0.9995 0.9981 0.9967 0.9954 0.9944 0.9937 0.9935 0.-2 y 0 0.001299 0.002469 0.003395 0.003986 0.0 186 0.003975 0.003377 0.002451 0.001288 0 x 1.008 1.007 1.06 1. 004 1.001 0.9981 0.9951 0.9925 0.9904 0.9891 0.9886 y 0 0.002912 0.005532 0.007598 0.008909 0.009340 0.008857 0.007513 0.005447 0.002860 0 4 x 1.019 1.018 1.015 1.010 1.004 0.9979 0.9917 0.9861 0.9817 0.9789 0.9780 y 0 0.006372 0.01210 0.01657 0.01937 0.02024 0.01913 0.01618 0.01171 0.006132 0 x 1.043 1.040 1.34 1.023 1.011 0.9972 0.9839 0.9722 0.9631 0.9574 0.9554.5 y 0.01404 0.02653 0.03615 0.04197 0.04354 0.04085 0.03434 0.02472 0.01292 0 06 x 1.096 1.091 1.075 1.052 1.023 0.9937 0.9658 0.9419 0.9238 0.9125 0.9087 y 0 o.03169 0.05941 0.07998 0.09150 0.09342 o.08635 0.07164 0.005105 0.02651 0 x 1.222 1.207 1.166 1.108 1.042 0.9781 0.9213 0.8754 0.8421 0.8221 0.8153 y 0 0.07597 0.1396 0.1826, 0.2019 0.1992 0.1784 0.1442 0.1008 0.05173 0 x 1.550 1.498 1.366 1.201 1.042 0.9087 o.8056 0.7308 0.6808 0.6521 0.6427 0.8 y 0 0.2104 0.3649 0.4415 0.4503 0.4130 0.3482 0.2687 0.1819 0.09158 0 x 2.736 2.359 1.686 1.168 0.8414 0.6428 0.5199 o.4434 0.3971 0.3721 0.3642 0.9 y 0 0.8663 1.178 1.122 0.9507 0.7637 0.5875 0.4262 0.2773 0.1366 0 x o ~ 0 0 00 0 0 0 0 0 0 1.0 y 0 6.302 3.072 1.959 1.374 o.9981 0.7252 0.5086 0.3243 o.1581 0

o O 0 000 *Yo~ a a I 6 6 6 6y o III Io P L kz-o c~~~~~~~0 I I 1 —Q

THE dn-FUNCTION, x + ly = dn(u + iv) k = sin 10~ = 0.173648; K = 1.582843, K' = 3.153385 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 K v K' x 1.000 0.9996 0.9985 0.9969 0.9947 o.9924 0.9900 0.9879 0.9862 0.9852 0o.9848 y 0 0 0 0 0 0 0 0 0 0 0 x 1.002 1.001 0.9998 0.9977 0.9952 0.9924 0.9895 0.9870 0.9850 0.9857 0.9833 y 0 0.001580 0.003002 0.004124 0.004835 0.005070 o.oo4808 0.004079 0.002958 0.001552 0 x 1.007 1.006 1.004 1.001 0.9968 0.9925 0.9879 0.9839 0.9808 0.9788 0.9781 0.2 2 y 0 0.003819 0.007251 0.009947 0.01165 0.01219 o.01154 0.009779 0.007082 0.005715 0 x 1.018 1.017 1.013 1.007 1.000 0.9921 0.9843 0.9774 0.9719 o.9685 0.9673 0. 3 y 0 0.007678 0.01455 0.01991 0.02324 0.02424 0.02288 0.01932 0.01396 0.007312 0 x 1.040 1.038 1.031 1.020 1.006 0.9913 0.9771 0.9646 0.9549 0.9488 0.9467 y 0 o.o1491 0.02816 0.03835 0.04448 o.o4609 0.04321 0.03628 0.02610 0.01365 0 x 1.083 1.078 1.064 1.042 1.016 0.9886 0.9626 0.9402 0.9232 0.9126 0.9090 ~.5 y 0 0.02915 0.05471 0.07376 o.08454 0.08648 0.08008 0.06654 0.04747 0.02467 0 x 1.169 1.158 1.128 1.083 1.031 0.9792 0.9324 0.8938 o.8654 0.8481 0.8422 o y 0 0.05891 0.1091 0.1442 O.1614 0.1612 0.1459 0.1190 0.08572 0.04313 0 x 1.351 1.323 1.249 1.149 1.043 0.9471 0.8670 0.8055 0.7626 0.7574 0.7291 ~07 y 0 0.1280 0.2299 0.2911 0.3108 0.2966 0.2582 0.2041 0.1404 0.07155 0 x 1.787 1.694 1.472 1.222 1.007 0.8424 0.7240 0.6426 0.5900 0.5605 0.5510 0,8 y 0 0.3261 0,5388 0.6158 0.5963 0.5247 0.4289 0.3238 0.2160 0.1079 0 x 3.274 2.636 1.684 1.078 0.7416 0.5511 0.4384 0.3701 0.3294 0. 077 0.3008 0.9 y 0 1.220 1.483 1.306 1.057 0.8254 0.6245 o.4485 0.2900 0.1424 0 x o 0 0 0 0 0 0 0 0 0 1.0 y 0 6.266 3.054 1.948 1.566 0.9924 0.7210 0.5056 0.5224 0.1572 0

ol, aq ll n co 0 c~~~~~~~~~~~~~~~~~~~~00o 0- ~~ 6 a 66 6 0 0 0 0 00 o, d ~ - IN~ O~~~~~~~~~~~~~~~~ II I I'aN ~ ~ ~ ~ ~ ~ OD O~~~~~~~~~~~~~~~~~ H 0N 0 0 1 I a 0 0~ 6 + E IV, IJ I~~~~~~~~~~N ~~~~~~~~~~~~~~~~~~~~~~~~o o rr,~~~~~~~~~~~~~~~~~~ t I 1~~~~~~~~~~~_ N I~~~~~C I I T,- 1_t I-Al C"~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~,.~~~~lx cnOII Oo~~~~~~~~~~~~~~~~~AJI o O~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~" I — rr, O~~~~~~~~~~~~~~~~~~~~~0 ci eq~C\ 0 03 O0 I'- ~ ~ ~t'.3 oJ ~-) -- 0 0~~~~~~~~~~~~~~~~~~~~~0 >1V F ~1 I 1,11, ~ ooo; > A I;01~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~( c~~~~~~~~~~~ Al0 IN v I 01 =3~~~~~~~~~~~~~~~~~ O~~~~~AII " 1\1 I Lo~~~~~~~~~~~~~~~~ J O.V _TT l ~ c 0.oo r' - r r7 U I 1 I IN I N I /F v

THE dn-FUNCTION, x + iy = dn(u + iv) k = sin 15~ = 0.258819; K = 1.598142, K' = 2.768063 u 0 0.1 0.2 0.5 0.4 0.5 0.6 0.7 0.8 0.9 1.0 K V K' x 1.000 0.9992 0.9967 0.9929 0.9881 0.9828 0.9776 0.9729 0.9691 0.9667 0.9659 y 0 0 0 0 0 0 0 0 0 0 0 x 1.003 1.002 0.9988 0.9944 0.9889 0.9828 0.9767 0.9713 0.9671 0.9643 0.9654 0.1 y o 0.003064 0.005813 0.007965 0.009312 0.009730 0.009197 0.007781 0.005628 0.002950 0 x 1.011 1.010 1.006 0.9995 0.9913 0.9826 0.9759 0.9662 0.9602 0.9564 0.9551 0.2 2 y 0 0.007118 o.01348 o.01844 0.02151 0.02241 0.02113 0.01783 0.01287 0.006741 0 x 1.029 1.027 1.020 1.009 0.9961 0.9819 0.9682 0.9560 0.9466 0.9406 0.9386 O.5 y 0 0.01353 0.02557 0.03482 0.04041 0.04188 0.03927 0.0 99 0.02375 0.01240 0 x 1.062 1.058 1.05 1.027 1.004 0.9800 0.9572 0.9374 0.9223 0.9128 0.9096 y 0 0.02468. 04639. 06271 0.07210 0.07400 0.06874 0.05727 o.04094 0.02130 0 x 1.122 1.114 1.090 1.056 1.016 0.9744 0.9363 o.9044 0.8806 0.8659 o.8610 0~5 y 0 0.04529 0.08436 0.1125 0.1271 0.1282 0.1172 0.09630 0.06812 0.05522 0 x 1.235 1.218 1.170 1.105 1.029 0.9580 0.8959 o.8464 0.8108 0.7895 0.7824 y 0.08631 0.1578 0.2048 0.2246 0.2198 0.1955 0.1571 0.1093 0.05597 0 x 1.461 1.419 1.310 1.170 t 1.051 0.9107 o.8157 0.7456 0.6979 0.6704 0.6614 0'7 y 0. 1780 0.3126 0.3842 0.3981 0.3698 0.3150 0.2450 0.1666 0.08414 0 x 1.981 1.843 1.535 1.215 0.9628 0.7828 0.6595 0.5774 0.5255 0.4968 0.4876 y 0 0.4325 0.6843 0.7463 0.6953 0.5944 0.4763 0.3547 0.2346. 1166 0 x 3.702 2.800 1.659 0.9962 0.6666 0.4878 0.5846 0.5229 0.2865 0.2671 0.2610 o0.9 y 0 1.513 1.686 1.410 1.109 0.8533 0.6398 0.4570 0.2946 0.1444 0 x o O o o 0 0 0 0 0 0 1.0 y 00 6.206 5.025 1.929 1.555 0.9828 0.7140 0.5007 0.5193 0.1557 0

O0 00 O O o a-~~~~~~~~~~~~~~~~~~~~~~~~~oo l, A —I ) I CD I o+ U, 10f040 I 0 II. 0oo 0I 0 ~ 0 0 0 0 0 -11~~~ xi DOIi + I 1A <": -^f^:::^^^7(=^$ ^^^ 7=^ ^^ ^^:aA' OQ 0 0 0,- 000 W - - ~~- A SJ- I II If

THE dn-FUNCTION, x + iy = dn(u + iv) k = sin 20~ = 0.342020; K = 1.620026, K' = 2.504550 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 K v K' x 1.000 0.9985 0.9941 0.9873 0.9787 0.9694 o.9601 0.9518 0.9453 o.9411 0.9397 y 0 0 0 0 0 0 0 0 0 0 0 x 1.004 1.002 0.9970 0.9894 — 0.9798 0.9693 0.9589 0.9497 0.9424 0.9378 0.9362 0o.l y o.o00486 0.009205 0.01257 0.01464 0.01522 0.01432 0.01207 0.008701 0.004552 0 x 1.016 1.014 1.007 0.9961 0.9830 0.9688 0.9549 0.9426 0.9330 0.9270 0.9249 0.2 y 0 0.01103 0.02083 0.02837 0.03291 0.03410 0.03197 0.02684 0.01931 0.01008 0 x 1.040 1.036 1.025 1.009 0.9889 0.9674 0.9468 0.9289 0.9151 o.go9064 0.9035 y 0 0.02024 0.03810 0.05159 O.a0544 0.06114 0.05691 0.04750 0.03400 0.01771 0 x 1.083 1.077 l. o0 1.031 0.9979 0.9639 0.9321 0.9050 0.8846 0.8720 0.8677 y 0 0.03546 b.o6629 0.08887 0.1011 0.1027 0.09447 0.07804 0.05542 0.02872 0 x 1.158 1.146 i.1 1.065 1.010 0.9547 0.9053 0.8648 0.8352 0.817-2 0.8112 0.5 y 0 0.06248 011154 0.1519 0.1693 0.1683 0.1518 0.1234 0.08657 o.o4454 0 x 1.295 1.271 1.i05 1.115 1.019 0.9306 0.8560 0.7983 0.7577 0.7337 0.7257 y 0 0.1146 o.2066 0.2632 0.2828 0.2715 0.2375 0.1884 0.1299 0.06615 0 x 1.559 1.501 1.354 1.175 1.008 0.8698 0.7652 0.6902 0.6404 0.6120 0.6028 Y 0 0.2282 0.3916 o.4678 0.4714 0.4279 0.3580 0.2747 0.1852 0.09305 0 x 2.151 1.963 1.565 1.190 0.9136 0.7265 0.6030 0.5228 0.4729 0.4455 0.4368 Y 0 0.5359 0.8127 0.8503 0.7675 o.6418 0.5066 0.3736 0.2455 0.1216' 0 x 4.071 2.894 1.573 0.9197 0.6036 0.4371 0.3426 0.2866 0.2557 0.2363 0.2308 y 0 1.771 1.831 1.473 1.136 0.8653 0.6449 0.4590 0.2952 0.1446 0 x oo o - 0 0 0 0 0 0 0 1. y 0 6.122 2.984 1.903 1.334 0.9694 0.7042 0.4938 0.3149 0.1535 0

0-~~~~~~~~~~~~~~~~~'Is o,~~~~~~~~~~~~~~~~~~~~~~~~~~~~ —4 /I~~~~~~~~~~~~~ O~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~1_1 -J Ii- P- re~~~~~~~~~~~~~~~~~~~~~~~ 1 C " II C~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ % 1 Al~~~~~~d I IL~~~~~~~~C I =Ily ~~~~~~~~~~~~~~~~~~~~~~~~~n o~_( I1 d 17~IlrT~1~7 1 r Y I-~TU~ in xn ~x x IXYIYT yl V nlA(L-A I Al V I \1 X I I I tI X I N V I I ILI I k _C IJI I I I " 1'' It II 6- I y I IN I- - ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ t IN YN~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~~~~~~ oe I~~~~O I ] - I I -,A V ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ Y in 77: O ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~J~~~~~~~~~~~~~~~4-'o, I Y~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~Q 114~~~~~~11 /VI I ~ ~ ~ ~ ~ ~ ~ ~ ~ If. I -4 -L. ICTJI I A 0~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~c 44-~~~~~~~~ o~~~~~~~~~~~~~~~~~~~~~- ~ -/ ) ( O d) ao re (D ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~5 <U O~~~~~-4) J O O O O O O O~~~~~~~~~~~~~~~~~~~~1 Nr v Y I)N v ZI Ix ~ ~ ~ ~ ~ ~ ~ ~ ~

THE dn-FUNCTION, x + iy = dn(u + iv) k = sin 250 = 0.422618; K = 1.648995, K' = 2.508787 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 K v K' x 1.000 0.9976 0.9906 0.9799 0.9666 0.9520 0.9376 0.9249 0.9149 0.9085 0.9063 y 0 0 0 0 0 0 0 0 0 0 0 x 1.005 1.002 0.9944 0.9826 0.9678 0.9518 0.9360 0.9221 0.9112 0.9043 0.9019 o.1 y 0 0.006921. 006.01776 0.02056 0.02126 0.01989 0.01667 0.01197 0.006245 0 x 1.021 1.017 1.007 0.9909 0.9716 0.9509 0.9308 0.9132 0.8996 0.8910 0.8881 2 y 0 0.01544 0.02906 0.03936 0.04537 0.04668 0.04346 0.03628 0.02597 0.01352 0 x 1.051 1.045 1.030 1.006 0.9782 0.9485 0.9204 0.8963 0.8779 0.8665 0.8626 y 0 0.02766 0.05183 o. 06970 0.07963 0.08119 0.07493 0.06208 0.04419 0.02293 0 x 1.103 1.094 1.069 1.031 0.9875 0.9429 0.9020 0.8679 0.8424 0.8268 0.8216 y 0 0.04711 0.08752 0.1162 0.1308 0.1313 0.1195 0.09781 o0.6899 0.03561 o x 1.193 1.176 1.132 1.068 0.9977 0.9294 0.8697 0.8219 0.7874 0.7667 0.7599.5 y 0 0.08064 0.1475 0.1916 0.2104 0.2060 0.1834 0.1474 0.1027 0.05256 0 x 1.350 1.318 1.232 1.118 1.002 0.8977 0.8132 0.7493 0.7053 0.6796 0.6712 y 0 0.1438 0.2556 0.3193 0.3362 0.3168 0.2729 0.2140 0.1463 0.07415 0 x 1.648 1.572 1.385 1.169 0.9764 0.8256 0.7i50 0.6378 0.5874 0.5590 0.5498 0.7 y 0 0.2791 0.4676 0.5430 0.5332 0.4741 0.3905 0.2963 0.1983 0.09924 0 x 2.304 2.060 1.575 1.153 0.8614 0.6728 0.5519 0.4748 0.4273 0.4015 0.3933 y 0 0.6376 0.9272 0.9337 0.8202 0.6735 0.5255 0.3846 0.2516 0.1243 0 x 4.401 2.938 1.496 0.8476 0.5482 0.3939 0.3073 0.2564 0.2266 0.2109 0.2059 y 0 1.999 1.935 1.509 1.147 0.8667 0.6431 0.4566 0.2932 0.1434 0 x o o0 0 0 0 0 0 0 0 0 y 6.014 2.931 1.869 1.311 0.9520 0.6915 0.4849 0. 092 0.1507 0

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THE dn-FUNCTION, x + iy = dn(u + iv) k = sin 300 = 0.500000; K = 1.685750, K' = 2.156516 u O 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 K V K' x 1.000 0.9965 o.9864 o.9708 0.9515 0.9306 0.9102 0.8921 0.8780 0,8691 0.8660 y 0 0 0 0 0 0 0 0 0 0 0 x 1.006 1.002 0.9909 0.9739 0.9529 0.9302 0.9081 0.8888 0.8737 0.8642 0.8609 0.1 y o 0.009201 0.01750 0.02340 0.02690 0.02761 0.02564 0.02135 0.01526 0.007935 0 x 1.025 1.020 1.005 0.9835 0.9569 0.9287 0.9016 0.8782 0.8602 0.8489 0.8450 0.2 0 0.02028 0.03802 0.05117 0.05850 0.05969 0.05512 0.04569 0.03253 0.01688 0 x 1.061 1.053 1.032 1.001 0.9637 0.9251 0.8890 o.8585 0.8354 0.8212 0.8164 ~- y o 0.03571 0.06655 0.08878 0.1004 o.1014 0.09266 0.07614 0.05386 0.02784 0 x 1.122 1.110 1.077 1.028 0.9723 0.9169 0.8671 0.8262 0.7962 0.7779 0.7718 0.4 y 0 0.05958 o.1099 0.1443 O.1604 0.1591 0.1451 0.1160 0.08125 0.04175 0 x 1.225 1.204 1.146 1.066 0.9799 0'.8989 0.8299 0.7758.73574 0.714'7 0.7071 0.5 y o 0.09977 0.1806 0.2312 0.2497 0.2409 0.2115 0.1683 0.1162.0.05924 0 x 1.402 1.360 1.252 1.113 0.9770 0.8599 0.7675 0.6994 0.6532 0.6266 0.6178 o'6 y o 0.1741 o.3047 0.3729 o.3846 o0.3559 0.3021 0.2543 0.1590 0.08023 0 x 1.730 1.633 1.403 1.151 0.9383 0.7784 0.6648 0.5873 0.5373 0.5094 0.5005 0.7 y 0 0.3310 0.5409 0.6104 0.5847 0.5100 0.4143 0.3114 0.2071 0.1033 0 x 2.443 2.136 1.566 1.106 0.8070 0.6207 0.5041 0.4309 0.3863 0.3621 0.3545 8 y 0 0.7383 1.029 0.9998 0.8574 o.6933 0.5356 0.3896 0.2538 0.1251 0 x 4.700 2.942 1.410 0.7788 0.4979 0.3555 0.2764 0.2301 0.2030 0.1887 0.1843 ~ y 0 2.201 2.007 1.525 1.145 0.8600 0.6359 0.4505 0.2889 O.1412 0 x ~ o O 0 0 ~ ~ ~ ~ ~ x 0 0 05,8.6 0 0 0 0 0 0 0 0 1.0 y 5.883 2.867 1.828 1.281 0.9306 0.6759 0.4738 0.3021 0.1472 0

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THE dn-FUNCTION, x + iy = dn(u + iv) k = sin 35~ = 0.573576; K = 1.731245, K' = 2.034715 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 K V K' x 1.000 0.9951 0.9811 0.9597 0.9334 0.9051 0.8776 0.8536 0.8349 0.8232 0.8192 y 0 0 0 00 0 0 0 0 0 0 x 1.001 1.002 0.9863 0.9631 0.9348 0.9044 0.8752 0.8497 0.8301.0.8177 0.8135 0.1 y 0 0.01168 0.02186 0.02936 0.03348 0.03405 0.03135 0.02591 0.01840 0.009535 0 x 1.029 1.022 1.003 0.9737 0.9388 0.9021 0.8674 0.8377 0.8150 o.8009 0.7961 2 y 0 0.02551 0.04758 o.06355 0.07198 0.07271 0.06651 0.05467 o.05867 0.01999 0 x 1.070 1.061 1.033 0.9925 o.9451 0.8969 0.8526 0.8156 0.7880 o.7710 0.7653 ~03 y 0 o.o4434 0.08210 0.1085 0.1214 0.1211 0.1095 0.08916 0.06263 0.03224 0 x 1.140 1.124 1.081 1.021 0.9523 0.8859. 8274 0.7803 0.7461 0.7255 0.7187 Y 0 0.07280 0.1331 0.1728 0.1894 0.1852 0o1646 0.1321 0.09185 0.04698 0 x 1.254 1.227 1.155 1.058 0.9561 0.8632 0.7860 0.7268 0.6854 0.6610 0.6530 05 y 0 0.1199 0.2145 0.2701 0.2868 0.2722 0.23558 0.1855 0.1272 0.06455 0 x 1.449 1.396 1.264 1.100 0.9455 0.8174 0.7193 0.6485 0.6012 0.5741 0.5653 Y 0 0.2056 0.3537 0.4235 0.4276 0.3886 0.3253 0.2497 0.1683 0.08454 0 x 1.805 1.684 1.409 1.123 0.8941 0.7288 0.6146 0.5380 0.4894 0.4624 0.4537 y 0 0.3840 0.6109 0.6697 0.6264 0.5367 0.4305 0.3208 0.2122 0.1054 0 x 2.570 2.194 1.540 1.051 0.7510 0.5700 0.4589 0.3901 0.3483 0.3259 0.3188 y 0 0.8379 1.118 1.050 0.8811 0.7030 0.5385 0.3896 0.2530 0.1245 0 x 4.971 2.908 1.319 0.7127 0.4512 0.3206 0.2484 0.2063 O.1818 0.1688 0.1648 y 0 2.378 2.050 1.524 1.133 0.8464 0.6239 0.4411 0.2826 0.1381 0 x o 0 0 0 0 0 0 0 0 0 0. o 5.728 2.791 1.779 1.247 0. 9051 0.6571 0.4605 0.2935 o0.1430 0

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THE dn-FUNCTION, x + iy = dn(u + iv) k = sin 40~ = 0.642788; K = 1.786769, K' = 1.935581 U 0 0.1 0.2 0.3 o,4 0.5 0.6 0.7 0.8 0.9 1.o K V K' x 1.000 0.9935 0.9748 0.9465 0.9120 0.8752 o.8400 0.8094 0.7858 0.7711 0.7660 y o00 0o 0 0 0 0 0 0 0 x 1.008 1.001 0.9806 0.9501 0.9133 0.8743 0.8372 0.8051 0.7806 0.7653 0.7601 0.1 y O 0.01434 0.02670 0.03556 0.04013 0.04038 0.03679 0.03013 0.02125 0.01097 0 x 1.033 1.024 0.9987 0.9613 0.9170 0.8711 0.8282 0.7919 0.7645 0.7475 0.7418 0.2 y 0 0.03112 0.05767 0.07630 0.08547 0.08533 0.07719 0.06285 0.04414 0.02272 0 x 1.079 1.067 1.032 0.9807 0.9223 0.8640 0.8113 0.7679 0.4359 0.7164 0.7099 0.3 y o 0.05353 0.09836 0.1285 0.1420 0.1398 0.1249 0.1007 0.07018 0.03595 0 x 1.156 1.137 1.083 1.009 0.9272. 0.8499 0.7832 0.7304 0.6926 0.6701 0.6626 y 0 0.08679 0.1572 0.2011 0.2171 0.2092 0.1834 0.1457 0.1005 0.05117 0 x 1.282 1.248 1.160 1.044.09264 0.8225 0.7384 0.6751 0.6315 0.6060 0.5977 5 y 0 0.1409 0.2488 0.3079 0.3208 0.2993 0.2557 0.1991 0.1355 0.06843 0 x 1.492 1.427 1.268 1.079 0.9075 0.7707 0.6688 0.5967 0.5491 0.5221 0.5133 y 0 0.2384 0.4022 0.4704 0.4647 0.4148 0.3425 0.2602 0.1742 0.08718 0 x 1.874 1.725 1.403 1.086 0.8448 0.6771 0.5642 0.4898 0.4429 0.4171 0.4088 y 0.4384 0.6772 0.7206 0.6583 0.5545 0.4396 0.3250 0.2138 0.1060 0 x 2.685 2.232 1.498 0.9905 0.6939 0.5204 0.4157 0.3514 0.3127 0.2919 0.2853 y 0 0.9364 1.194 1.085 0.8928 0.7037 0.5349 0.3851 0.2493 0.1225 0 x 5.218 2.841 1.224 o.6488 0.4074 0.2881 0.2226 0.1845 0.1622 0.1505 0.1468 y 0 2.528 2.069 1.508 1.112 0.8265 0.6075 0.4287 0.2743 0.1339 0 x o 0 0o o 0 0 0 00 y 1 5.550 2.703 1.722 1.206 0.8752 0.6350 0.4448 0.2834 0.1380 0

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THE dn-FUNCTION, x + iy = dn(u + iv) k = sin 45~ = 0.707107; K = K' = 1.854075 u 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 K v K' x 1.000 0.9915 0.9673 0.9309 0.8870 0.8409 0.7971 0.7596 0.7310 0.7132 0.7071 y o 0 0 00 0 0 0 0 0 0 x 1.009 0.9996 0.9736 0.9346 0.8881 0.8396 0.7940 0.7550 0.7255 0.7072 0.7010 0. 1 0y 0 0.01718 0.03177 o.04188 0.04670 0.04638 o.04174 0.03383 0.02367 0.01215 0 x 1.036 1.025 0.9930 0.9460 0.8912 0.8353 0.7839 0.7409 0.7088 0.6890 0.6824 y 0 0.03708 0.06820 0.08923 0.09863 0.09713. 08676 o 06988. 04868 0.02493 o x 1.087 1.071 1.028 0.9652 o.8950 0.8261 0.7651 0.7156 0.6796 0.6577 o.6504 y 0 0.06326 0.1152 0.1486 0.1617 0.1568 0.1382 0.1102 0.07619 0.05884 0 x 1.171 1.147 1.081 0.9919 0.8968 0.8088 0.7346 0.6768 0.6360 0.6118 0.6038 y 0.1015 O.1818 0.2290 0.2429 0.2302 0.1990 0.1563 o.1069 0.05417 0 x 1.307 1.265 1.159 1.023 o.8905 0.7769 0.6872 0.6210 0.5759 0.5497 0.5412 o0.5 0y 0 o.1630 0.2835. 3438 0.3511. 3218 0.2709 0.2086 0.1409 0.07084 0 x 1.532 1.453 1.265 1.049 0.8633 0.7202 0.6162 0.5440 0.4969 0.4703 0.4617 y 0 0.2724 0.4497 0.5129 0.4953 0.4341 0.3535 0.2660 0.1769 o.08819 0 x 1.936 1.756 1.384 1.040 0.7909 0.6238 0.5138 0.4423 0.3976 0.3731 0.3653 0.7 y 0 0.4939 0.7389 0.7623 0.6804 0.5639 0.4420 0.3243 0.2123 0.1049 0 x 2.789 2.250 1.442 0.9244 0.6360 0.4718 0.3741 0.3145 0.2788 0.2596 0.2535 y 0 1.033 1.256 1.106 0.8931 0.6961 0.5253 0.3764 0.2429 0.1191 0 x 5.440 2.743 1.126 0.5869 0.3659 0.2577 0.1985 0.1640 0.1439 0.1333 0.1300 y 0 2.651 2.065 1.478 1.081 o.8004 0.5867 0.4132 0.2640 0.1288 0 x o 0 0 0 0 0 0 0 0 0 1.0 y o 5.347 2.604 1.658 1.160 0.8409 0.6095 0.4266 0.2716 0.1322 0

CC I zrV 00~, ~~ 0, f i' l 1; I I IX o) - 6 o 6 6 o 0 6 o 6 ~~I 711-~~~~~.\'4 — 11 I I II I / I I /-, x.,, [ [ [ V,.-.~ -, l -' — II I/, I In I 4 Arr I 4I~~~~~~~~~~~~~o~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~X -: —~ ~ ~ ~ ~~~~~~ Ul)~~~~..... 6 0 0 0~~~~~ I/ ],,,[ ],~ [,]~.,~~~>

THE dn-FUNCTION, x + iy = dn(u + iv) k = sin 50~ = 0.766044; K = 1.935581, K' = 1.786769 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 v K' x 1.000 0.9892 0.9584 0.9126 0.8582 0.8017 0.7490 0.7043 0.6707 0.6498 0.6428 y 0 0 o o o o o o 0 o o x 1.009 0.9978 0.9650 0.9163 o.8590 0.8001 0.7455 o.6996 0.6651 0.6439 0.6368 y 0 0.02020 0.05704 0.04825 0.05299 0.05184 0.04599 0.03682 0.02553 0.01303 0 x 1.039 1.025 0.9852 0.9274 0.8609 0.7945 0.7344 o.6848 o.6483 0.6259 0.6184 0.2 y 0 o.04341 0.07912 0.1021 0.1111 0.1077 0.09478 0.07541 0.05206 0.02651 0 x 1.094 1.075 1.021 0.9456 0.8626 0.7832 0.7141 0.6589 0.6192 0.5953 0.5873 y 0 0.07358 0.1325 0.1685 0.1800 0.1716 0.1490 0.1173 0.08038 0.04075 0 x 1.185 1.155 1.075 0.9697 0.8607 0.7626 o.6816 0.6197 0.5765 0.5510 0.5426 y 0 0.1171 0.2068 0.2558 0.2661 0.2476 0.2107 0.1635 0.1109 0.05589 0 x 1.329 1.279 1.151 0.9956 0.8486 0.7266 0.6328 o.5646 0.5187 0.492-3 0.4837 y 0 0.1863 0.3181 0.3772 0.3768 0.3388 0.2809 0.2139 0.1433 0.07172 0 x 1.567 1.471 1.250 1.011 0.8153 0.6662 o.5619 0.4906 0.4445 o.4186 0.4102 y 0 0.3078 0.4958 0.5500 0.5186 0.4461 0.3583 0.2669 0.1763 0.08757 0 x 1.992 1.776 1.352 0.9853 0.7329 0.5691 0.4634 0.3954 0.3532 0.3301 0.3228 y 0 0.5507 0.7950 0.7942 0.6924 0.5647 0.4378 0.3188 0.2077 0.1023 0 x 2.882 2.248 1.372 0.8537 0.5777 0.4241 0.3338 0.2790 0.2462 0.2286 0.2230 y 0 1.126 1.304 1.113 0.8824 0.6804 0.5099 0.5636 0.2340 0.1145 0 x 5.659 2.617 1.025 0.5267 0.5263 0.2290 0.1757 0.1447 0.1266 0.1170. 1140 y 0 2.744 2.038 1.436 1.043 0.7684 0.5615 0.3945 0.2517 0.1227 0 x oo o o o ooooooo xy 5. 1210 0 0 0 0 0 0 0 0 0 1.0 Y m 5.121 2.492 1.585 1.108 0.8017 o.5803 o.4056 0.2580 0.1255 0

>rx ( _'o-rl.-.........~....... \____ ~ _____ /I.. - X — ~ —_ —1 — f /., 7-: ~ 0!, 6 6 6 6 6 6 1>r'll o / - -: - / 7-/, _!? _'R z: -— t —-~L_...~-~-ttl J "/i~''"k~ J/ "" /./,/ /' ~_'_~/" //'Z' //,,'='~L / I -,; /. - A,,I /.- - T,., - Y >1 4P \z I I.:_0': c~~~~~~~~ c~~~~~~~~~~ 0~~~~-L 0 ~ ~~~~~~~~~~~~- -,..._.0 0 -::~~i ~+~,,,,,,,~,~.,.~...~.~._~_:=_,-___~ —~_:-.~>, -....,,

THE dn-FUNCTION, x + iy = dn(u + iv) k = sin 55~ = 0.819152; K = 2.034715, K'- = 1.731245 0 O.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 K K' x 1.000 0.98635 0.9478 0.8911 0.8249 0.7573 0.6954 0.6436 0.6051 0.5815 0.5736 y 0 0 0 0 0 0 0 0 0 0 x 1.010 0.9956 0.9546 0.8947 0.8253 0.7553 0.6916 o.6388 0.5998 0.5759 0.5678 0.1- y O 0.02342 0.04250 0.05451 0.05883 0.05651 o.04930 0.03892 0.02670 0.01354 0 x 1.042 1.024 0.9751 o.9048 0.8258 0.7483 0.6796 0.6239 0.5832 0.5586 0.5504 0.2 y 0 0.05014 0.09036 0.1147 0.1225 0.1165 0.1008 0.07911 0.05405 0.02734 0 x 1.101 1.077 1.011 0.9212 0.8248 0.7348 0.6581 0.5979 0.5551 0.5295 0.5210 3 y 0 o.08454 0.1503 0.1874 0.1963 0.1835 0.166 0.1216 0.08251 0.04158 o x 1.197 1.160 1.065 0.9416 0.8187 0.7111 0.6245 0.5593 0.5143 o.4880 0.4794 Y 0 0.1336 q.2322 0.2810 0.2858 0.2606 0.2180 0.1669 0.1122 0.05622 0 x 1.349 1.288 1!137 0.9606 0.8003 0.6718 0.5752 0.5062 0.4602 0.4338 0.4253 0.5 y 0 0.2107 0.;3523 0.4072 0.3969 0.3497 0.2853 0.2146 0.1426 0.07099 0 x 1.598 1.483 1.g26 0.9644 0.7575 0.6089 0.5060 0.4365 0.3919 0.3669 0.3589 y 0.3448 0.5396 0.5807 0.5337 0.4503 0.3565 0.2628 0.1724 0.08530 0 x 2.041 1.784 1.308 0.9228 0.6714 0.5133 0.4130 0.3491 0.3096 0.2880 0.2811 0.7 y 0 o.6086 0.8443 0.8152 0.6940 0.5569 0.4269 0.3084 0.1999 0.09821 0 x 2.965 2.223 1.289 0.7794 0.5192 0. 3773 0.2947 0. 2446 0.2147 0. 1985 0. 1935 y 0 1.216 1.335 1.104 0.8608 0.6567 0.4886 0.3467 0.2223 0.1086 0 x 5.815 2.464 0.9235 0.4683 0.2885 0.2015 0.1540 0.1263 0.1100 0.1014 0.09864 0~9 y 0 2.806 1.991 1.381 0.9956 0.7300 0.5315 0.3724 0.2371 0.1154 0 x 0 0 0 0 0 0 0 0 0 0 y 4.870 2.67 1.50757 5470 816 2421178 y g 4.870 2.367 1.503 1.049 0.7573 0.5470 0.3816 0.2423 0.1178 0

U K 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0,9 1.0 1.01.4 0.9 O o't -'I k I I r019 0.8 ]). lt~ T I1 II I] \1I II Iv nlK7~\n, h~~Q~Y`~PI~Y n~~IIIY\ II Ii u II /I Id~til I I I III II II T-CII-llrdrrIrr roy KI05 11 l~Y~l~lh~~ lrLtjNI -ID~~~~3K l O.1 0 0 0.5 i0.6.... E~~~~~~~~ 80. ~~~~~~~~~~~~~~~~~~~~~~~~~07 IN~~ ~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~.,,,-,,,.." 0.6~~~~~~~~~~k= i 0.819

THE dn-FUNCTION, x + iy = dn(u + iv) k = sin 60~ = 0.866025; K = 2.156516, K' = 1.685750 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 K v K' x 1.000 0.9829 0.9351 0.8657 0.7863 0.7071 0.6359 0.5775 0.5347 0.5087 0.5000 y 0 0 0 0 0 0 0 0 0 0 0 x 1.011 0.9925 0.9417 0.8688 0.7861 0.7045 0.6319 0.5727 0.5296 0.5034 0.4947 0.1 y 0 0.02686 0.04812 0.06055 0.06394 0.06009 0.05140 0.03992 0.02706 0.01362 0 x 1.045 1.023 0.9621 0.8776 0.7849 0.6963 0.6194 0.5581 0.5139 0.4875 0.4786 ~0.2 y 0 0.05736 0.1019 0.1268 0.1324 0.1231 0.1044 0.08063 0.05444 0.02734 0 x 1.107 1.077 0.9972 0.8912 0.7809 0.6806 0.5972 0.5328 0.4875 0.4607 0.4518 0.3 y 0 0.09626 o.1683 0.2052 0.2098 0.1918 0.1605 0.1227 0.08230 0.04119 0 x 1.207 1.162 1.048 0.9066 0.7700 0.6542 0.5632 0.4958 0.4498 0.4230 0.4143 y 0 0.1512 0.2576 0.3038 0.3012 0.2683 0.2203 0.1662 0.1105 0.05504 0 x 1.366 1.292 1.115 0.9173 0.7455 0.6124 0.5147 0.4458 0.4003 0.37-44 0.3660 0'5 y 0 0.2367 0.3858 0.4328 0.4105 0.3536 0.2834 0.2103 0.1385 0.06859 0 x 1.625 1.487 1.192 0.9088 0.6962 0.5487 0.4486 0.3817 0.3391 0.3153 0.3076 6 y 0 0.3835 0.5804 0.6036 0.5397 0.4460 0.3477 0.2536 0.1651 0.08131 0 x 2.083 1.779 1.249 0.8525 0.6067 0.4565 0.3626 0.3032 0.2665 0.2464 0.2400 0.7 y 0.6676 0.8852 0.8241 0.6844 0.5400 0.4091 0.2931 0.1888 0.09247 0.8 x 3.037 2.175 1.194 0.7019 0.4606 0.3313 0.2565 0.2112 0.1840 0.1693 0.1647 o.8 y 0 1.301 1.349 1.081 0.8281 0.6247 0.4611 0.3253 0.2078 0.1013 0 x 5.967 2.285 0.8208 0.4114 0.2521 0.1753 0.1332 0.1085 0.09399 0.08623 0.08379 0'.9 y 0 2.833 1.922 1.314 0.9395 0.6850 0.4965 0.3466 0.2200 0.1069 0 x oo O O O O0 0 O O 1.0 y 4 4.592 2.229 1.412 0.9822 0.7071 0.5091 0.3541 0.2243 0.1089 0

u K 1.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 _ | ll L L \ \ V rs i j F S; 2 / 1 1 z _11 IX I I I I I I I TI \1'1 I\ I\ I' \1~ IN I LYt 1.0|t|11 1111111 1110. 1 Sl N-r -1 1: r7- - I I F XXFW o A k SA;C _,y 0.2 1 X V rol \~ CX \m X1 1 11 X 1AS W \1 1 0.8 X \ 11 1 I N J s 1 A' *~~~~/Nl X X k \1 VA ~~~~~~~~~~~~~~~~~~~~~0.8 -T_~~~ I si 60~ X\ 8660,, 0.7 X I Ij -)l I I I \,/I\ N \1^, 1A N, ~~~~~~~~~~~~~~0.7 0.6 "00 X 1 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~0.6 -.0~~~~~~~~~~~~~~~~~~~~1. 0. 5 I/N ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~0.5 v 0.4 00,' 04:6.7o ~ 0.. 00 C. The dn-function, z~x~iy~dn~u~iv)=~l- k' sn'(u~iv) N X~~~~ i 0"=086

THE dn-FUNCTION, x + iy = dn(u + iv) k = sin 65 = 0.906508; K = 2.308787, K' = 1.648995 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 K K' x 1.000 0.9786 0.9194 0.8353 0.7413 o.6501 0.5701 0.5060 0.4597 0.4 419 0.4226 y 0 0 O O 0 O O 0 O 0 O x 1.011 0.9885 0.9258 0.8377 0.7405 0.6471 o.5660 0.5014 0.4549 0.4271 0.4179.1 y o 0.0306 0.05390 0.06622 0.06805 0.06227 0.05202 0.05963 0.02649 0.01522 0 x 1.047 1.019 0.9453 o.8446 0.75753 0.676 0.5553 0.4874 0.4407 0.410 0.4058.2 y 0 o.o6519 0.1137 0.1379 0.1400 0.1268 O.1051 0.07959 0.05301 0.02641 0 x 1.111 1.075 0.9784 0.8544 0.7298 0.6200 0.5310 o.4656 0.4168 0.3893 0.3803 0.3 y 0 0.1090 0.1866 0.2212 0.2197 0.1955 0.1599 0.1200 0.07949 0.05948 0 x 1.216 1.161 1.025 o.8656 0.7140 0.5915 0.4976 0.4292 0.3830 0.5563 0.3476 0.4 y 0.1702 0.2828 0.5255 0.3108 0.2696 0.2166 0.1607 0.1056 0.05224 0 x 1.381 1.291 1.084 0.8648 0.6837 0.5483 0.4511 0.3836 0.5594 0.5142 0.5061 0.5 y 0 0.2645 0.4179 0.4525 0.4160 0.3493 0.2745 0.2007 0.1508 0.06440 0 x 1.649 1.482 1.145 0.8437 0.6293 0.4855 0.3897 0.3263 0.2860 0.2656 0.2564 o0.6 y 0 0.4244 0.6169 0.6171 0.5352 0.4323 o.5314 0.2587 0.1541 0.07550 0 x 2.120 1.757 1.175 0.7747 0.5589 0.3988 0.35122 0.2575 0.2258 0.2055 o.1994 0.7 0y 0.7278 0.9156 0.8192 0.6626 0.5134 0.3839 0.2723 0.1743 0.08501 0 x 3.097 2.100 1.087 0.6216 o.4019 0.2860 0.2190 0.1784 0.1539. 1406 0.1564 0.8 y 0 1.379 1.343 1.042 0.7856 0.5838 0.4270 0.2991 0.1901 0.09256 0 x 6.097 2.081 0.7174 0.3558 0.2169 0.1500 0.1131 0.09136 0.07843 0.07151 o. 6932 0.9 0y 2.823 1.831 1.235 0.8756 0.6526 0.4557 0.5165 0.2001 0.09701 0 x o 0 0 0 0 0 0 0 0 0 0 1.0 y o 4.286 0.2076 1.310 0.9074 0.6501 0.4657 0.5225 0.2036 0.09860 0

u K 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.0 JOAY\\\Lh 119 11 I I I I I II I z I 1 1 I II 11 I I I I I I I 11 h I nu i 1 I I I 11 11 I i i 11 111.0 I 1 I~T~ —~ IIICII 0~ C~ol "lT I lIll lN^' I I| |.J 11 I I I I — TT~l^ ^Ell 0.9 0.9 K~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~'?. 0.38,'a | 7Eb E | 10.8 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~0.7 ____\^_ ~ ~ I11\XI ]I V^_x ^ N — L- ^ - --- 4n -4 ^ ^ — -vJ ITL^ ^ _ ^ i _\ _^ g ~ _y _ ^ / _ 0. 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 U The dn-function, z=x+iy=dn(uFiv)=VI- k\ n\Nu+iv) ~ k sin 65~ =0.190631 07 07 The dn-functi~Lon, z I xi=duiv=/I -A ~ n(u v 0.5~~~~~~~~~ = i06~ 096

THE dn-FUNCTION, x + iy = dn(u + iv) k = sin 70~ = 0.99693; K = 2.504550, K' = 1.620026 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 K v K' x 1.000 0.9730 0.8995 0.7979 o.6880 0.5848 0.4971 0.4286 0.3802 0.3515 0.3420 y 0 0 0 0 0 0 0 0 0 0 0 x 1.012 0.9830 0.9055 0.7995 0.6864 0.5815 0.4931 0.4244 0.3761 0.3475 0.3381 y 0 0.03475 0.05985 0.07123 0.07071 0.06259 0.05079 0.03781 0.02486 0.01229 0 x 1.048 1.014 0.9233 0.8037 0.6810 0.5709 0.4806 0.4117 0.3637 0.3355 0.3262 y 0 0.07387 0.1258 0.1476 0.1446 0.1266 0.1020 0.07558 0.04953 0.02444 0 x 1.115 1.o69 0.9530 0.8087 0.6698 0.5519 0.4590 0.3901 0.3430 0.3156 0.3067 ~03 y 0 0.1230 0.2050 0.2345 0.2245 0.1933 0.1539 0.1151 0.07377 0.03631 0 x 1.223 1.155 0.9937 0.8105 0.6494 0.5223 0.4274 0.3595 0.3141 0.2881 0.2796 4 y 0 0.1912 0.3074 0.3380 0.3131 0.2631 0.2061 0.1499 0.09718 0.04767 0 x 1.393 1.282 1.041 0.8014 0.6140 0.4791 0.3844 0.3195 0.2772. 0.2533 0.2456 0.5 y 0 0.2949 0.4477 0.4644 0.4115 0.3354 0.2576 0.1852 0.1192 0.5825 0 x 1.667 1.465 1.083 0.7679 0.5563 0.4191 0.3292 0.2701 0.2327 0.2118 0.2051 y 0 0.4682 0.6472 0.6185 0.5181 0.4079 0.3066 0.2176 0.1390 0.06770 0 x 2.150 1.717 1.085 0.6889 0.4681 0.3400 0.2615 0.2120 0.1813 0.1645 0.1591 0'7 y 0 0.7891 0.9324 0.7981 0.6270 0.4759 0.3503 0.2456 0.1558 0.07562 0 x 3.147 1.992 0.9681 0.5384 0.3431 0.2411 0.1821 0.1461 0.1243 0.1124 0.1087 8 y 0 1.446 1.314 0.9850 0.7257 0.5328 0.3852 0.2674 0.1687 0.08163 0 x 6.203 1.851 0.6133 0.3014 0.1827 0.1254 0.09358 0.07460 0.06322 0.05708 0.05514 ~-9 y 0 2.768 1.715 1.136 0.7965 0.5712 0.4079 0.2812 0.1768 0.08539 0 x 0 0 0 0 0 0 0 0 0 0 1.905 1.196.8225 848.4158.286 0.1796 8667 Y " 3.946 1.905 1.196 0.8225 0.5848 0.4158 0.2860 0.1796 0.08667 0

u K 0 010 2 0.3 0.4.~.Q.... 1.0 9,8.c 0.8 o.~ -cc-H-r~~cu -~0, 0.6- O.1 N ~ ~[~,l't, ~,t~ NI I I I/"-I-dJ1 ] II I I, - I' O W ) I. I O C__'_ C___. _CF__ N O J 1195 i|,|| ~,X, ~*Nt,|I V X 0. 0.5 o. INJN\ o m-Ltq S fb tPt-L 0.4' -.II 1I I 1X 1\ 1 1 1/1 1 11i IWI i - 1 _ _ _ _ _ I I\1 I r - 1 77 I k I 1 1 1\ 1 1/1 I I fI I I 1 1:~~~, lllxlSilVIVIf -Iri \ A11E 1W \11X1ENJS111iE11l1 N~i l~ -I I v "I Eliy~ f! 1\~I\d~AIIIVLII 0. 1 N\N^ I i i \ - I I I 0r7 I \ I i I I / I I 1 1 II - 1 1 1 T 0 0 0.3 O.4...... 0.2 OI y 11t. - o t NY \1 /I S~llXll" ->k~k /Wl \ IA Nli 1/1* X$d1 N\11\ I/! Ni lll tf4' %,044X4 S l t 4 X > 1 2|! $ 1 \ g~~~~~~~~~~~~~~~~11 0,1 IO. I 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 u K The dn-function, z= x+iy= dn(u+iv) =\ - k2 snZ(u+ iv) k = sin 70~=09397

THE dn-FUNCTION, x + iy = dn(u + iv) k = sin 75~ = 0.965926; K = 2.768063, K' = 1.598142 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 V K' x 1.000 0.9653 0.8731 0.7501 0.6229 0.5087 0.4155 0.3450 0.2964 0.2681 0.2588 y 0 0 0 0 0 0 0 0 0 0 0 x 1.012 0.9752 0.8782 0.7506 0.6205 0.5051 0.4117 0.3414 0.2931 0.2650 0.2557 0.1 y 0 0.03954 0.06595 0.07514 0.07121 0.06038 0.04724 0.03417 0.02201 0.01074 0 x 1.050 1.006 0.8933 0.7515 0.6129 0.4940 0.4000 0.3304 0.2830 0.2555 0.2466 0.2 y 0 0.08388 0.1380 0.1547 0.1447 0.1214 0.09441 0.06802 0.04370 0.02131 0 x 1.118 1.060 0.9178 0.7507 0.5982 0.4745 0.3802 0.3120 0.2663 0.2400 0.2314 0.3 y 0 0.1392 0.2231 0.2433 0.2223 0.1836 0.1412 0.1011 0.06473 0.03150 0 x 1.229 1.144 0.9495 0.7439 0.5736 0.4449 0.3516 0.2862 0.2431 0.2186 0.2106 0.4 y 0 0.2151 0.3308 0.3455 0.3053 0.2467 0.1872 0.1329 0.08468 0.04111 0 x 1.402 1.264 0.9824 0.7240 0.5346 0.4036 0.3138 0.2530 0.2138 0.1917 0.1846 0.5 y 0 0.3291 0.4736 0.4652 0.3939 0.3097 0.2312 0.1626 0.1030 0.04990 0 x 1.682 1.433 1.002 0.6793 0.4762 0.3488 0.2665, 0.2128 0.1788 0.1599 0.1539 6 y 0 0.5160 0.6682 0.6040 0.4855 0.3704 0.2717 0.1892 0.1193 0.05760 0 x 2.173 1.649 0.9739 0.5939 0.3934 0.2794 0.2200 0.1661 0.1389 0.1239 0.1191 ~07 y 0 0.8513 0.9308 0.7568 0.5746 0.4251 0.3067 0.2116 0.1327 0.06395 0 x 3.186 1.844 0.8351 0.4519 0.2834 0.1960 0.1452 0.1140 0.09496 0.08455 0.08124 y 0 1.498 1.256 0.9063 0.6517 0.4695 0.3340 0.2287 0.1429 0.06869 0 x 6.286 1.592 0.5077 0.2475 0.1491 0.1012 0.07431 0.05805 0.04821 0.04287 0.04118 0.9 y 0 2.660 1.571 1.021 0.7051 0.4986 0.3514 0.2394 0.1492 0.07163 0 x o 0 0 0 O 0 0 0 0 0 0 1.0. y ~ 3.564 1.710 1.065 0.7242 0.5087 0.3574 0.2431 0.1513 0.07263 0

u K 0 0.1 0.2 0.3 0.4 0.5 0.6 07 0.8.9 1.0 1.0......... 1. 0 0.9 0t~tf'~"' "' " "" ~ " " "" " " " ". 9 0SL~~~~~~~~~~~~J 7T]VX 80.5~ ~ ~ ~~~k=sn7- 0.95 0 9II7-LI _ 0.9 [1 1 k\1\I l~X 6 11[lll~~~~~~\A 1 4~~~~~~1I1 - 0.48 ~I~ o V 2= sin 75"=0.26N0 0~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~.71 0O.6 I 0. 0. A.. 0.60.080910 ~~~~~~~~~~k=sn7~0.96A

THE dn-FUNCTION, x + iy = dn(u + iv) k = sin80~ = 0.984808; K = 3.153385, K' = 1.582843 u0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 K v K' x 1.000 0.9537 0.8345 0.6845 0.5391 o.4167 0.3221 0.2537 0.2081 0.1821 0.1736 y 0 0 0 0 0 0 0 0 0 0 0 x 1.012 0.9632 0.8383 0.6836 0.5361 0.4131 0.3188 0.2508 0.2056 0.1799 0.1715 1 y O 0.04548 0.07213 0.07697 0.06824 0.05451 0.04058 0.02824 0.01769 0.008494 0 x 1.051 0.9924 0.8492 0.6804 0.5264 0.4022 0.3088 0.2423 0.1983 0.1733 0.1653 0.2 y 0.09624 0.1501 0.1573 0.1376 0.1090 0.08072 0.05602 0.05504 0.01681 0 x 1.121 1.074 0.8659 0.6727 0.5090 0.3834 0.2919 0.2280 0.1862 0.1626 0.1550 0.3 y 0 0.1590 0.2405 0.2443 0.2090 0.1632 0.1199 0.08285 0.05168 0.02477 0 x 1.233 1.121 0.8849 0.6567 0.4816 0.3560 0.2682 0.2083 0.1696 0.1479 0.1409 ~*4 y o0 0.2441 0.3510 0.3409 0.2826 0.2165 0.1574 0.1081 0.06726 0.03218 0 x 1.409 1.231 0.8987 0.6263 0.4415 0.3192 0.2576 0.1833 0.1487 0.1294 0.1255 0.5 y 0 0.5696 0.4918 0.4485 0.3575 0.2679 0.1923 0.1512 0.08137 0.05887 0 x 1.693 1.375 0.8919 0.5729 0.3860 0.2724 0.2002 0.1534 0.1240 0.1078 0.1026 6 y 0 0.5698 0.6734 0.5660 0.4311 0.3153 0.2236 0.1515 0.9362 0.04466 0 x 2.189 1.542 0.8353 0.4867 0.3127 0.2156 0.1565 0.1192 0.09607 0.08337 0.07931 ~0'7 y 0 0.9131 0.9012 0.6871 0.4992 o.3566 0.2498 0.1683 0.1036 0.04936 0 x 3.214 1.639 0.6842 0.3599 0.2215 0.1497 0.1076 0.08154 0.06555 0.05681 0.05403 o0.8 y O 1.524 1.160 0.7981 0.5553 0.3899 0.2698 0.1808 0.1111 0.05284 0 x 6.345 1.298 0.3985 0.1929 0.1151 0.07675 0.05482.o04141 0.03323 0.02878 0.02737 0.9 y 0 2.479 1.387 0.8785 0.5928 0.4096 0.2823 0.1886 0.1157 0.05498 0 x 0 0 0 0 0 0 0 0 0 0 1..482.9082.467 2865 1912 117 0.05570 y 00 3.118 1.482 0.9082 o.6o6o 0.4167 0.2865 0.1912 0.1172 0.05570 0

o, a, r- ~D ~n ~ t> P4 o~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~00 P t I I p I I I I I i I I I I I I I IP I I I I I I I P o, d~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ I4-~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~~~~~~r 0. IA C;~~~~~~~~XII C I A ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ Lc 11 I A'01 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ C I ~~IY InII Oo~~II J oo,~~ ~~ ~~~~~~~~~~~ o~~~~~~~~~~~~~~~~~~~oo Y X J ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~~X v VIA~~~~ C I I i I =3 -4- C;~~~~~~~~~~~~~~~~~ 0- A v ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~~~~~~AI Y IV oo, 11 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ II 11 A 1 Y 11 A CT~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~c n~~~~~~~~ 01o,1- - -7,7 n _-s-O O I 44_T_ N~~~~~~~~~~~~~~~~~ I I ~ ~ ~ ~ ~ ~ ~ ~ ~ c A Al Z o o, LInn u Z' orI v I -A C

THE dn-FUNCTION, x + iy = dn(u + iv) k = sin 85' = 0.996195; K = 3.831742, K' = 1.573792 u 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 K V K' x 1.000 0.9313 0.7661 0.5790 0.4176 0.2952 0.2087 0.1505 0.1138 o. 0938 o.08749 ~ y o o o o o o o o o o o x 1.012 0.9398 0.7676 0.5766 0.4141 0.2922 0.2063 0.1487 0.1124 0.09243 0.08609 0.1 y O 0.05407 0.07779 0.07375 0.05841 0.04233 0.02910 0.01902.119 o. 0.005318 o x 1.051 0.9655 0.7716 0.5687 0.4037 0.2830 0.1992 0.1434 0.1083 o.o8903 0.08290 ~0.2 y 0 0.1140 o.1605 o.1493 o.1168 o.08412 0.05765 0.03764 0.02251 0.01051 o x 1.122 1.010 0.7759 0.5537 0.3855 0.2676 0.1874 0.1346 0.1015 0.08343 0.07769 o.5 y 0 0.1871 0.2531 0.2280 0.1750 0.1248 0.08510 0.05542 0.03312 0.01545 0 x 1.235 1.075 0.7764 0.5289 0.3588 0.2458 0.1711 0.1225 0.09227 0.07579 0.07036 y 0 0.2840 0.3614 0.5108 0.2324 0.1635 0.1108 0.07197 0.04294 0.02003 0 x 1.413 1.161 0.7651 0.4907 0.3226 0.2177 0.1505 0.1073 0.08074 o. 06627 0.06169 0'5 y 0 0.4226 0.4904 0.3970 0.2878 0.1994 0.1342 0.08686 0.05175 0.02412 0 x 1.699 1.264 0.7290 o.4346 0.2761 0.1834 0.1258 o.08947 0.06719 0.05512 0.05129 y 0 0.6334 0.6436 0.4842 0.3392 0.2314 0.1546 0.09970 0.05931 0.02762 0 x 2.199 1.354 0.6487 0.3567 0.2191 0.1433 o0.09771. 06927 0.05195 0.04259 0.03965 0.7 y 0 o.9683 0.8170 0.5671 0.3839 0.2581 0.1713 0.1101 0.06542 0.03045 0 x 3.231 1.331 0.5022 0.2555 0.1524 0.09853 0.06680 0.04724 0.03559 0.02900 0.02697 y 0 1.493 0.9923 0.6373 0.4191 0.2783 0. 1837 0.1178 0.06992 0.03253 0 x 6.381 0.9424 0.2788 0.1339 0.07827 0.05019 0.03391 0.02395 0.01792 0.01468 0.01336 0.9 y 0 2.169 1.151 0.6851 0.4416 0.2909 0.1913 0.1225 0.07267 0.03380 0 x.o o o o 0 00 0 0 0 y 2.5461.84 7021 0.4494.2952 1941.042 y 2.546 1.184 0.7021 0.4494 0.2952 0.1939 0.1241 0.07560 0.05425 0

U K 10 0. 0.2 O.3 0.4 0.5 0.6 0.7 0.8 0.9 1-1 I'.0 0.9 ] 0. 9 087 1 I IR\Ih111 111\g 1 \20 I I I I U IA I 1 / IhIINIIIYIrI1l Ih i n irx r 4 9 w I ur r\I wI t nrI + 4 o 8 n 0,8 0.8 t1\\N 3'T -f t AXIIA I I 11X NY!II S>f~kc~li eli Iv A~~~~~~~~~~~~~~~ 0 16 1 \YY1\\51 1.1z1 L _1 11 1 11l _I 1 1 1 MI _ vr L_ _ _ I I\ I A1/ \'1 I,r 1/ 1 1 I\V I 1\1 1 RI 1 1 1 LY I m i 1Ni 1 T1 r A N V1 1wR 1 Uf 1i~o 1w 1 1 Iws Ir 11 1i 1 -- 0.4 I 0I.4 1111 I II \ 1 i X'Il\! 1 1 I z I'4I'lA' lIllW8.I Y 1 > v I]!!i L,~ IN I ~/II IIll ]~IJ~~ K~3 11 I II II/ I I I I I lN I I ^ I 1 1 L I \1 I in I I \1/ I r 1\r 0.3 tS 2 f ei 0.3 0.7 A I/ 0.7 0,1 0,1 I1111 I Ihii I I1~ l'e;- ~1t'''';'""''''''"' / \-'''"- A l\ 1Y'1''' " " AV/\''''' \ " O O 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 U K The dn-function, z= x+iy =dn(u+iv)=\/ k2 sn2(u+iv) k = sin 85 = 0.9962

THE dn-FUNCTION, x + iy = dn(u + iv) k = sin 89~ = 0.999848; K = 5.434910, K' = 1.570916 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 K K' x 1.000 o.8686 0.6057 0.3774 0.2249 0.1321 0.07760 0.04624 0.02881 0.02009 0.01745 0 y 0 0 0 0 0 0 0 0 0 0 0 x 1.012 0.8740 0.6037 0.3741 0.2224 0.1305 0.07665 0.04567 0.02846 0.01985 0.01724.1 y o o0.06859 0.07605 0.05481 0.03422 0.02032 0.01179 0.006692 0.003585 0.001557 0 x 1.051 0.8902 0.5970 0.3639 0.2149 0.1258 0.07384 0.04399 0.02740 0.01911 0.01660 0.2 y 0 0.1433 0.1543 0.1094 0.06785 0.04018 0.02331 0.01322 0.007082 0.003076 0 x 1.122 0.9164 0.5858 0.3464 0.2025 0.1181 0.06922 0.04122 0.02568 0.01790 0.01555 0.3 y 0 0.2314 0.2366 0.1633 0.1002 0.05915 0.03427 0.01943 0.01041 0.004520 0 x 1.236 0.9504 0.5611 0.3211 0.1852 0.1075 0.06290 0.03743 0.02332 0.01626 0.01412 o0.4 y 0 0.3421 0.3243 0.2159 0.1307 0.07676 0.04440 0.02516 0.01347 0.005852 0 x 1.414 0.9861 0.5245 0.2873 0.1631 0.09423 0.05503 0.03243 0.02038 0.01421 0.01234 0.5 y 0 o.4886 0.4172 0.2658 0.1585 0.09260 0.05347 0.03028 0.01621 0.007040 0 x 1.701 1.008 0.4685 0.2446 0.1367 0.07854 0.04579 0.02721 0.01694 0.01181 0.01026 y 0 0.6878 0.5129 0.3115 0.1828 0.1062 0.06123 0.03465 0.01855 /0.008056 0 x 2.203 0.9831 0.3879 0.1932 0.1064 0.06081 0.03539 0.02102 0.01309 0.009123 0.007923 0o.7 y 0 0.9561 0.6057 0.3508 0.2028 0.1173 0.06749 0.03818 0.02043 0.008872 0 x 3.236 0.8449 0.2801 0.1339 0.07282 0.04147 0.02411 0.01431 0.008908 0.006210 0.005393 y 0 1.289 0.6858 0.3812 0.2178 0.1254 0.07209 0.04076 0.02181 0.009471 0 x 6.392 0.5145 0.1475 0.06856 0.03700 0.02102 0.01221 0.007247 0.004509 0.003143 0.002730'.9 y 0 1.610 0.7411 0.4005 0.2270 0.1304 0.07489 0.04234 0.02265 0.009836 0 x o 0 0 0 0 0 0 0 0 0 0 1.0 y o 1.752 0.7609 0.4071 0.2301 0.1321 0.07584 0.04287 0.02293 0.009959 0

U K 100 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 g i 8 g y ~.~~~~~~~~~0 It~~~~4 I I II I. 0 7 M t i t q: 1 L 0 s~~~~~~~~- _T _17 t 0 0: t 00'' 0.9.9 0.8 _.8 0.7..7 0.6g1 t l t!0 Ak X( 4 | _ S 0 < j~ ta 0;0kl- jA004iH [\2t~l~l\\lihil 48 -.6 I1~~~~~~~~~~~~~~~~~~I V~~~~~~~~~~~~~~~~~~~~~~~ K' 0.5 4 + 0 i \ X W - t /t >! 32 X X 0 tC +;t= t 0 t0.5 ~ 0.4 Il 1 A \ S III \ S /\~t+ h \E~ II \l1ADl' IX1 t 0.4 0.3;0.3 0.2 0.2 z t X 0 t 4; =- 4 0 X PL 3 0. 1;A 0 oI 0.1 oM~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~i;GES Xti..... -0! 00.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0I9 1.0 U K The dn-function, z = x+iy = dn(u+iv)=V/' - k2sn2(u+ iv) k = sin 89~= 0.9999

APPENDIX IV THE E-FUNCTION

ER EI THE E-FUNCTION, ER + iEI = E(u + iv); ER = E- Ei = K' -E' k = sin 1~ = 0.0174524; K = 1.570916, K' = 5.454910, K' - E' = 4.434158; E = 1.570677, E' = 1.000752 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 _v K' 0 E 0 0.1000 0.2000 0.3000 0.4000 0.5000 0.6000 0.7000 0.8000 0.9000 1.000 E 0 0 0 0 0 0 0 O O0 0. E. 0 0.1000 0.2000 0.3001 0.4001 0.5001 0.6001 0.7001 0.8000 0.9000 1.000 1 EI 0.1226 0.1226 0.1226 0.1226 0.1226 0.1226 0.1225 0.1225 0.1225 0.1225 0.1225 0.2 E. 0 0.1001 0.2001 0.3002 0.4002 0.5002 0.6002 0.7002 0.8001 0.9001 1.000 Ef 0.2452 0.2452 0.2452 0.2451 0.2451 0.2451 0.2451 0.2451 0.2450 0.2450 0.2450 0. 0 0.1002 p.2004 0.3005 o.4006 0.5006 o.6o60 0.7005 0.8004 0.9002 1. 00 Ej 0.3679 0.3679 0.3678 0.3678 0.3677 0.3677 0.3676 0.3675 0.3675 0.3674 0.3674 E 0 o0.1006 0.2011 0.3015 0.4018 0.5019 O.6018 0.7015 O.8011 0.9006 1.000 Ef 0.4909 0.4908 0o4907 0.4906 0.4904 0.4902 0.4900 0.4898 0.4897 0.4896 0.4895. EA 0 0.1017 0.0335 0.3045 0.4052 0.5056 0.6053 0.7045 0.9032 0.9017 1.000 EI 0.6147. 6146. 6144. 6139 O.6134. 6128 0.6121 O.6116 0.6112 0.6109 0.6108.6 EA 0 0.1052 0.2999 0.3135 0.4156 0.5165 0.6155 0.7131 0.8095 0.9050 1.000 6 E 0.7412 0.7410 0.7401 0.7388 0.7371 0.7353 0.7335 0.7319 0.7307 0.7298 0.7295 0.7 E 0 0.1162 0.2306 0.3413 0.4469 0.5488 0.6453 0.7378 0.8270 O.9140 1.000 E 0.8758 0.8750 0.8721 0.8680 0.8627 0.8573 0.8521 0.8476 0.8441 0.8419 0.8412 0.8 E 0 0.1563. 3027 0.4333 0.5443 0.6430 0.7271 0.8022 0.8711 0.9364 1.000 Ef 1.058 1.035 1.023 1.008 0.9910 0.9746 0.9604 0.9490 0.9409 0.9362 0.9343 0.9 E 0 0.810 o.6443 0.7845 0.8457 0.8856 0.9089 0.9300 0.9521 0.8756 1.000 9 Ef 1.332 1.301 1.229 1.156 1.099 1.057 1.029 1.010 0.9979 0.9913 0.9892 EA o 4.119 2.159 1.549 1.276 1.137 1.063 1.024 1.007 1.001 1.000 EI 1 1.000 1.000 1.000 1.000 1.000 1.000 1. 1.000 1.000 1.000

0-o oo 6 6 n o 6 10!Dl i: J: A W~~ 0 0 w H X~l~i V W X 4 4 W i A tS t t9A 1_ =|S o ffI S ] i W | I I.4 0 ao o l ll! X6X0 0 I 0 6 O ol 11,/li 11 II \ J 1 IX I I _ T r - - = =-= 1- I I I I} I 111 l I 11 1 1 IWKIA'1\IIIIjM /M 1 L~llI """"' i'q e FI I I - I i rrl 11 T Ii|11111111ltt111 I~~~ kAW jX~~~~~~~~~~~~~~~~II- I olIllddlll 1 1 —-4 -it - -11 I Y I I -1 r- - - - - - -4 6: j D.1 14 J J oIIN~~~~~~~~~~~~~~~~~~~~~~~ai IT I 0 IA IN I I I o.~~~~~~~~~~~ I O. -- 0

ER EI THE E-FUNCTION, ER + 1EI = E(u + iv); E = -, Ei = I k = sin 5~ = 0.0871557; K = 1.573792, K' - 3.831742, K' -E' = 2.819078; E = 1.567809, E' = 1.012664 K 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.o v K' E' 0. 1004 0.2007 0.3010 O. 4012 0.5012. 6012 0.7010 O.8007 0.9004 1.000 0 E 0 0 0 0 0 0 0 0 0 0 0 ER0.1 ~ o 0.1005 0.2009 0.3013 0.4015 0.5016 0.6015 0o.7013 0.8009 0.9005 1.000 Ef 0.1360 0.1359 0.1359 0.1357 0.1356 0.1354 0.1352 0.1351 0.1349 0.1349 0.1348 0.2 E 0 0.1009 0.2017 0,3024 o.4028. 5029 0.6028 0.7024 0.8017 0.9009 1.000 Ef 0.2723 0.2722 0.2720 0.2717 0.2713 0.2708 0.2703 0.2699 0.2696 o.2694 0.2693 0.3 E1 0 0. 1019 0.2036 0.3049 o.4058 0.5061 o.6058 0.7049 o.8035 0.9019 1.000 Ef 0.4095 0.4094 o 4089 o.4082 0.4072 o.4062 o.4052 o.4043 0.4035 0.4031 0.4029 EA' 0 0.1041 0.2077 0.3106 0.4124 0.5130 0.6122 0.7104 0.8075 0.9039 1.000 Ef 0.5489 0.5485 0.5475 0.5458 0.5438 0.5415 0.5393 0.5374 0.5358 0.5349 0.5345. E 0 0.1089 0.2169 0.3231 0.4268 0.5278 0.6261 0.7219 0.8158 0.9082 1000 Ef 0.6928. 6920. 6896 o. 6860 o. 6815. 6767 0.6720 0.6679 0.6646 o. 6626. 6619 0.6 ER 0~ 0.1202 0.2379 0.3510 o0.4584 0.5596 0.6551 0.7457 0.8326 0.9169 1.000 Ef 0.8473 0.8453 0.8397 0.8314 0.8214 0.8109 0.8010 0.7925 0.7861 0.7821 0.7807 R0.7 ~ 0 o.1485 0.2891 0.4164 0.5288 0.6270 0.7138 0.7920 0.8643 0.9330 1.000 -EI 1.027 1.022 1.008 0.9868 0.965 0.9407 0.206 0.90go43 0.8925 o.8854 0.8830 E0.8 A 0 0.2342 0.4327 0.5816 0.6872 0.7634 0.8221 0.8714 o.9160 o.9584 1.000 Ef 1.279 1.261 1.214 1.155 1.099 1.052 1.015 0.9885 0.9707 0.9605 0.9572 E' 0 0o.6525 0,9511 1.016 1.006 0.9871 0.9747 0.9719 0.9769 0.9871 1.000 0.9 E 1.835 1.702 1.463 1.279 1.163 1.093 1.049 1.022 1.005 0.9966 0.9938 E1.0 4.119 2.159 1.549 1.276 1.136 1.063 1.024 1.007 1.001 1.000 E d1.0 f _ 1. 000 1. 1.000 1.000 1.000 1.000 1.000 1.000

0~' ~ ~ ~ ~' ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~o, o o ~ C -' > 1 I I/ 1II\ fll1\ ~ III I~r II l 0 0 0 I I I X~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~I O d~~~~~~ f~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ III LJC O O C C 0 o~~~~~~~~~~~~~~~~~~~ o to co " C~~~~~~~~~~~~~~~~~~~~~~~~- L MC O. t- ~I L 1 NY 0 L II I p _ -Z ------ - a C;~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~~~~~I LJ I II. I L ~ ~ ~ I x /I I A- Li~~~~~~~W - 0 o I Y1, v I v IM ILI 01 f 0o:3 i-Ij - I rd v 1\1 I ~ ~ ~ ~ ~ ~ ~ ~ ~ o C5DI/V t -C Ov i II u I rr~ii it x / / /I IV VI [ I;K It1 I L- 4I II III II1 -1 I iIII -4 -4-1 J I II:I L O (3! a) a)I IZ1c/I 11 C11 I I/ )-I 0! F W A- I loni III zlk~~~~~~~~~~~1

ER EI THE E-FUNCTION, ER + iEI = E(u + iv); EA = -, EI = K E' k = sin 10~ = 0.173648; K = 1.582843, K' = 3.153385, K' - E' = 2.113271; E= 1.558887, E' = 1.040114 u 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 v K' 0 E' 0 0.1015 0.2029 0.3040 0.4046 0.5049. 6046 0.7039 0.8029 0.9015 1.000 E 0 0 0 0 0 0 0 0 0 0 0 EV O O O O O O 0 E' 0 0.1018 0.2035 0.3048 0.4056 0.5059 0.6056 0.7047 0.8034 0.9018 1.000 E 0.1494 0.1493 0.1489 0.1484 0.1477 0.1469 0.1462 0.1455 o.1450 0.1447 0.1446 0.2 E 0 0.1029 0.2055 0.3075 O.4088 0.5092 0.6087 0.7074 0.8054 0.9928 1.000 Ei 0.2997 0.2994 0.2986 0.2973 0.2957 0.2939 0.2921 0.2905 0.2893 0.2885 0.2882 E' 0 0.1052 0.2098 0.3134 0.4156 0.5163 0.6154 0.7130 0.8094 0.9049 t.000 E~ 0.4525 0.4519 0.4502 0.4476 o.4443 0.4407 0.4372 0.4341 0.4316 0.4301 0.4295 ER 0 0.1097 0.2183 0.3249 0.4289 0.5300 0.6281 0.7236 0.8170 0.9089 1.000 Ei O. 6102 0.6090 0.6057. 6005 0.5942 0.5873 0.5807 0.5749 0.5703 0.5675 o.5665 0.5 E 0 0.1188 0.2352 0.3475 0.4544 0.5557 0.6516 0.7429 0.8306 0.9159 1.000 Ef 0.7777 0.7753 0.7685 0.7583 0.7460 0.7330 0.7207 0.7102 0.7022 0.6972 0.6955 0 E 0 0.1378 0.2700 0.3926 0.5036 0.6035 0.6937 0.7764 0.8538 0.9277 1.000 E_ 0.9654 0.9602 0.9456 0.9244 0.9000 0.8755 0.8534 0.8352 0.8217 0.8135 0.8108 0.7 E 0 0.1822 0.3475'0.4868 0.5994 0.6903 0.7657 0.8310 0.8901 0.9458 1.000 E 1.198 1.185 1.150 1.103 1.053 1.007 0.9694 0.9403 0.9200 0.9081 O.9042 0.8 E 0.3092 0.5457 0.6951 0.7826 0.8366 0.8752 0.9078 0.9386 0.9692 1.000 E 1.552 1.508 1.403 1.284 1.183 1.105 1.049 1.010 0.9853 0.9713 0.9668 E O0 0.8829 1.151 1.138 1.078 1.030 1.001 0.9877 O.9860 0.9913 1.000 9 EI 2.403 2.101 1.650 1.363 1.204 1.114 1.060 1.028 1.009 0.9986 0.9954 1.0 E o 4.119 2.159 1.549 1.276 1.137 1.062 1.024 1.007 1.001 1.000 EI co 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

~i< o C)P 0 0 c~~~~ 0 ~-,T~rl11T~ll 1 III -I r11 II T1 I I EII71 1T B II~1\1 I n II~ YL\TnIY, Y1711I1 I I A X 1NX9.1 II I - I O O~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~o 3;- O i I IIII II I I II I ~ II LU L1Iii II II I I ILU_ II I LIIII 1l I Ikn I IIIILILI 1 II_1 Il I\ 1YI\C I -I- I vn, Y I I/'A I vI r All/ 11 1 7Iv -4+_I I' II C) J, I x 4 o 1 llr~ T III ill ~~Il ll-rr- 1 ILL~~ —~-II I 1 7n1.I rllYI II-1Ir/l [nlY Il~ln/ iIA I llit~~flrA A AIP~,~ r 0 - ----- i I i III I J O' i 1 A 1I IL III 111 Y I lc I c IIII II ~W ~ I L IL(% Ay~l~ I / U I II~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~Ik (D~~~~~~I 1I 7~ —— ~.-ln r I -lIIII ~IT rIIII ~rIrrrII- nrrI\rI n~Irr r -nIur~ ~ lr m IUI M. 7 71 1- I1 n-n T Ill Ii I I I I I L 1 11 1 III I iI I Y lK hiI IIu I c f - O u~ rr Ilr I 1 rlrlr III' "'' 1-LL —-- rlll r-I I II -rnlr 71 1 1l1 II-111 rr n l I rll rIr~l~ l I~r 1 I lI\ I r Ivr~rr ~r r~l I\I~~fl~If Ill I I eA 1- rr I r (D a S~~~Drl 11711T 111 1 r 1 IIIIIIIIII IIIIIIIL ILLIL LIIII L _I _L L -_1\ IN~\I/lI IIIunIn~ Io o~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ "IAW_"11r CL ~ ~ ~ ~ c l1 r 1 1 - ll rll ~ I1 1 11I IiI1I IIII )IIIIIjII I \)IIIIM \ I1 IY \ bo~~~~~ —-, o o o o o o ~~~~~~~~~~~~~~~~~~~~~~~'-`b~~~~~~~~~~-~~~~ ~~~~:-b~~~~/ \ cut Y k\n oj D 0-4 IP~~~~~~~~~J

E EI THE E-FUNCTION, ER + EI = E(u + iv); E - -, El - k - sin 15* = 0.258819; K = 1.598142, K' = 2.768063, K' - E' = 1.691658; E = 1.544150, E' = 1.076405 K 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.o v K' E 0 0.1034 0.2065 0.3090 0.4105 0.5110 0.6105 0.7089 0.8064 0.9034 1.000 0 0 0 0 0 0 0 0 0 0 0 0.1 EA 0 0.1040 0.2075 0.3104 0.4121 0.5127 0.6120 0.7102 0.8074 0.9059 1.000 Ef 0.1639 0.1636 0.1628 0.1615 0.1598 0.1580 o.1563 o.1547 0.155 0.1527 o.1524 0.2 0 0.1058 0.2109 0.3150 0.4175 0.5182 0.6172 0.7146 0.8105 0.9055 1.000 0 E 0.3297 0.3290 0.3270 0.3259 0.3201 0.3160 0.3120 0.3084 0.3055 0.3057 0.3031 0.3 EA 0 0.1094 0.2178 0.3243 0.4282 0.5293 0.6275 0.7231 0.8166 0.9087 1.000 E 0.4997 0.4983 o.4945 0.4887 0.4815 0.4737 0.4662 0.4596 0.4544 0.4512 0.4501.4 E 0.1164 0.2308 0.3416 0.4479 0.5492 0.6457 0.7381 0.8272 0.9142 1.000 Ef 0.6780 0.6755 0.6684 0.6576 o.6445 0.6307 0.6176 0.6062 0.5975 0.5921 0.5902 0.5 E 0.1296 0.2551 0.3734 0.4830 0.5838 0.6766 0.7630 0.8445 0.9230 1.000 Ei 0.8721 0.8672 0.8536 o.8334 0.8097 0.7855 0.7652 0.7445 0.7306 0.7220 0.7191 0.6 0 0.1560 0.3024 0.4330 0.5458 0.6427 0.7270 0.8021 0.8711 0.9364 1.000 Ef6 1.097 1.087 1.059 1.020 0.9759 0.9359 0.8973 0.8681 0.8472 0.8346 0.8305 0.7 E 0.2154 0.4026 0.5490 0.6580 0.7397 0.8043 0.8588 0.9080 0.9546 1.000 E- 1.389 1.364 1.300 1.217 1.135 1.064 1.008 0.9665 0.9384 0.9222 0.9169 08 EA O 0 3802 0.6433 0.7834 0.8504 0.8851 0.9086 0.9299 0.9520 0.9756 1.000 Ef 1.855 1.773 1.590 1.402 1.253 1.147 1.074 1.025 0.9945 0.9776 0.9722 0.9 E o 0 1.080 1.292 1.213 1.118 1.053 1.014 0.9961 0.9908 0.9936 1.000 E 5.030 2.497 1.811 1.431 1.236 1.130 1.069 1.053 1.011 0.9998 0.9963 1.0 Ei' 4.121 2.159 1.549 1.276 1.137 1.063 1.024 1.007 1;001 1.000 Ei0 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

> 6 a, cO~~~~~~~~~~~~W:3 o o~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~C) 0 C C =Ilx ~~~~~I~~I I~~~lrU LL~~~~~r Ur 4w "/w III I X ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ " I A - o a a a a 0 a 6 a X X~~~~~~~~~~~~~~~' 0 \. \/ N k IL I\II I I iI X1IT rl~Yfl~~~TnT-~;~TnTT,~ v I I~~~~~~~~~~~ u ~ ~ ~ ~ ~ ~ ~ u AuII Iv r II\ L IIlr I I!I /N Yi1 I Inrm~m~77rr r r rr rT~LT~t~~( o I~1Jln._I I INII I/yl/~yyyyn\vlvr I1~I ~lvr I1.I\ Iurr I I ILILILUII 1IIII I 1 rrr irmI rTTI I ~7-1ID I1, Z_ - I LttH O O~~~~~~~~~~~~~~~~~~~~~~'r 0 X O InLdn u I 1:1 4. —I- I I I L

ER E THE E-FUNCTIOM, ER + =EI - E(u + iv); ER = -g, EI, K E, k a sin 20~ = 0.342020; K - 1.620026, K' = 2.504550, K' - E' = 1.386172; E = 1.523799, E' = 1.118378 K 0 0.1 0.2 0.5 0.4 0.5 0.6 0.7 0.8 0.9 1.0 KK K' E 0 0.1062 0.2118 0.3162 0.4189 0.5198 0.6187 0.7159 0.8115 0.9060 1.000 0 0 0 o o 0 0 0 0 0 0 0.1 0 0.1070 0.2133 0.3182 0.4212 0.5222 0.6209 0.7177 0.8128 0.9067 1.000 E 0.1811 0.1806 0.1789 0.1765 0.1752 0.1698 0.1664 0.1654 o.1611 0.1597 0.1592 EA0. 0.1096 0.2181 0.3247 0.4287 0.5299 0.6281 0.7237 0.8171 0.9089 1.000 0.2 Ei 0.5651 0.5368 o. 600 0.3542 0. 470 0.3395 o.5518 o0.255 0.5202 0.5169 0. 158 0.3 0 0.1147 0.2277 0.3376 0.4434 0.5447 0.6417 0.7349 0.8250 0.9130 1.000 5 E 0.5556 0.5551 0.5460 0.5553 0.5222 o.5083 0.4950 o.4835 0.4746 0.4691 0o. 4672 0.4 E 0 0.1242 0.2452 0.3606 0.4691 0.5703 0.6647 0.7535 0.8380 0.9197 1.000 E 0.7585 0.7540 0.7411 0.7220 0.6993 0.6757 0.6557 0.6550 0.6210 0.6125 0.6094 E05 0 0.1415 0.2768 0.4011 0.5128 0.6122 0.7013 0.7824 0.8578 0.9298 1.000 ~.5 Ef 0.9841 0.9755 0.9518 0.9173 0.8780 0.8390 0.8059 0.7755 0.7545 0.7416 0.7575 0.6 0 0.1755 0.3362 0.4736 0.5866 0.6793 0.7569 0.8245 0.8859 0.9438 1.000 Ef 1.253 1.256 1.188 1.124 1.055 0.9915 0.9581 0.8968 0.8677 0.8506 0.8449 0.7 0 0.2500 0.4574 0.6076 0.7100 0.7815 0.8356 0.8808 0.9219 0.9613 1.000 Ef 1.614 1.572 1.466 1.338 1.217 1.118 1.042 0.9885 0.9527 0.9325 0.9257 E0 0.4518 0.7336 0.8584 0.9041 0.9216 0.9329 0.9456 0.9614 0.9800 1.000 8 EI 2.212 2.076 1.1789 1.518 1.519 1.184 1.095 1.057 1.002 0.9819 0.9756 E0 B 0 1.262 1.402 1.267 1.146 1.068 1.024 1.002 0.9940 0.9950 1.000 E 5.767 2.918 1.965 1.495 1.265 1.145 1077 1.077 1.013 1.001 0.9967 1.0 EA o 4.118 2.158 1.549 1.276 1.136 1.062 1.024 1,007 1.001 1.000 EIf * 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.00o 1.000 1.000

0~~~~~ 0~~~~~~~~~~~~~~~~I 0 — H m O L ru P UI;T, 03 (D O~~~~~~~~~~~~~~~~~~ I00 o CI ~~~~~ 0 ~. m~rl + - - I (D Ir'__ ~o - PO F~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~0 ~ C) I~~~~~~~~(, )- IN (0 -T - c:L g + to ~~~~t~~~_J ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~'0 I0 o ~~~~~~21 P P~~~~~~~~~~~~~~~~~~~~~~~~~~~"1I rV~~~~~~~~~C) m/a~~~~~~~~~~~O I OD II, C ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~It ~ ~ ~ ~ ILClX P,,~~~~~~~~~~~~~~~~~~~~~~~~~~~ P_ O Z

ER EI THE E-FUNCTION, ER + iEI = E(u + iv); E = E-, Ef =' - k - sin 25~ = 0.422618; K = 1.648995, K' = 2.308787, K' - E' = 1.144959; E = 1.498115, E' = 1.163828 U 0 0.1 0.2 o.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 K K'' 0 0.1099 0.2188 0.3257 0.4300 0.5313 0.6295 0.7249 0.8180 0.9094 1.000 0 0 0 0 0 0 0 0 0 0 0 0.1 E 0 0.1109 0.2207 0.3283 0.4330 0.5343 0.6323 0.7273 0.8197 0.9103 1.000 Ei 0.2023 0.2015 0.1984 0.1940 0.1885 0.1827 0.1771 0.1721 0.1683 0.1659 0.1651 0.2 E 0 0.1144 0.2271 0.3368 0.4427 0.5441 0.6413 0,7347 0.8249 0.9130 1.000 Ef 0.4087 0.4064 0.3998 0.3899 0.3778 0.3650 0.3527 0.3420 0.3338 0.3287 0.3269 0.3 A 0 0.1211 0.2595 0.3533 0.4611 0.5626 0.6580 0.7482 0.8344 0.9179 1.000 Ef 0.6244 0.6201 0.6081 0.5901 0.5686 0.5461 0.5249 0.5069 0.4932 0.4847 0.4818 0.4 EA 0 0.1331 0.2616 0.3820 0.4925 0.5931 0.6849 0.7697 0.8492 0.9254 1.000 Ef 0.8575 0.8497 0.8285 0.7974 0.7611 0.7244 0.6908 0.6629 0.6423 0.6296 0.6254 0.5 E 0 0.1549 0.3005 0.4308 0.5438 0.6410 0.7257 0.8013 0.8706 0.9362 1.000 Ef 1.121 1.107 1.069 1.014 0.9540 0.8958 0.8450 0.8045 0.7754 0.7579 0.7521 0. E 0 0.1967 0.3720 0.5150 0.6266 0.7138 0.7844 0.8447 0.8990 0.9502 1.000 EI 1.444 1.416 1.341 1.242 1.141 1.051 0.9783 0.9233 0.8855 0.8655 0.8562 0.7 0 0.2867 0.5131 0.6657 0.7574 0.8180 0.8621 0.8991 0.9355 0.9667 1.000 E 1.888 1.822 1.659 1.470 1.303 1.171 1.075 1.008 0.9648 0.9403 0.9325 0.8 E. O 0.5259 0.8196 0.9241 0.9485 0.9507 0.9518 0.9576 0.9685 0.9833 1.000. E 2.645 2.432 2.008 1.640 1.386 1.221 1.116 1.49 1.008 0.9853 0.9782 0.9 EA 0 1.435 1.492 1.308 1.167 1.079 1.030 1.005 0.9962 0.9960 1.000 E 4.660 3.383 2.123 1.557 1.295 1.161 1.085 1.041 1.015 1.001 0.9971 1.0 E' Go 4.115 2.157 1.548 1.276 1.136 1.062 1.024 1.007 1.001 1.000 Ei.0 1000. 1.000 1. 1.000 1.000 1.000 1.000 1.000

U K 0 0.1 0.2 0.3 0.4 0.5 E 0.6 0.7 0.8 0.9 1.0 1.0 1.0 k= 0.1 O.4226 06 0.1 II1 1 0.2 0.3 ~ 0. 0.5 0.6 0 0.8 0.9 1 ~~~~~~~0.31 ~ ~ ~ 0,i NN'Co o OI O 0 0. 1 0.2 0.3 0,4 0.5 0,6 0,7 0,8 0.9 1.0 k = sn 25'-0.4226

ER El THE E-FUNCTION, ER* +iEI = E(u + iv); EA = -, EI = K _ k sin 350 = 0.500000; K = 1.685750, K' = 2.156516, K' - E' = 0.945460; E = 1.467462, E = 1.211056 u 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 o.8 0.9 1.0 K' Ei 0 0.1146 0.2276 0.3377 0.4439 0.5456 0.6429 0.7562 0.8261 0.9136 1.000 E 0 0 0 0 0 0 0 0 0 0 0 0. EA 0 0.1159 0.2301 0.3410 0.4476 0.5494 0.6463 0.7389 0.8280 0.9146 1.000 0.1 Eo 0.2290 0.2273 0.2226 0.2154 0.2066 0.1974 0.1886 0.1810 0.1752 0.1716 0.1704 E02 0 0.1202 0.2381 P.3516 0.4594 0.5611 0.6569 0.7475 0.8340 0.9177 1.000.2 E 0.4636 0.4599 0.4492 0.4333 o.4141 0.3941 0.3752 0.3591 0.3469 0. 393 o. 5567 E 0 0.1286 0.2535 0.5717 o.4815 0.5828 0.6762 0.7629 0.8447 0.9232 1.000 0 Eif 0.7109 0.7041 o.6848 o.6564 0.6229 0.5887 0.5570 0.5506 o.5108 0.4987 0.4946 E 0.1435 0.2803 0.4058 0.5181 0.6176 0.7062 0.7865 o.8608 0.9313 1.000 ~4 E 0.9814 0.9694 0.9359 0.8877 0.8328 0.7785 0.7501 0.6908 0.6621 0.6448 0.6390 E 0 0.1699 0.3268 0.4626 0.5761 0.6704 0.7500 0.8197 0.8829 0.9423 1.000 05 E 1.293 1.271 1.212 1.129 1.041 0.9580 0.8879 0.8333 0.7947 0.7720 0.7644 E06 0 0.2200 0.4102 0.5576 0.6661 0.7468 0.8099 0.8631 0.9109 0.9560 1.000 o.6 Ei 1.682 1.638 1.525 1.580 1.257 1.115 1.019 0.9491 0.9016 0.8743 0.8653 E0 0.3265 0.5703 0.7184 0.8013 0.8505 0.8852 0.9146 0.9429 0.9713 1.000 7 E 2.231 2.128 1.885 1.619 1.396 1.228 1.109 1.028 0.9758 0.9468 0.9375 E08 O 0.6038 0.9025 0.9831 0.9863 0.9747 0.9671 0.9673 0.9743 0.8960 1.000 Ef 3.185 2.860 2.257 1.771 1.456 1.259 1.17 1.060 1.013 0.9881 o.9801 E0 A 0 1.601 1.569 1.540 1.182 1.088 1.035 1.008 0.9979 0.9968 1.000 09 E 5.772 3.912 2.292 1.625 1.328 1.178 1.094 1.045 1.017 1.002 0.9974 E1' 4.110 2.155 1.547 1.275 1.136 1.062 1.024 1.007 1.001 1.000 1 Ef G 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

U K 10 0.1 0.2 0.3 0.4 0.5 E 0.6 07 0. 0.9 The EXfunction. ER+iE E(uiv)= E(0) fdn2codc4 0.9k 3 k=0.8.~~~~~~~~~ sin 30~= 0.5000.8 0.7 0. 6 Xl'N X \l_.8 09.O K

ER EI THE E-FUNCTION, ER + iEI = E(u + iv); E = -, Ef E= K' I k = sin 35~ = 0.573576; K = 1.731245, K' = 2.034715, K' -E' = 0.776035; E = 1.432291, E' = 1.258680 K 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.o K' E' o 0.1205 0.2387 0.3526 0.4610 0.5631 0.6591 0.7496 0.8357 0.9186 1.000 0 E 0 0 0 0 0 0 0 0 0 0 0 o O O o o o o o O O O 0.1 EA 0 0.1221 0.2417 0.3566 0.4654 0.5675 0.6630 0.7527 0.8378 0.9197 1.000 Ei 0.2634 0.2607 0.2532 0.2420 0.2286 0.2146 0.2015 0.1904 0.1820 0.1769 0.1751 E02 0.1274 0.2513 0.3692 0.4792 0.5810 0.6750 0.7623 0.8444 0.9231 1.000 Ef 0.5543 0.5284 0.5117 0.4870 0.4578 0.4278 0.4001 0.3769 0.3596 0.3490 0.3454 g, E 0 0.1375 0.2698 0.3929 o.5048 0.6055 0.6963 0.7791 0.8559 0.9289 1.000 E 0.8222 0.8115 0.7816 0.7381 0.6879 0.6377 0.5924 0.5552 0.5279 0.5113 0.5057.4 E 0 o0.1554 0.3016 0.4325 0.5462 0.6439 0.7287 o.8040 0.8726 0.9373 1.000 E 1.141 1.122 1.071 0.9982 0.9177 0.8402 0.7750 0.7195 0.6812 o.6583 0.6507 0 EA 0 0.1869 0.3558 0.4969 0.6100 0.7002 0.7743 0.8378 0.8948 0.9482 31.000 0 5 E 1.514 1.480 1.390 1.268 1.142 1.028 0.9339 0.8625 0.8131 0.7842 0.7747 0.6 E' 0 0.2458 0.4512 0.6016 0.7055 0.7786 0.8340 0.8802 0.9217 0.9613 1.000 Ef 1.987 1.920 1.751 1.543 1.347 1.186 1.063 0.9751 0.9168 o.8836 0.8728 0.7 E 0.3698'0.6296 0.7719 0.8423 0.8800 0.9056 0.9282 0.9512 0.9752 1.000 E 2.668 2.512 2.159 1.792 1.499 1.288 1.144 1.047 o.9862 0.9525 0.9417.8 E 0 0.6863 0.9830 1.036 1.019 0.9949 0.9798 0.9752 0.9790 0.9882 1.000 Ef 3.873 3.389 2.546 1.918 1.532.01 1.159 1.072 1.019 0.9905 0.9816.9 E o 1.761 1.654 1.366 1.194 1.094 1.039 1.011 0.9993 0.9974 1.000 E 7.187 4.528 2.481 1.700 1.36.196.4 1.050 1.019 1.003 0.9976 1.0 E = 4.100 2.151 1.545 1.273 1.135 1.062 1.024 1.007 1.001 1.000 Ef 1.000 1.000 1. 000 1. 000 1.000000 1.000 1.000 1.000

u K 0 0.1 0.2 0.3 0.4 0.5 E'= 1.0-70.6 0.7 0.8 0.9 1.0 1.0 1.0 r0i-1 _\ VI KV hr tt " T =T IUt rL —tt T I Ili' 11:!^! *' 1iu^^-o9I U ~~E0.8tttXf-tt+ttt Ylr Ii v \ 1 I4 0.8E 0The E function0. ER+i EI = E(U + v) = E ( ) = dn7 d> Contours of ER=-,and EI = k= sin 35 =0.5736 k = sin 35" =0.5761

ER E_ THE E-FUNCTION, ER + iEI = E(u + iv);ER =, EI = K, -E' k - sin 40~ = 0.642788; K = 1.786769, K' = 1.935581, K' - E' = 0.630042; E = 1.593140, E' = 1.305539 o0 0.1 0.2 o.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 V v K' E. 0 0.1277 0.2522 0.3707 0.4815 0.5840 0.6783 0.7654 0.8468 0.9244 1.000' E 0 0 0 0 0 0 0 0 0 0 0 0. EA 0 0.1297 0.2557 0.3753 0.4866 0.5888 0.6825 0.7687 0.8491 0.9256 1.000 E1 0.3088 0.3047 0.2930 0.2758 0.2556 0.2350 0.2161 0.2005 0.1888 0.1817 0.1793 0.2 E 0 01360 0.2672 0.3900 0.5024 0.6039 0.6956 0.7790 0.8561 0.9291 1.000 Ef 0.6277 0.6185 0.5927 0.5550 0.5115 0.4679 0.4284 O.3960 O.3723 0.3579 0.3531 0.3 0 o.1481 0.2889 0.4173 0.5311 0.6308 0.7184 0.7966 0.8680 0.9350 1.000 Ef 0.9689 0.9524 0.9066 0.8411 0.7674 0.6954 0.6321 0.5813 0.5447 0.5228 0.5155 0.4 0.0 1692 0.3258 0.4623 0.5768 0.6720 0.7524 0.8221 0.8848 0.9434 1.000 Ei 1.351 1.322 1.244 1.136 1.020 0.9121 0.8207 0.7497 0.6999 0.6705 0.6608 E 0 0.2061 0.3880 0.5339 0.6454 0.7307 0.7985 0.8556 0.9065 0.9540 1.000 Ei 1.804 1.752 1.616 1.440 1.263 1.108 0.9846 0.8931 0.8310 0.7952 0.7835 E0.6 0.2747 0.4955 0.6471 0.7447 0.8094 0.8569 0.8961 0.9318 0.9661 1.000 Ei 2.388 2.286 2.035 1.741 1.475 1.265 1.110 1.002 o.9315. 8918 0.8790 0.7 0.4174 0.6912 0.8243 0.8808 0.9068 0.92^39 0.9403 0.9586 0.9788 1.000 Ef 3.241 3.006 2.496 1.996 1.618 1.356 1.182 1.068 0.9964 0.9575 0.9451.8 EA 0 0.7743 1.061 1.085 1.047 1.012 0.9906 0.9820 0.9830 0.9900 1.000 Ef 4.773 4.054 2.891 2.088 1.619 1.347 1.184 1.084 1.025 0.9928 0.9828 0.9 EA 0 1.916 1.689 1.387 1.204 1.099 1.042 1.012 1.000 0.9979 1.000 E 9.038 5.265 2.699 1.788 1.406 1.218 1.115 1.056 1.021 1.005 0.9978 1. E 4.085 2.144 1.541 1.271 1.134 1.061 1.024 1.007 1.001 1.000 Ef 1 1.000 1.000 1.000 1.000 1. 1.000 1.000 1.000 1.000 1.000

U K 0.7 07 0 0.1 0.2 0.3 0.4 0.5 0 E.O6 0 P.7, 0.8 0.9..o --- -0.5 0.5 K' -_`+ t+ / 1 ^-YI --— I 0.4 0. I I A I 0.4 0 O.I 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 2. K The E-function, ER +iEi =E(u+iv)=E( ) = jdn Idc N_ EI Contours of Es i E and EI K -= E k= sin 40~ =0.6428

ER EI THE E-FUNCTION, ER + iEI = E(u + iv); EA = E, EI = K' - E' k = sin 45~ = 0.707107; K = K' = 1.854075, K' - E' = 0.503431; E = E' = 1.350644 U 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 K' 0 E' 0 0.1365 0.2685 0.3924 0.5059 0.6084 0.7004 0.7835 0.8596 0.9310 1.000 E' 0 0 0 0 0 0 0 0 0 0 0 0.1 EA 0. 0.1388 0.2727 0.3977 0.5116 0.6137 0.7050 0.7870 0.8619 0.9321 1.000 Ef 0.3704 0.3640 0.3460 0.3198 0.2898 0.2599 0.2332 0.2115 0.1957 0.1862 0.1831 0.2 0 0.1463 0.2860 0.4144 0.5292 0.6301 0.7188 0.7976 0.8690 0.9357 1.000 E 0.7542 0.7399 0.7003 0.6454 0.5792 0.5164 0.4612 0.4170 0.3852 0.3662 0.3599 0.3 ER 0 0.1605 0.3111 0.4453 0.5608 0.6589 0.7426 0.8155 0.8809 0.9416 1.000 Ef 1.168 1.142 1.072 0.9740 0.8667 0.7649 0.6779 0.6099 0.5618 0.5334 0.5240 o.4 EA 0 0.1852 0.3535 0.4954 0.6101 0.7020 0.7771 0.8409 0.8974 0.9496 1.000 EI 1.635 1.590 1.472 1.313 1.147 0.9978 0.8752 0.7825 0.7187 0.6816 0.6695 E0 0 0.2281 0.4239 0.5737 0.6824 0.7618 0.8228 0.8733 0.9179 0.9596 1.000 EI 2.195 2.116 1.913 1.657 1.410 1.202 1.042 0.9262 0.8491 0.8051 0.7909 0.6 ER 0 0.3072 0.5435 0.6943 0.7837 0.8392 0.8786 0.9112 O.9413 0.9707 1.000 Ef 2.928 2.773 2.403 1.987 1.629 1.358 1.164 1.03i 0.9465 0.8993 0.8842 0.7 E 0 0.4700 0.7552 0.8757 0.9169 0.9313 0.9404 0.9512 0.9653 0.8919 1.000 E 4.013 3.658 2.922 2.245 1.758 1.454 1.224 1.090 1.007 0.9621 0.9479 0.8 ER 0 0.8682 1.137 1.128 1.072 1.027 0.9997 0.9878 0.9865 0.9917 1.000 Ef 5.987 4.917 3.313 2.290 1.722 1.401 1.212 1.098 1.031 0.9950 0.9838 E.9 0 2.066 1.734 1.402 1.210 1.102 1.044 1.014 1.001 0.9984 1.000 Ei 11.53 6.173 2.961 1.893 1.456 -1.244 1.128 1.062 1.024 1.004 0.9979 1.0 EA 0 4.062 2.133 1.535 1.268 1.132 1.060 1.023 1.007 1.001 1.000 EI 0 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

u K 0 0.1 0.2 0.3 0.4 0.5 E' 1.70. 6 0.7 0.8 0.9 1.0 0 7 1.0 0 0 0 E t - i -; t - E a d'K' E ~~~~~~~~~k0.9 0 0.8 0.9 1.0 0.7 0.7 NMI 0 - 0 0.k 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 A.0 U The E-function. E +iE =E(u iv)- E() = dn dw _~ Contours of E E and E I. 0in Ak=sin45~=0.7071 i ~~~~~otuso.=-k-T - n = K-Z)o~~~ = 10 si k5 =I 77

ER EI THE E-FUNCTION, ER + iEI = E(u + iv); EA = E-, Ej = K' - E k = sin 50~ =0.766044; K = 1.935581, K' = 1.786769, K' -E' = 0.393629; E = 1.305539, E' = 1.393140 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 K' E' 0 O 0.1472 0.2882 0.4183 0.5347 0.6368 0.7259 0.8040 0.8739 09 9383 1.000 E 0 0 0 0 0 0 0 0 0 0 0 I EA 0 0.1499 0.2930 0.4243 0.5408 0.6425 0.7305 0.8075 0.8762 0.9394 1.000 E1 0.4568 0.4467 0.4187 0.3787 0.3341 0.2910 0.2535 0.2240 0.2029 0.1905 0.1864 0.2 EA 0.1586 0.3084 0.4430 0.5600 0.6597 0.7447 0.8181 0.8832. 9429 1.000 Ef 0.9315 0.9091 0.8476 0.7613 o.6666 0.5768 0.5002 0.4405 0.3986 0.3739 0.3658 0.3 E 0 0.1752 0.3371 0.4774. 5941 0.6897 0.7688 0.8358 0.8946 0.9485 1.000 EI 1.446 1.406 1.298 1.151 0.9942 0.8508 0.7321 0.6418 0.5795 0.5433 0.5314 E 0.2038 0.3851 0.5324 0.6463 0.7338 0.8030 0.8603 0.9102 0.9560 1.000 EI 2.032 1.963 1.782 1.546 1.309 1.103 0.9392 0.8191 0.7383 0.6920 0.6769 5 EA 0 0.2534 0.4639 0.6167 0.7210 0.7934 0.8471 0.8908 0.92935 0.9652 1.000 E~ 2.743 2.620 2.312 1.940 1.595 1.316 1.109 0.9630 0.8678 0.8144 0.7972 0.6 EA 0 0.3441 0.5955 0.7429 0.8224 0.8681 0.8994 0.9255 0.9503 0.9751 1.000 EI 3.683 3.444 2.891 2.302 1.820 1.468 1.226 1.064 0.9623 0.9064 0.8885 0.7 0 0.5284 0.8215 0.9256 0.9505 0.9537 0.9554 O.9611 0.9713 O.9848 1.000 7 E 5.091 4.547 3.477 2.556 1.929 1.528 1.274 1.115 1.019 0.9666 0.9502 8 Et 0 0.9683 1.209 1.167 1.093 1.039 1.007 0.9927 0.9895 0.9931 1.000 EI 7.680 6.069 3.848 2.539 1.846 1.465 1.245 1.115 1.038 0.9973 0.9846 0.9 E 0 2.207 1.771 1.413 1.214 1.104 1.045 1.015 1.002 0.9988 1,000 EI 15.00 7.329 3.286 2.023 1.158 1.276 1.145 1.070 1.027 1.005 0.9980 E 00 4.027 2.118 1.526 1.263 1.129 1.059 1.023 1.006 1.001 1.000 E 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

U K I.O.._. 0q.T5 E: =0,-70 6 1 1 1 fini iE i I i1 i) 1 l P91 11101 10 0.9 0.I8 K' 0E a E' E1 0.3 E k sin 5~~~~~~~~~~~~~~~~~0.30 0.20 0 r i ttS L I;S g 0.2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 U K The E -function, ER +iEI =E(u+iv)= E () = | dn2d "EAr, E\ Contours of E = - and E = K'- E k= sin 50~ =0.7660

ER EI THE E-FUNCTION, ER + IEI = E(u + iv); EA = -, EI = K', -, k = sin 55~ = 0.819152; K = 2.034715, K' = 1.751245, K' -E' = 0.298954; E = 1.258680, E' = 1.452291 U 0 0.1 0.2 0.5 0.4 0.5 0.6 0.7 0.8 0.9 1.0 K K' E' 0.1602 0.3120 0.4490 0.5682 0.6694 0.7546 0.8268 0.8897 0.9463 1.000 E 0 0 0 0 0 0 0 0 0 0 0 E; O O O O O O O O O O O 0 E 0.1633 0.5175 0.4557 0.5748 0.6752 0.7593 0.8303 0.8919 0.9474 1.000 E01 E 05831 0.5668 0.5224 o.4606 0.3935 0.35510 0.2785 0.2584 0.2107 0.1945 0.1893 EA 0.1754 0.5549 0.4764 0.5953 0.6930 0.7734 0.8405 0.8984 0.9506 1.000 2 1.191 1.155 1.057 0.9245 0.7832 0.6542 0.5479 0.4678 0.4129 0.5812 0.5709 EA o 0.1926 0.3674 o.5140 0.6512 0.7255 -0.7970 0.8573 0.9091 0.9557 1.000 0. Ef 1.852 1.789 1.619 1.395 1.163 0.9601 0.7980 0.6787 0.5985 0.5526 0.5377 E0 0 0.2257 0.4214 0.5735 0.6854 0.7675 0.8299. 8805 o.9254 0.9626 1.000 E.4 2.612 2.502 2.220 1.864 1.521 1.255 1.017 0.8613 0.7595 0.7018 0.6855 0 E 0.2826 0.5087 0.6628 0.7611 0.8254 0.8714 0.9082 0.9405 0.9706 1.000 0 E5 543 5. 47 2.871 2.320 1.855 1.459 1.189 1.006 0.8881 0.8252 0.8025 0. E 0 O.5860 O.6516 0.7926 0.8605 0.8958 0.9192 0.9592 0.9589 0.9795 1.000 0 E6 4.784 4.406 5.566 2.719 2.063 1.66 1.1.00 1.102 0.9796 0.9152 0.8922 EA 0.5955 0.8895 0.9755 0.9815 0.9757 0.9688 0.9701 0.9769 0.9875 1.000 Ej 6.663 5.810 4.228 2.961 2.145 1.645 1.335 1.145 1.051 0.9710 0.9521 E 0 1.075 1.277 1.199 1.110 1.049 1.014 0.9970 0.9922 0.9945 1.000 E8 10.15 7.668 4.551 2.858 2.004 1.546 1.287 1.154 1.046 0.9997 0.9852 E 0 2.557 1.796 1.417 1.214.4 45 1.104 1.045002 0.999 1.000 0 9 E 20.06 8.858 5.709 2.193 1.599 1.316 1.165 1.080 1.031 1.006 0.9981. E o 35.976 2.095 1.513 1.255 1.125 1.057 1.022 1.006 1.001 1.000 1 E0 E 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

u K 0 0.1 0.2 0.3 0.4 0.5 E' 1. 0.6 0.7 0.8 0.9. 1.0 \ \w. IIT. Contours of ER=^.and EI = K, IE k sin 55 0.8192 A IKL- X 0,1 0.6 0 0 K Contours of E.= E,and E I K'- I

ER EI THE E-FUNCTION, ER + EI = E(u + iv); EA = -, E = K' - E k = sin 60~ = 0.866025; K = 2.156516, K' = 1.685750, K' - E' = 0.218288; E = 1.211056, E' = 1.467462 K 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 K v K' E O 0.1760 o.3406 0.4855 0.6072 0.7064 0.7866 0.8519 0.9067 0.9550 1.000 EI 0 0 0 0 0 0 0 0 0 0 0 0.1 EA 0.1797 0.3468 0.4927 0.6141 0.7123 0.7911 0.8551 0.9088 0.9559 1.000 E~ 0.7778 0.7507 0.6778 0.5792 0.4763 0.3843 0.3103 0.2557 0.2192 0.1984 0.1917 0.2 E 0 0.1914 0.3665 0.5153 0.6355 0.7301 o.8046 o.8646 0.9147 0.9587 1.000 E 1.590 1.530 1.371 1.161 0.9454 0.7570 0.6084 0.5005 0.4288 O.3883 0.3753 0.3 EB 0 0.2135 0.4030 0.5559 0.6726 0.7602 0.8270 0.8800 0.9242 0.9633 1.000 E 2.479 2.373 2.097 1.743 1.396 1.105 0.8813 0.7230 0.6196 0.5617 0.5431.4 E 0 0.2515 0.4629 0.6191 0.7275 o.8028 0.8577 0.9007 0.9368 0.9692 1.000 Ef 3.507 3.325 2.868 2.317 1.811 1.409 1.114 0.9119 0.7827 0.7114 0.6886 0 E 0 0.3166 0.5587 0.7119 0.8023 0.8575 0.8955 0.9253 0.9515 0.9761 1.000 E0 4.774 4.452 3 691 2.856 2.159 1.645 1.291 1.057 0.9110 0.8319 0.8069 0.6 BA 0 0.4339 0.7119 0.8430 0.8975 0.9222 0.9379 0.9521 0.967.2 0.9833 1.000 E6 6.481 5.859 4.542 3.298 2.389 1.784 1.394 1.148 0.9995 0.9201 0.8951 E0 0. 6657 0.9582 1.018 1.009 0.9914 0.9807 0.9782 0.9821 0.9901 1.000 E0 9.082 7.695 5.292 3.512 2.432 1.793 1.412 1.182 1.047 0.9758 0.9536 0.8 E 0 1.187 1.338 1.225 1.122 1.056 1.018 1.000 0.9946 0.9957 1.000 Eo 13.94 10.00 5.523 3.290 2.213 1.652 1.340 1.160 1.056 1.002 0.9858 0.9 A 0 2.451 1.809 1.415 1.211 1.102 1.044 1.015 1.003 0.9994 1.000 E 27.85 10.99 4.287 2.422 1.707 1.370 1.192 1.092 1.036 1.007 0.9982.0 E 53.903 2.062 1.495 1.245 1.119 1.054'1.021 1.006 1.001 1.000 Ef 0 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

U K 0 0.1 0.2 0.3 0.4 0.5 E =.o07 0.6 0.9 1.0 01,0 The- E fT I c I I i I IEIRIEIuIv E1u)Ifdn IdI 0:2~~~~~~~~~~~~V- __ -Lt I f:Li — _I_ RKT E: - n, ER +;EI I( Contours of E' ER and E =K E ~~k ~~~~~~= ~sin 60~0 ~ 08660=0.8 0 0.5 V JJ_ 0.4 k 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8660 0 I E, EN Contours of E. =- -.and Ex= KI A I

ER EI THE E-FUNCTION, ER + iEI = E(u + iv); EA = -, Ef = K' E k = sin 65~ = 0.906308; K = 2.308787, K' = 1.648995, K' - E' = 0.150880; E = 1.163828, E' = 1.498115 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 K' V K' O E 0 O 0.1955 0.3753 0.5286 0.6521 0.7481 0.8217 0.8789 0.9249 0.9640 1.000 E 0 0 0 O 0 0 0 0 0 0 0 0.1 ER 0 0.1998 0.3822 0 5364 0.6592 0.7538 0.8259 0.8817 0.9266 0.9649 1.000 EI 1.101 1.053 0.9271 0.7624 0.5985 0.4591 0.3525 0.2775 0.2291 0.2023 0.1937 E0 0 0.2132 0.4043 0.5605 0.6809 0.7709 0.8383 0.8901 0.9516 0.9672 1.000 Ef 2.254 2.148 1.874 1.523 1.183. 9008 0.6887 o0.5415 0.4473 0.3954. 3789 0.3 E 0 0.2388 0.4448 0.6035 0.7181 0.7996 0.8587 0.9035 0.9397 0.9710 1,000 Ef 3.520 3.333 2.858 2.276 1.737 1.305 0.9916 0.7785 0.644i 0.5708 0.5476 E. 0 0.2824 0.5108 0.6693 0.7722 0.8394 0.8861 0.9213 0.9502 0.9758 1.000 Ef 4.991 4.671 3.893 3.003 2.231 1.649 1.243 0.9756 0.8103 0.7211 0.6931 E0.5 0 0.3566 0.6144 0.7638 0.8440 0.8894 0.9191 0.9421 0.9623 0.9814 1.000 EJ 6.817 6.253 4.975 3.657 2.622 1.901 1.424 1.121 0.9383 0.8410 0.8105 0.6 EA 0 0.4891 0.7761 0.8928 0.9324 0.9467 0.9552 0.9642 0.9749 0.9871 1.000 EI 9.293 8.211 6.046 4.149 2.850 2.027 1.518 1.207 1.024 0.9275 0.8975 0.7 0 0.7462 1.026 1.058 1.032 1.006 0.9909 0.9854 0.9868 0.9924 1.000 E 13.09 10.71 6.897 4.311 2.835 1.998 1.514 1.229 1.066 0.9811 0.9549 0.8 EA 0 1.303 1.389 1.243 1.129 1.060 1.022 1.003 0.9966 0.9968 1.000 Ef 20.23 13.63 6.952 3.909 2.508 1.798 1.412 1.193 1.069 1.006 0.9862 0.9 EA 0 2.543 1.805 1.403 1.202 1.097 1.042 1.015 1.003 0.9996 1.000 EI 40.76 14.15 5.129 2.754 1.860 1.445 1.229 1.109 1.043 1.009 0.9982 1. E EA 3.799 2.015 1.469 1.230 1.111 1.050 1.019 1.005 1.001 1.000 E1..000 1.000 1.000 1.000 1.000 1.000 1.000 1000 1.000 1.000 1.000 1.000 1.000

u K 0 0.1 0.2 0.3 0.4 0.5 Ei= 1.070.6 0.7 0.8 09 0.9 4 _S — N,, -~~~~~~~~~~~~~~ ~~0. 9 0.811~~~~~~~~~ itB IX4SX @R~~~~~~~~~~~ INX 1R IX I 1RSC E0.8 0.6 r -1,-rFf - - ifl -~ 1 \i In I \c/I I 1 [I I 9N0 20iFl 0: 0.3 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00 K The E-function, E. + iE, =E(u+iv) = E () = /dn2)dw Contours of ER = E'and E I K'- EIk = sin 65~ =0.9063

ER EI THE E-FUNCTION, ER + EI = E(u + iv); ER = -, E = K' -E' k = sin 70~ = 0.939693; K = 2.504550, K' = 1.620026, K' - E' = 0.0962267; E = 1.118378, E' = 1.523799 u u 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 K' 0 ER 0 0.2199 0.4177 0.5797 0.7036 0.7942 0.8595 0.9072 0.9436 0.9733 1.000 E' 0 0 0 0 0 0 0 0 0 0 0 0.1 EA 0 0.2248 0.4254 0.5879 0.7106 0.7994 0.8631 0.9096 0.9449 0.9739 1.000 Ej 1.697 1.604 1.366 1.070 0.7928 0.5714. 4124 0.3065 0.2413 0.2063 0.1954 E02 0 0.2403 0.4498 0.6132 0.7318 0.8152 0.8739 0.9164 0.9489 0.9758 1.000 Ej 3.475 3.271 2.756 2.131 1.561 1.116 0.8024 0.5961 0.4701 0.4029 0.3819 0.3 E O 0.2697 0.4944 0.6576 0.7678 0.8412 0.8913 0.9274 0.9553 0.9786 1.000 Ef 5.436 5.075 4.189 3.164 2.272 1.605 1.147 0.8526 0.6746 0.5805 0.5512.4 EA 0 0.3197 0.5660 0.7243 0.8191 0.8767 0.9144 0.9417 0.9634 O.9823 1.000 E 7.723 7.107 5.672 4.131 2.885 2.004 1.425 1.061 0.8449 0.7317 0.6966 E05 0 0.4041 O.6762 O.8177 0.8853 0.9201. 9417 0.9582 0.9727 0.9865 1.000 EI 10.58 9.497 7.176 4.958 3,339 2.277 1.613 1.208 0.9730 0.8510 0.8133 0.6 E 0. 5530 0.8433 0.9405 0.9641. 9685 0.9709 0.9754 0.9822 0.9907 1.000 Ej 14.47 12.42 8.582 5.509 3.555 2.386 1.694 1.287 1.055 0.9360 0.8994 E0 0 0.8355 1.090 1.090 1.049 1.017 0.9991 0.9916 0.9911 0.9946 1.000 EI 20.48 16.01 9.542 5.570 3.449 2.301 1.660 1.295 1.091 0.9875 0.9558.8 EH 0 1.420 1.426 1.250 1.129 1.060 1.023 1.005 0.9983 0.9978 1.000 EI 31.81 19.83 9.250 4.876 2.958 2.015 1.515 1.239 1.087 1.010 0.9865 E.9 0 2.601 1.781 1.381 1.188 1.089 1.039 1.014 1.003 0.9998 1.000 E 64.52 19.30 6.471 3.275 2.096 1.558 1.282 1.133 1.051 1.011 0.9983 E1 A 53.653 1.950 1.433 1.210 1.100 1.044 1.017 1.005 1.001 1.000 EI 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

u K 0 0.1 0.2 0.3 0.4 0.5 E.. 0.6 0.7 0.8 0.9 1.0 KK 1.0 snI I OI1.0 The E-function. E- 1+iE I E(u+iv)=E()) =dn'd 04~~~k= sin 70 =.0.9397 0,8 0,8 I1111141F~t~tt1S t YIm2J!Illlll-.Z\0\ N \ NXI3 01 0. 0.2 0.3 0.4 0.5 0.6 0,7 0.8]0.9 1,0 Contours of ER= E- and EI=K, z

ER Ei THE E-FUNCTION, ER + iEI = E(u + iv); ER = E, EI = E,_ I k = sin 75~ = 0.965926, K = 2.768063, K' = 1.598142, K' - E' = 0.0539915; E = 1.076405, E'= 1.544150 K 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 K V K' 5E' O 0.2512 0.4705 0.6408 0.7621 0.8445 0.8989 0.9359 0.9621 0.9824 1.000 0 E' 0 0 0 0 0 0 0 0 0 0 0 EA 0 0.2568 0.4790 0.6491 0.7687 0.8488 0.9018 0.9376 0.9631 0.9828 1.000 0.1 EI 2.984 2.776 2.261 1.661 1.141 0.7589 0.5057 0.3487 0.2576 0.2110 0.1967 E 0 0.2748 o.5056 0.6746 0.7884 0.8623 0.9104 0.9427 0.9659 0.9840 1.000 0.2 Ef 6\116 5.658 4.547 5.290 2.233 1.474 0.9794 0.6759 0.5008 0.4115 0.3842 E. o 0.3087 0.5538 0.7188 0.8215 0.8842 0.9240 0.9508 0.9703 0.9860 1.000 0~3 Ef 9.577 8.768 6.874 4.843 3.220 2.101 1.390 0.9611 0.7162 0.5918 0.5540 E' 0 0.3659 0.6300 0.7859 0.8672 0.9136 0.9419 0.9612 0.9759 0.9884 1.000 ~04 Ef 13.63 12.25 9.226 6.245 4.055 2.591 1.707 1.186.8924 0.7443 0.6993 0 E 0 0.4617 o.7444 0.8720 0.9247 0.9487 0.9626 0. 9730 0.9823 0.9912 1.000 E5 18.71 16.51 11.52 7.358 4.584 2.897 1.907 1.558 1.022 0.8634 0.8155 El 0 0.6278 0.9116 0.9853 0.9908 0.9867 0.9843 0.9852 0.9888 0.9940 1.000 E 25.66 21.15 13.48 7.981 4.772 2.975 1.970 1.407 1100.9470 EA o 0.9558 1.146 1.112 1.059 1.024 1.005 0.9965 0.9948 0.9966 1.000 0.7 EI 36.43 26.82 14.55 7.824 4.504 2.800 1.891 1.395 1.128 0.9961 0.9566 Et 0 1.531 1.441 1.242 1.121 1.056 1.022 1.006 0.9996 0.9987 1.000 Ef 56.83 32.02 15.48 6.596 3.751 2.575 1.681 1.310 1.112 1.016 0.9868 E 0 2.609 1.729 1.344 1.165 1.077 1.053 1.012 1.003 1.000 1.000 9 115.8 28.93 8.915 4.201 2.503 1.746 1.368 1.170 1.065 1.014 0.9983. E' 3.447 1.858 1.383 1.181 1.085 1.037 1.014 1.004 1.000 1.000 1 E0 f c 1.000 1.000 1. 1.000 1. 00 000 1.00000 1.000 1.000 1. 000

u K 0 0.1 0.2 0.3 0.4 0.5 EI= 1.0-7 0.6 0.7 0.8 0.9 1.0 1.0 1._ 09i7 f E ^^^ ^#UIXrl~ t l f c Ift t f 0.7 X 3 1 1 I W _ I 1 7 _r F __n _ _ _ rr _~i _n r _ _vI _ Nrx i iv in A ll iy E 1 I.L\"\I ", "I Ll 1 1 1 1 1 7JI.l C1_11]. _ Z x S Z v U _ r g c g _ _ _ 7 _ _ r _ 0. ioX]t: n } \ ~z i s _ Ys - ts - C- c L I 4I I:111 I I I I II__ 0: 6I: 0.2 -.N I A A 1 I 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 K The E-function,. ER~iEi=E(u~iv)= E(w) =/ dn2ldI Contours of E =E a nd Ez = K,- Il k= sin 75~ =0.9659

ER El THE E-FUNCTION, ER + iEI = E(u + iv); ER = j-, Ej = K' - E' k = sin 80~ = 0.984808; K = 5.155585, K' = 1.582843, K' - E' = 0.0239556; E = 1.040114, E' = 1.558887 u o 0.1 0.2 0.5 0.4 0.5 0.6 0.7 0.8 0.9 1.0 K K' E' o 0.2958 0.5589 0.7148 0.8285 0.8972 0.9585 0.9632 0.9792 0.9906 1.000 E 0 0 0 0 0 0 0 0 0o 0 0 El 0 O. 5004 0.5481 0.7228 0.8558 0.9007 0.9405 0.9643 0.9797 0.9908 1.000. E 6.661 6.o45 4.604 3.080 1.905 1.154 0.6771 0.4205 0.2831 0.2171 0.1976 0 E 0 0.5215 0.5764 0.7470 0.8505 0.9108 0.9461 0.9675 0.9814 0.9915 1.000 02 E 15.66 12.51 9.218 6.062 3.699 2.190 1.305 0.8115. 5489 0.4229 o.3858 E0 0 o 0.3604 0.6270 0.7882 0.8778 0.9271 0.9555 0.9724 0.9859 0.9925 1.000 03 Ej 21.41 19.02 15.82 8.822 5.274 5.089 1.856 1.146 0.7816 0.6075 0.5560.4 E 0 0 4260 0.7055 0.8475 0.9148 0.9485 0.9671 0.9788 0.9871 0.9939 1.000 EI 30.50 26.47 18.51 11.18 6.500 5.756 2.228 1.401 0.9681 0.7619 0.7012 E0 0 0.5543 0.8190 0.9245 0.9598 0.9734 0.9806 0.9859 0.9907 0.9954 1.000 0 E 41.95 54.95 22.41 12.88 7.236 4.124 2.450 1.561 1.100 0.8812 0.8171. E 0 0.7176.0.9772 1.017 1.010 0.9997 0.9945 0.9932 0.9944 0.9969 1.000 Eo6 57.62 44.70 25.50 15.58 7.344 4.142 2.482 1.616 1.172 0.9652 0.9019 E 0 1. 041 1.184 1.118 1.059 1.025 1.007 0.9998 0.9977 0.9985 1.000 E0 82.00 55.23 26.41 12.85 6.724 3,790 2.522 1.571 1.189 1.009 0.9571 0 E 0 1.624 1.424 1.215 1.102 1.046 1.018 1.005 1.000 0.9994 1.000 E8 128.3 62.71 23.30 10.40 5.358 3.092 1.991 1.436 1.156 1.025 0.9870 E 0 2.559 1.658 1.286 1.152 1.059 1.025.0 1.002 1.000 1.000 0 9 E 262.5 51.84 14.52 6.247 3.362 2.121 1.550 1.236 1.087 1.018 0.9983 1. E co 3.152 1.728 1.512 1.142 1.064 1.027 1.010 1.005 1.000 1.000 1.0 E 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

U K 0 0.1 0.2 0.3 0.4 0.5 Ej= 1.0 0.6 0.7 0.8 0.9 1.0 1.0 1.0 0.2 0.3 0 0.7 0.8 0. K0.9.K 0.72 Z j R SX! g!gtt t~5i:EtkWtk s800.94 v i I St tt E 0.4 0.5t4 U~8I jg Ir / \ I Z )r /11, 1E, I ~~~~~~0.28 ~ ~ = -A in AA =X Q~Zl Conours of ER In EI=KEIE YX,~~~~~= i 0~094

ER EI THE E-FUNCTION, ER + iEI = E(u + iv); ER = E, EI K' --' k = sin 85~ = 0.996195; K = 3.831742, K' = 1.573792, K' - E' = 0.00598306; E = 1.012664, E' = 1.567809 U 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 v K' E O 0.3610 0.6576 0.8094 0.9031 0.9507 0.9745 0.9865 0.9930 0.9970 1.000 EI 0 0 0 0 0 0 0 0 0 0 0 I E' E 0 0.688 0.6468 0.8160 0.9068 0.9526 0.9754 0.9869 0. 9932 0.9971 1.000 1 E 26.52 22.93 15.41 8.747 4.532 2.261 1.130 0.5881 0.3365 0.2282 0.1982 0. E 0 0. 935 0.6751 0.8359 0.9180 0.9583 0.9781 0.9882 0.9938 0.9973 1.000 EI.4o 46.54 30.60 17.05 8.734 4.335 2.164 1.130 0.6502 0.4440 o.3868 0.3 E 0 0.4392 0.7246 0.8687 0.9359 0.9673 0.9824 0.9902 0.9947 0.9976 1.000 E 85.30 71.53 45.22 24.41 12.28 6.045 3.016 1.582 0.9202 0.6359 0.5571 E04 0 o. 5148 0.7984 0.9140 0.9595 0.9788 0.9879 0.9928 0.9958 0.9981 1.000 E 121.6 98.57 58.60 30.22 14.84 7.235 3.610 1.911 1.130 0.7951 0.7024 0.5 E 0 0.6361 0.9002 0.9701 0.9870 0.9919 0.9940 0.9956 0.9971 0.9986 1.000 E 167.3 128.1 69.52 33.79 16.12 7.784 3.897 2.094 1.268 0.9156 0.8180 E 0 0.8326 1.032 1.034 1.016 1.005 1.000 0.9985 0.9984 0.9990 1.000 E 230.2 159.6 75.87 34.34 15.90 7.622 3.851 2.120 1.332 0.9955 0.9025 E 0 1.155 1.1898 44 1.08 1.044 1.01 0.9995 0.9995 1.000 E 328.1 188.3 74.49 31.21 14.07 6.747 3.481 1.996 1.323 1.037 0.9574 E.8 0 1.667 1.352 1.156 1.067 1.028 1.010 1.003 1.000 0.9998 1.000 E 514.2 197.7 61.79 24.14 10.73 5.238 2.830 1.743 1.253 1.045 0.9871 0.9' o 0 2.332 1.482 1.196 1.083 1.034 1.013 1.004 1.001 1.000 1.000 EI 1054 145.7 36.07 15.60 6.198 3.248 1.970 1.396 1.138 1.029 0.9984 1.0 E 2.703 1.533 1.211 1.088 1.037 1.014 1.005 1.001 1. 000 1.000 EI o 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

u K 0 0.1 0.2 0.3 0.4 El__o___ 0.5 l 0,,6, ID _ 7 _ 1 1. 0. 910 0 0.9 T011 I x 1).8j 11 I I I I I I 1 I 0.8 0,6 v L t lrt i -~l \=st=-S==cVL:hE^: —:3^?"~::: v V I 1 I 0 O.I 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0. I I. ~~~k=80.419 - k=si-5~=.9 6 o. N I vXo M,, I I I 11UP IFT-17E,and E, K,_ E0.6 I I IIA _0-6 2 I I Y N I 01.6~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ A Ill I X X -T~~ The E-function. E. + i~~~~~~~~~~~~z =E~~~u~~iv) = E (~)) = fdn~)d~)~~~~~~ANL UALA_~~~~~~~~~~~~ EZ 7 N~~~Conor of EI=E, Ez,EE T::~~~~~~~ = Ii 5 096

ER E_ THE E-FUNCTION, ER + iEI = E(u + iv); E = E, E = K' - E k = sin 89~ = 0.999848; K = 5.434910, K' = 1.570916, K' - E' = 0.000239249; E = 1.000752, E' = 1.570677 U 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 K V K' 0:. 0 O.4955 0.7955 0.9256 0.9740 0.9909 0.9967 0.9988 0.9995 0.9998 1.000 E 0 0 0 0 0 0 0 0 0 0 0 E 0 o.5046 0.8025 0.9288 0.9752 0.9915 0.9969 0.9988 0.9995 0.9998 1.000 Ef 662.1 496.4 259.1 92.52 52.72 11.28 3.890 1.585 0.579 0.2624 0.1984 E0 0 0.5337 0.8241 0.9585 0.9787 0.9925 0.9973 0.9989 0.9996 0.9998 1.000 E02 1558 998.8 467.1 177.4 62.45 21.49 7.416 2.641 1.055 0.5089 0.3871 0 EA 0 0.5864 0.8603 0.9556 0.9842 0.9944 0.9979 0.9992 0.9996 0.9999 1.000 0 Ef 2150 1510 671.1 248.2 86.46 29.66 10.24 3.660 1.446 0.7252 0.5575 E E 0 0 0.6698 0.9106 0. 9734 0.9912 0.9969 0.9988 0.9994 0.9997 0.9999 1.000. E 5037 2028 385.2 297.9 102.4 55.00 12.09 4.350 1.748 0.8999 0.7028 E 0 0.7952 3.9758 0.9965 0.9992 0.9995 0.9997 0.9998 0.0998 0.9999 1.000 Ei 4180 2552 959.2 520.7 108.6 56.98 12.80 4.655 1.918 1.026 0.8183 0. E 0 0.9782 1.o46 1.021 1.007 1.002 1.001 1.000 0.9999 0.9999 1.000 E6 5752 2962 959 9 512.5 104.5 55.39 12.31 4.553 1.948 1.100 0.9028 0. E 0 1.255 1.122 1.045 1.015 1.005 1.001 1.000 1.000 1.000 1.000 Ef 8205. 3181 875.6 271.9 89.53 30.36 10.67 4.064 1.847 1.125 0.9575 E' 0 1.559 1.190 1.062 1.021 1.007 1.002 1.001 1.000 1.000 1.000.8 Ef 12870 2918 675.0 201.4 65.71 22.3498.4.244 1.633 1.109 0.9871 0. E 0 1.875 1.238 1.075 1.024 1.008 1.002 1.001 1.000 1.000 1.000 E9 26390 1846 569.7 107.6 55.15 12.27 4.713 2.185 1.358 1.062 0.9984 E. E M 2.016 1.256 1.079 1.026 1.008 1.005 1.001 1.000 1.000 1.000 1.0 E l 00 1.00.1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

u K 0 0.1 0.2 0.3 0.4 0.5 E i 1. 0.6 0.7. 0.8 0.9 1.0 1.0 1.0 V ~r~ttfi~t~thurttl~f~~fCntur o E E E1'4 0.5' j" 1 1 t ^ ^ AL 1 /\ I n' \ 1 ^ ^ ^^ ^^11 _ l__ 1_ll _ _ lll I' 1\ I 171013I 1 1 I 1 1 I ll" I Ii l^ III lil^ IEE ll/ I l/ l ll /1 I n 0'5 1W r i W t i\r 1 1 X i- I s I 1J ^:iT1 YX17 I i IY^ 1f1: 1 1T: I/ I 1/ 11A 17 1 1t 0.8 ^ ^S.::::^::: ^.z84:, h'~'3 o o 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 K The E-function, ER+;EI-E(u+;iv)=E(w ) = Jdn2Mdc Contours of ER Eand EI= K_ E k0.= sin 89 =0.9999

UNIVERSITY OF MICHIGAN 3 9015 03026 75491 3 9015 03026 7549