THE UNIVERSITY OF MICHIGAN INDUSTRY PROGRAM OF THE COLLEGE OF ENGINEERING NOTES ON MATHEMATICS FOR ENGINEERS Prepared by the American Nuclear Society. University of Michigan Student Branch October, 1959 IP-394

ACKNOWLEDGEMEi These notes were prepared by the University of Michigan Student Branch of the American Nuclear Society, Committee on Math Notes, which included R. W. Albrecht, J. M. Carpenter, D. L. Galbraith, E. H. Klevans, and R. J. Macky all students in the University of Michigan Department of Nuclear Engineering. Assistance was provided by Professors William Kerr and Paul Zweifel.

FOREWORD This set of notes has been compiled with one primary objective in mind: to provide, in one volume, a handy reference for a large number of the commonlyused mathematical formulae, and to do so consistently with respect to notation, definition and normalization. Many of us keep these results available to us in an excessive number of references, in which notation or normalization varies) or formulae are so spread out that they are difficult to find, and their use is time-consuming. Short explanations are included, with some examples, to serve two purposes: first, to recall to the user some of the ideas which may have slipped his mind since his detailed study of the material; second, for those who have never studied the material, to make its use at least plausible, and to help in his study of references. No claim can be made that all the results anyone ever uses are here, but it is hoped that a sufficient quantity of material is included to make necessary only infrequent use of other references, except for integral tables, etc. for elementary work. Of course, the user may find it desirable to add some pages of his own. Finally, it is recommended that those unfamiliar with the theory at any point not blindly apply the formulae herein, for this is risky business at best; a text should be studied (and, of course, understood) first.

INDEX Subject Page ORTHOGONAL FUNCTIONS Importance of Orthogonal Functions Generation of Orthogonal Functions4Use of Orthogonal Functions8 Operational Properties of Some Common Sets of Orthogonal Functions 10 I. Fourier Series 10 A. Boundary Value Problem 10 B. Orthogonal Set 11 C. Expansion in Fourier Series 11 D. Normalization Factors 11 E. Orthonormal Set 11 F. Fourier Series, Range 0 x' L 11 G. Expansion in Half-range Series 12 H. Full Range Series 12 II. Legendre Polynomials 13 A. Generating Function 13 B. Recurrence Relations 14 C. Differential Equation Satisfied by P(x) 14 D. Rodriques Formula 14 E. Normalizing Factor 14 F. Orthogonality 14 G. Expansion in Legendre Polynomials 14 H. Normalized Legendre Polynomials 15

-iiI. Expansion in Normalized Polynomials 15 J. A Few Low-degree Legendre Polynomials and Respective Norm 15 K. Integral Representation of PI(x) 15 L, Bounds on P (x) 15 References on Legendre Polynomials 15 III.Associated Legendre Functions 16 A. Definition 16 B. Recurrence Relations 16 C. Differential Equation Satisfied by P? (x) 16 D. Expression of P? (cos @) in terms of Pm 17 E. Normalizing Factor 17 F. References 17 IV,, Spherical Harmonics 18 A. Definition of m (S-) 18 B. Expression of P (cos Q) in terms of the Spherical Harmonics 18 C. Orthonormality of ym (.) 18 D. Expansion in Spherical Harmonics 19 E. Differential Equation Satisfied by Y ) 19 F. Some Low-order Spherical Harmonics 19 G. "A Useful Relationship" 19 H. References 20 V. Laguerre Polynomials 20 A. Derivative Definition 20 B. Generating Function 20 C. Differential Equation Satisfied by LI )(x) 21

-iiiD. Orthogonality, Range 0 x L o 21 E. Expansion in Laguerre Polynomials 21 F. Expansion of x in Laguerre Polynomials 21 G. Recurrence Relations 21 VI. Bessel Functions 23 A. Differential Equation Satisfied by Bessel Functions 23 B General Solutions 23 C. Series Representation 24 D. Properties of Bessel Functions 24 E. Generating Function 26 F. Recursion Formulae 26 G. Differential Formulae 26 H. Orthogonality, Range 26 I Expansion in Bessel Functions 27 J. Bessel Integral Form 27 VII.Modified Bessel Functions 27 A. Differential Equation Satisfied 27 B. General Solutions 27 C. Relation of Modified Bessel to Bessel 28 D. Properties of In 28 References 30 THE LAPLACE TRANSFORMATION 32 I. Introduction 32 A. Description 32 B. Definition 32 C. Existence Conditions 32 D. Analyticity 3 E. Theorems 33

-ivF. Further Properties 34 II. Examples A. Solving Simultaneous Equations 34 B. Electric Circuit Example 36 C. Transfer Functions 36 i- III.Inverse Transformations A. Heaviside Methods 38 B. The Inversion Integral 40 IV. Table of Transforms 44 Appendix A - Analyticity 47 Appendix B Cauchy's Integral Formula 48 Appendix C - Calculation of Residues 50 I. Laurent Series 50 II. Residues 51 III. Determination of Residues 51 Appendix D - Regular and Singular Points 52 FOURIER TRANSFORMS I. Definitions 54 A. Basic Definitions 54 B. Range of Definition 54 C. Existence Conditions 55 II. Fundamental Properties 55 A. Transforms of Derivatives 55 B. Relations Among Infinite Range Transforms 57 C. Transforms of Functions of Two Variables 57 D. Fourier Exponential Transforms of Functions of Three Variables 58

-YIII. Summary of Fourier Transform Formulae 60 A. Finite Transforms 6 B. Infinite Transforms 61 IV. Types of Problems to which Fourier Transform May be Applied 64 A. General Discussion 64 B. Example 64 V. Inversion of Fourier Transforms 69 A. Finite Range 69 B. Infinite Range 70 C. Inversion of Fourier Exponential Transforms 71 VI. Table of Transforms 74 MISCELLAEOUS IDENTITIES, DEFINITIONS, FUNCTIONS AND NOTATIONS I. Leibnitz Rule 77 II. General Solution of First Order Linear Differential Equations 77 III. Identities in Vector Analysis 78 IV. Coordinate Systems 80 V. Index Notation 81 VI. Examples of Use of Index Notation 83 A. Handy Symbols83 B. Relationships in Index Notation 8 C. Identities in &ij and ijk 84 VII. The Dirac Delta "Function" 84 VIII. Gamma Function 86 IX. Error Function 86

-viNOTES AND CONVERSION FACTORS I. Electrical Units 88 A. Electrostatic CGS System 88 B. Electromagnetic CGS System 88 C. Practical System 89D. Energy Relationships 89 II. Physical Constants and Conversion Factors; Dimensional Analysis 90

-1ORTHOGONAL FUNCTIONS In general, orthogonal functions arise in the solution of certain boundary-value problems. The use of the properties of orthogonal functions may often greatly simplify and systematize the solution to such a problem, in addition to providing a natural way of making approximate solutions. Let us first make clear the concept of orthogonality. We begin at what may seem an improbable starting point. Two vectors, A and B in a three-dimensional space are said to be "orthogonal" if the dot (inner) product A-B vanishes: A-B ABl + A2B2+ A3B3 A iBi = 0 1=1 This is easily generalized to a space with more than three dimensions. In an n-dimensional space the concept of orthogonality is unchanged, except that then the sum is over n terms AiBi, and we have for orthogonality n A'B = AB1 + + A2B2 +... AiBi = 0. Now one may think of the components of a vector, A1, A2, A5, as the values of a (real) function at three values of its argument; say A1 = f(rl): A2 = f(r2), A5 = f(r5)X or, in terms which will make our efforts here more clear, Ai - f(ri) (ri = r, r2, r3). That is, r has the values rli r2, r3, and to get Ai, put ri in f(r). We may now think of r as having any number) say n, possible values in some range, so that f(r) evaluated at the various r's generates an ndimensional vector. The step to considering a function as an infinitelymany-dimensional vector is now a natural onej we allow n to increase without bound, r taking all values in its range. Let r have some range afrtb in which it takes on n values such that r. - r 1 =TAr. and suppose two such n-dimensional vectors f(rj) and 3 j - 3

-2g(ri) are thus generated. The inner product of f with g is generalized as n < f(r) g(r) r. j=1 Above, for A'B, r. takes on only the discrete values 1, 2, 35 so A r is J always unity in this simple case. Now let us consider the case as n->OD r taking all values between a and b. The inner product, if it exists, is then lim f(r.) g(rj ) r. n -cl J1 j J With proper restrictions on\r. (maxAr.j-0) and on the range of r(arLb), a J this is just the limit occuring in the definition of the ordinary integral. Thus we say that the inner product of f(r) with g(r), which is often denoted (fg), is (fg) = f(r) g(r) d r. Of course, the range could be infinite. This discussion constitutes the generalization of the dot, or inner, product, to functions. Two functions are then by definition orthogonal over the range sarob when be (fg) = f(r) g(r) dr = 0 As we shall see, this definition is subject to generalization by the inclusion of a "weight function" p(r), with which b (f,g) - p(r) f(r) g(r) dr. a Here f and g are said to be orthogonal "with respect to weight function p'" over the range a-r-b. The function t(r) may of course be unity. Only the function f(r) = 0 a'-r!b is orthogonal to itself. In general we denote the inner product of a function with itself as (f, f) = Jf(r2) dr = N2 and call it the norm of the function.

-5Orthogonal sets may or may not possess two other properties, normality and completeness. A set of orthogonal functions Ui (u)t is said to be 2 normal or orthonormal if N. = 1 for all i. A set of functions {Ui(u)} orthogonal on the interval u l u - u is complete if there exists no other function orthogonal to all the I^ on the same interval, with respect to the same weight function, if one is involved. Importance of Orthogonal Functions The importance of orthogonal sets in mathematical physics may perhaps be indicated by further considerations of their analogs, orthogonal coordnate vectors. It is true that any N-dimensional vector may be defined in terms of its components along N coordinates, provided that no more than two of the reference coordinates are coplanar. But if the reference coordinates are orthogonal, e.g., Cartesian coordinates) the equations take a particularly simple form. The situation is somewhat similar when it is desired to expand a function in terms of a set of other functions -- it is much simpler if the set is orthogonal. Completeness is another important property. It is apparent that no two reference axes will suffice for the definition of a vector in 3-dimensional space. The set of two references axes is not complete in ordinary space, since a third coordinate can be added which is orthogonal to both of them. Addition of this coordinate makes the set complete. The situation with orthogonal functions is exactly analogous. Some authors define a complete set as a set in t of which antion defindinterms of which any other function dined on the same inval can be expressed. Some more common sets of orthogonal functions are the sines and cosines, Bessel functions, Legendre polynomials, associated Legendre functions, spherical harmonicsj Laguerre and Hermite polynomials; operational properties of which are listed in these notes.

-4Generation of Orthogonal Functions In the mathematical formulation of physical problems, one often encounters partial differential equations or integro-differential equations (which contain not only derivatives of functions but also integrals of functions), with which are associated a set of boundary conditions. If the equation and its boundary conditions are such as to be "separable" in one of the variables, one may attempt to apply the method of "separation of variables". Suppose we (admittedly rather abstractly) represent our equation ~J F (UV, v, w... wheree jis an operator involving the variables u, v, w,..., applied to the function F(uv,...). For example, $fmight be something like so that when is applied to a function F the equation appears = t e-ZI / J)A The process of separation proceeds as follows. One attempts to find a variable, say L, such that if it is assumed that F(u,v,w,...) may be written F(u,v,w) = Uay) ( --- ) then the equation <^Ff = @i0(.) (f>^^o can be written Here,2 is an operator involving only C, and [is an operator involving only pTj_._o. Returning to the example, assume F~zi~vWY^ (J 5(^w)

then = #wa64~5 A C) 7w -We can, in this case, if 1Jq t, divide by /U to get a6/ fZ V 6 ^j) j K(Y) / zw/00 W9 which can be rearranged --' z w= w- ~_y i',L p as we wished. In this case the separation has been successful, for on the left are functions of u only, while on the right stand only functions of v and w. Now suppose we were to vary u, fixing v and w. Then the right side would not change since it does not involve u, and is therefore a constant. We therefore state this fact wU d 1t _ Aa -onsta, We may ( J where / is a constant, called the "separation constant". We may choose it at our discretion. We now have two equations, where only one existed before:. - )0 4,o (7fz) z f* /K t ( w ) Q w')d'= O In order to effect a solution using this method, not only the equation, but also the boundary conditions must be separable. Back in the example, say the boundary conditions are F I i, t, w O ) h Introducing our assumption as to the form of F, that is. F(Ca ^^)= Cu^)^^,^)W

-6these become O c i.) Z- Z For O; we can divide byd to get which are separated, they do not involve >Tand sQ. By the process of separation of variables we have, from the original equation involving us v, w generated a new set of equations, some of which (those involving u) are a complete problem. 2U a,) o, d. (zuri-^.) f /+, f^(,')$ W ^ = O This was our objective in applying the method of separation of variables. The process may be repeated on the remaining equation or performed on another variable. Now if, after separation, the u- equation can be put in the form d rr(u) dUl - [q(u) + i p (u) U = 0 (1) du du and the boundary conditions in the form U + aaU(' (U) + aOU' (u) +'U(u (2) = 0 (2) bU(ul) + b2U' (u1) + P1U(u2) + P2U' (u2) = 0 where als, b's, a's, and P's are constants (some may be zero) then the system *o of differential equation and boundary conditions is called a "Sturm-Liouville.system". It will be noted that the Sturm-Liouville system is very general, and includes many important equations as special cases, for example the wave equation with those boundary conditions which are commonly applied. pronounced LEE-oo-vil, NOT LOO-i-vil.

-T7 This system under quite general conditions generates a complete set of orthogonal functions) one for each of an infinite, discrete set of values of the parameter p. One finds the values of y, called "eigenvalues", for which solutions exists and the solution functions corresponding to these eigenvalues, called "eigenfunctions". If the eigenvalues are/,l/2) I! and the corresponding eigenfunctions are U1 ()1U2 (),.., then in general the functions are orthogonal over the range 1 to 2, with respect to weight function p(U,> that is, jf1 p( (U Ui(tL) U.()dtL = Ni bi' Here..i is the Kronecker delta) and p(U) is the same as in equation (1). One must take care to find all possible eigenvalues (when the equation and boundary conditions are written exactly as in (1) and (2) they are all real). When all eigenvalues are found, the set of eigenfunctions is complete, and any function reasonably well behaved between I1 and _2 may be represented in terms of them. Say we seek an expansion of a function f(C) in terms of our eigenfunctions, f(U) = 2 f u(v.) i where fi are a set of constants. Multiply on the right and left by Up(t) and integrate with respect to U in the range ~ to L Ult L (2 hi i- P(M) U. ( U( d( As a consequence of the orthogonality of the Ui()'s, defined in equation (3) this becomes v 2 2 4fiNi 6ij' fjNj

We solve then for the f.'s; CIL2 i= 2 U. ( f(L) d) = - (ujf)( 33N U1 J In order to justify the switching of the order of summation and integration here, and to guarantee the existence of (U.jf), we usually require that f(X.) be absolutely integrable, i.e., t42 f L(.) duo exists. U1 It is to be noted that the outline of procedure above requires the solution of an ordinary differential equation, perhaps not an easy task, but one hopes not as difficult as the problem of solving the partial differential equation. Very often the eigenfunctions which fit a given problem are known, and so this process can be bypassed. Different sets of eigenfunctions have different sets of operational properties, that is, sets of relationships between members, which may be found useful. We note in conclusion that sets of orthogonal functions are generated by other means than by Sturm-Liouville systems; by sets of differential equations and boundary conditions which are not of Sturm-Liouville type, and by integral equations, to mention two. Use of Orthogonal Functions Let us consider a linear partial differential equation outlining the elimination of one variable from the equation. Under rather general conditions, we may expand F in an infinite series of orthagonal functions which we assume known; F(uvw,...) = w fi(yw(...) U(u) i^O J

-9where u2 -fi~ W...) -1 J P(4) F(u,v,w,...) U.(uA) dA.. Ni tl The formula for the coefficient f. follows immediately from multiplying the first equation by p() Ui(t) and integrating. Let the original partial differential equation be represented abstractly. F(uvw)...) = S(uv,v,...) where e is a linear differential operator and. F merely represents that part of the differential equation that involves F. Assume F to be expanded in a series in fiUiJ multiply the equation by p(UL)U. (u) and integrate over U from U to 6L. 1 2 ^P= s ) S oz flUi= S aC2 0-2 2 P(ita) U (U) fUdjL- Sp(U)U. (Lo = N.t) Ji (t4 1 i=LSP()Uj() U1 where L2 so that cSv S(u,v,w,...) = S. Uj. j 33 Now, by using the operational properties of the U.'s one reduces the equations (an infinite set, one for each i) to a set in the f.is and s's. The particular steps taken depend upon the exact nature of the operator t, and the set of equations may be coupled, i.e., f's with several indices may appear in the same equation (for example, f. - fi, fi+l ) These equations do not involve derivatives with respect to variable to, and we have gained in this respect. But we have to contend with the infinite set of equations.

-10All is not lost at this point, for, as it turns out, the series for F, (called a generalized Fourier series because of the manner in which the coefficients of Ui's are chosen in the expansion for F), is the most rapidly convergent series possible in the Ui's. Thus solving for only the coefficients of the leading terms in the series may enable us to obtain a satisfactory approximation to F. Also, very often, we are interested only in one or two of the fi's on physical or other grounds. Operational Properties of Some Common Sets of Orthogonal Functions There follow now a few pages on which are outlined, rather concisely, some basic operational properties of some commonly occuring sets of orthogonal functions. The familiar set of sines and cosines used in the construction of Fourier series is included as an example. These lists of properties contained in these notes are by no means complete, though they may suffice for the solution of many problems. The references listed with each section give detailed derivations, more extensive lists of properties, more discussions of the method and its limitations, or examples of the use of orthogonal functions. It is recommended- that one unfamiliar with these functions read in some of these references, in order to avoid the pitfalls of using mathematics beyond the realm of its applicability. These notes have been assembled mainly for reference. Of special interest may be the tables in Margenau and Murphy, page 254, which lists twelve special cases of the Sturm-Liouville equation with the name of the orthogonal set which satisfies each one. I. Fourier Series, Range _- x ^: A. Boundary Value Problem satisfied by sines and cosines: d2f + k2f = 0 (k real) dx2 f(4) = f(?), fl(-J) = f'(Q)

-11 - B. Orthogonal Set sin n2 cos for,_ x,> n = 0,1,2,... C. Expansion in Fourier Series F(x) a= a + (an cos nox + bn sin n )x ) z n=z / where a =1 F() cos nvx dx n=0,1,2... bn 1 F(x) sin nitx dx n=1,2,... D. Normalization Factors. N cos = N sin = n = 1,2... n n N cos = 2; E. Orthonormal Set 1, 1 cos ncx, 1 sin nTrx 2-, x, n=1,2,... F. Fourier Series, range 0 L x L L. One may wish to expand a function defined on an interval 0 = x o L, in a Fourier series. One may choose at his discretion either of two ways, expanding either in a series of sines or one of cosines. The sine- series expansion yields an odd functiony the cosine- serie.s an even function, when the series is considered as a continuation of the function outside the range 0- x _ L. An "odd function" is one such that F(x) = -F(-x), an "even function" such that F(x) = F(-x), as below.

-12/, \ I, x Odd Function Even Function Of course) either series represents the function on the interval 0 x S L. Note that on this range either.sin nrx or 1, cos njx n = 1,2)..., is a complete set, thus the two possible expansions. G. Expansion in "Half-Range" Series F(x) = ao an cos nrx Z n=l L or F(x) = b sin ntx, n=l n where an 2 F(x) cos nix dx, n = 01,22,... bn 2 F(x) sin ntx dx, n = 2,.. )J L H. Commnent on Expansions of "Odd" and "Even" Functions in Full-Range Series Consider the coefficients in a full-range expansion for an even function, i.e., F(x) such that F(x) = F(-x) an =1 F(x) cos nzrx dx = x snrrd —, x sn~x 1 [ F(x) cos nx dx dx+ F(x) cos njrx dx putting -x for x in the first integral, = lfF(-x) cos - (-dx) + F(x) coS ntx dx o~ ^-} JL ^ Jo

-13use F(-x) F(x), co = os( ngx and switch order of integration A-.j = 1 F(x)x) cos nxx dx dx an = 2 F(x) cos ndx 0 in general. bn - 1 (F(x) sin n-rx dx =1 PF(x) sin nrtx dx + (F(x) sin nx dxJ put -x for x, in first integral, = 1 (F(-X) sin -nrx (-dx) + IF(x) sin ngx dx use F(x) = F(-x), sin/-ntx= -sin ntax, and switch order of integrations. = 1 - F(x) sin nxx dx + fF(x) sin nxx dx =0 all n = 12.. Thus, the coefficients of the sine terms in the full range expansion of an even function are all zero; the cosinre coefficients do not, of course, vanish for all n.'he situation is much the same with respect to coefricients in the full range expansion of odd functions, except that in this case it is the coefficients of the cosine terms which vanish, for n = 0,12... II. Legendre Polynomials (Range -1 x l 1) A. Generating Function o H(x,y) = _1 = P (x)y (1 - 2xy + y2)l/2 -o Then P (x) = 1 H y=O ~

-14B. Recurrence Relations 1. + 1) P+ (x) - (2A+ 1) x P,(x) +,P (x) =0 2. Pi (x) - P (X) + XP1 (x) = C. Differential Equation Satisfied by P,'x) 1. (1 - x2) R" (x) - 2 x (x) + 2 (+ 1) (x)= 0 (ian integer) Very often x = cos 9. 2. Normal Form aP + V(1+i) (1 - x2) +1 P = aX2 (l-x) D. Rodrigues' s Formula P(x) = i d (x2 1) 2 2' dx E. Normalizing Factor Norm of P (x) is = f(Pp(X) x ) dx = 2 p -1 " 2J-+ 1 F~ Orthogonality 1 J P(x) P (x) dx = ee where eh is the Kronecker delta, defined e/,= -l,> G. Expansion in Legendre Polynomials Any function f(x) which is defined over the range -1! x - 1, and which is absolutely integrable over this range may be expanded in an infinite series of Legendre Polynomials, f(x) = P (x) where 1 f~:= 22 + 1 ( f(x) P (x) dx 2 1 = 1 ( f(x) P (x) dx 2 -1

-15H. Normalized Legendre Polynomials P (x) = P2(x) JZA -!.. I. Expansion in Normalized Polynomials g(x) =, P(x) where f1 gp )* g (x) P (x) dx. J. A Few Low-Degree Legendre Polynomials and Respective Norms WO(X)1 No 2L~2 P (x) =1 N0 = 2 P1(x) = x N = 2/3 P(x) = 1/2(3x2 - 1) 2 =2/5 2 P3(x) =....i'^....) N = 2/7 P4(x) = 1/8(35x4 - 30x2 + 3) N4 = 2/9 P5(x) = l/8(63x5 - 70x3 + 15x) N = 2/11 P6(x) = 1/16(231x6 - 515x4 + 105x - 5) N6 = /13 K. Integral Representation of Pk(x) (X() 1 x + 1) CO'S ) L. Bounds on (x) For -l1i x!-l, 1P(x)| l, ally. References on Legendre Polynomials 1. D. Jackson; tFourier Series and Orthogonal PolynomialsY, Carus Math. Monographs, Number 6, 1941, pp. 45-68. 2. H. Margenau and G. M. Murphy; "The Mathematics of Physics and Chemistry", D. Van Nostrand, First Edition, 1943, pp. 94-109. 5. A. Gb Webster; "Partial Differential Equations of Math. Phys."t GE. Stechert and Company, 1927, pp. 302-320. 4. RV. Churchill; "Fourier Series and Boundary Value ProblemsT"T McGraw Hill, 1941) pp.175-201. (for discussion of concept of orthogonality, see Chap. III). 5. E.. Jahnke and F. Emde; "Tables of Functions", Dover Publications, 1945, pp. 107-125 (Lists some properties and tabulates functions).

III. Associated Legendre Functions A. Definition of the Associated Legendre Functions (Associated Legendre Polynomials m Iml 1. Pm (x) =(l-x') dl P (x) (O-m 4:) dx i n (km -1) 2. p (X) - (i - x2) d (x2 1 B. Recurrence Relations 1. P (x) +[2m x (+ 1) (m- 0 ___P_ ()-= 0 M M m+l 2. xP (x) = (+ m) 1 (x) + ( - m + 1) (x) 2)+1 2 2 + 1 21/2 m m+l m+. (1 - 2 P(x) = P (x) - J (x) 2%+ 1. Pm+l 2m 4. pm+1 2m - li-( m) P (+ m) P (x) + C(e l) pm (x (2 +1) L -? +1 J - (Q +) - m(m 13 p.m-l(x) C. Differential Equation Satisfied by PM (x) 1. (1 - x2) d2j -j 2 x dP + 1) - m2 = 0 dx2 dx 1 - x2 2. Since P1 (x) is defined on the interval -1. x4 1, in physical applications FP' (x) is often associated with an angle 9 through the relation x = cos 0. Then the equation satisfied by ~1 (x) may be found in the form d2P (x) + cot Q d. (x) +2 + 1) - m2 ~ (x) =0 d2 dQ 2 [1 d.d sin d] P (x) +E(+l) m2 pm(x) = 0

-17D. Expression of Pe (cos 9) in terms of Pi (The Addition Theorem) 1. Expression e, 3l and e2 ( denote, respectively) the polar and azimuthal angles of two lines passing through the origin, then 0, the angle between these two lines, is given by cos g = cos 1 cos02 + sin0 sin 6 cos (<l -)2) see figre. With these definitions, P (cos g) may be expressed P, (cos 0) = P (cose ) P(coso ) +.2 I'Q ^ xI i m j7_ii__7) _ 1. 2, If (x) is defined in a slightly different manner, allowing negative values for m, pm(x) = (1 - 2) dml (x) (Iml Z| ) then the expansion may bP written P (cos 0) = os(- ) (cos(1 ) Pm(eos m (m1-<2) E. Normalizing Factor,f12 [1 j(xP 2 d = 2 +M)l F -I 2 1 22 m)R F. References 1. H. Margenan and G. M. Murphy; "The Mathematics of Physics and Chemistry, D. Van Nostrand, 1st edition (1934).

-182 A. G. Webster; "Partial Differential Equations of Math. Physics", G. E. Stechert and Company (1927). 3. D. K. Holmes and R. V. Meghreblian, "Notes on Reactor Analysis, Part II Theory", US.A.E.C, Document CF- 4-7-88 (Part II), August 1955, pp. 164-165. 4. E. Jahnke and F. Emde;"Tables of Functions", Dover Publications (1945). V-.i Spherical Harmonics A. Definition of m(fl) The spherical harmonics are a complete, orthonormal set of complex functions of two variables, defined on the unit sphere. Below, the vector symbol will be used to denote a pair of variables,, 4here taken to be, respectively, the polar and azimuthal angles specifying a point on the unit sphere with reference to a coordinate system at its center. With these conventions, the functions ( L) are defined 2 imP ym (l) = 2R| + 1 m ) (cose)eim<P also, it defines/= cos, Iml..e'e +mi ) 21! d+Im I Note: = (-1) Y where * denotes complex conjugate) B. Expression of P3 (cos Q) in Terms of the Spherical Harmonics Define l, f1}l i.e.JQ., and t2, ioe.,Q_2 as the polar and azimuthal angles specifying two points on the unit sphere, with respect to a coordinate system at its center. Denote by Q the angle between the lines drawn from each point to the origin of the coordinates. Then. m=j P (cos ) = r y C. Orthonormality of ym (L) jk ( -) - () d where the integral over vector JL_ indicates a double integration over the full ranges of &), tp; -_X Z.~ 1, O~ ^(QL2n. 2 "- 2

-19D. Expansion in Spherical Harmonics Any functions, perhaps complex, of the variables & and (, i.e.,, absolutely integrable over< and j, can be expanded in terms of the functins y (g)F(Z) = Fm (Q), =O mZ- - M-s - ~ where E. Differential Equation Satisfied by Ym (2L): t (sinma)n + 1 J +(e+l) (sine) Y sin e (x integer) ) 19si n&o Assume Y = [D(qp))(cos Q) and say 2Y -m 2Vt; impose the conditions Y bounded at cos9 = + 1 Q 2 (makes e an integer) Y single-valued in ( (makes m an integer). F. Some Low-Order Spherical Harmonics y (a ) 1 O ((40).) Y (.) = in2 cosin 1' P^ 1- i e1 1/2 t in e G. A Useful Relationship If the vector 2Lis considered to represent a point on the unit sphere) its components can be represented _LX = sin icos ^ = sinsin CQ CZ. -z cose

-20If a new set of components be constructed, -1= 1 (Q - iQy) =1 sin O(cos ( i sin ) ~~Qo0 =~2,z cos - i = sin(cos + Iy) 1 si sins ) then these are readily seen to be expressible as the k = 1 spherical harmonics (see E), 1/2 a 1 =( 4) Y -o E1927 ) 1/2 1 H... References 1. H. Margenan and G. M, Murphy: "The Mathematics of Chemistry and Physics" D. Van Nostrand, 1st edition, (1934). 2. A- G. Webster; "Partial Differential Equations of Mathematical Physics", G, E. Stechert, (1927). 5. E. Jahnke and F. Emde; "Tables of Functions", Dover Publications, 1945, pp. 107-125 (Lists some properties and tabulates functions). 4. D. K. Holmes and R. V. Meghreblian; "Notes on Reactor Analysis, Part II, Theory",, U.S.A.E.C. Document CF-54-7-88 (Part II), August 1955. 5. L. I Schiff, "Quantum Mechanics", 2nd Edition, McGraw Hill, 1955, p. 735. 6. Whittaker and Watson; "Modern Analysis"; 4th edition) Cambridge University Press (1927), pp. 391-356. V. Laguerre Polynomials A. Derivative Definition e'") -~x a~ (xa+= n x) Ln () = (-1) x e d (x ne-) dx1n

-21-22B. Generating Function -xt H(xt) = (1 - t)-(+) e r (x) t n=O n! -xt "7 Thus Ln) (x) = adn (1-t) ) e -t t-O dtn C. Differential Equation Satisfied by 1j() (x): x d2y + ( - x + 1) dy + n Yn = 0 n 0,1,2,... dx2 dx D. Orthogonality, range 0' xL C o xa e x( (x) L (x) dx = N2 nn where Nn2 = (-1) n! (n + + 1) E. Expansion in Laguerre Polynomials F(x) = Z"fn(n (x) n=0 where fn - 1 xc e F(x) Ln( (x) dx n m F. Expansion of x in Laguerre polynomials. m.-7 ( \ xm = (-l)n mI / ( + m + 1) L x) n= nn! r (n + a + 1) G. Recurrence Relations a. (x - 2n - a- 1) L() (x) = Ln+ (x) + n(n + c) n (a) (x) b. L' ( ) (x) = (n + l) L ( (x) - (x) n~ ( n nL

-25VI. Bessel Functions A. Differential Equation Satisfied by Bessel Functions. 1. d2 +1 dy + (1 2) y _ dx2 x dx C or.2. 1 d (x dy + (1 - QLj y = O x d dx/ 2 (An extensive listing of other equations satisfied by Bessel functions is given in.Reference 2.) B. General Solution of Above Equations. y =A J (x) + BJ n(x) ( non-integral) y = A J (x) + BN (x) (n integral) n n (N (x) is frequently represented by the symbol Yn(x). where A and B are arbitrary constants. J (x) = Bessel Functions of the first kind of nth order. Nn(x) = *Neumann Functions, or Bessel Functions of the second kind. For non-integers, N (x) = J (x) cosj/X - J (x) sin p E For )= ), an integer, the above expression reduces to** rl 1 n+2r N (x) =2 J (x) log 1 x - 1 (-1) x F() + F (v +r) /7 n2 IT! =n+r! -1 (n-w-1 -n 2 z). r where F(I) = 21 1 * In reference 4, these are called Weber functions ** This is shown in reference 8, pg. 577.

-.24A third function which sometimes finds use is the Hankel function, or Bessel function of the third kind. There are two such functions, defined by (1) H v (x) = J (X) + i N (x) (v unrestricted) H (x) = J (x) - i N(x) Then, for integers, we see that a s-olution to Bessel's equation will be y = AXH(In (x) + A 2) (x) where A1 and A2 are arbitrary constants and may be complex. These functions bear the same relation to the Bessel function.-J (x) and Nl(x) as the functions exp (+_y.) bear to cos x and sin )x. They satisfy the same differential equation and recursion relations as J, (x). Their importance results from the fact that they alone vanish for an infinite complex argument, vis. H(1) if the imaginary part of the argument is positive, H(2) if it is negative, i.e, lim H (0t,= O lim H re JO. r- AV co re From the.above equations, we can also write J (x) = [ H (x) + (1) (x) (1) unrestricted) N (x) 2 H (2x) - 1 (1) (x) N (x b (X) H ( C. Series Representation for Bessel Functions of the first kind of Order n.. Z-1 r a r Jn(x) = (-1) (x) O2 You 22r+ r (1 +n+r) D. Properties of Bessel Functions I. Jn(x) = (-1)r) Jn (x) 2. J (x) = (-1 n (-x) n n 5. Bounds on Jh(x) a. (,J(x)j _ 1 (n = 0,1,2...) x? 0

-254. Limits for J (x) where n equal zero or positive integer. a. lim J (x) = nn b. lim Jn(X) = 0 c. lim J (x) = x x -m 2'1 5. Limits for N (x) for n = 0 or positive integer and x real. a. lim N (x) 0 x -d n b. lim Nn(x)= (n l) (2) n2-1. x -- O x c. lim N(x) = - 2, (C= 1.781) x-o t - 6. Graphs of J (x) and N (x) h g / 2 R 4 s 6 7 8 q po ii Z ^ PI i Variation of J0(x), J, (x) and - DJ (x) with x. t~~~~~~~~~r~~Z

-26czCI?o 2 3 5 6 7 8 $ /~ Variations of N(x) and N (r) with x., E. Generating Function exp [_ (t - 17 = Jn(x) tn (n integral) 2 V t/J h=-oo F. Recurs ion Fomulae a. 2J'n(x) = _l() - ( b. 2J (x) _Jnl + Jn+l c. xJn'(x) = J n-l - _n Jn (X) = x J (x ) - xJn (x) G. Differential Formulae: b. d n[x ( x xn Jn1 1. f x <^ ( X) Jn (kx) dx = jNnj2 where are ptsitive roots of the equation Jn,(\c) = and iknj2 = C2 n (IC) 2

-272. f x i (Xn x) Jn (Xx) dx -= Mn where X are the positive roots of the equation ( c).J'n (Xc) ne -hJn(XC) or its equivalent (n + h) Jn (c) - Xc Jn+1 (ec) where h is some constant? 0 n = 0,1>2... I. Expansion in Be:ssel Functions f(x) = WAj.J(Xnjx) / j n nJ where A, will be represented either by c2[n+1 ( c) 2 A when J n( njc) Q b. or A. = + 2L ct J.'^ J ^4l1 a) ~ x f fx)) J(. x) x when XcJn (Xc) = -h Jn(Xc) J. Bessel Integral Form Jn(x) = 1_ f cos (n 9 - x sin @) d9 (n = 0.,2...) o VIIx Modified Bessel Functions A. Differential Equation satisfied by Modified Bessel Functions. 2dx x dx 2 d~y + 1 dy -(1 + y=0 B. General Solution of Above Equation. y = AIv(x) + BI0(x) (Inon-integral) y = AI (x) + BKg(x) (n integral)

28where A and B are arbitrary constants. IK(x) Modified Bessel Function of the first kind of nth rder. K (x) = Modified Bessel Function of the second kind of nth order. For non-integers, KO(x) =,[I (x) - I (x) 1 x 2) - - L -.sin -)= 1 H(1) (Hx) = i H()(-ix) For integers, n+1~n rv -n+n-2r Kn(x) = (-1) 2 I (x) log (1/2 x) + (-1 ) r(-r (1/2 x)n r=o ciD + (-1)n 1 r (1/2 ) F(r) + F( + r) r O(n +~ r / \ where r F(r) = 1_ 1 S=l S C. Relation of Modified Bessel Function of First Kind to Bessel Function of First Kind. For VI unrestricted, I (x) = JJ(ix) = i H (2) - H((x X- (1/4 X2)r 2lr(^+1) _ r! (1i+1) r where (()r (() (( + 1)......... (c+ r - 1) D. Properties of'I (where n is integral). 1. In(X) = IT(x) 2. 2I' (x) I1(x) + In+(x) 5. 2n In(x) = In(x) - In x) x

-294. x In' (x) = x In.l(x) - nIn(x) 5. x In' (x) = nl (x) + x (x) I6. (x) (X) S' i.^ ~'Kot o / 3 9 6 Variation of K (x) and K (x) with x 4 ~b',' 0 / 3 ) a 6 Variation of Iox) and I.(x) with x

-50*Comment on notation of Jahnke and Ermde. 1. A general cylindrical function Zp(x) is defined on page 144 by zpgx) = CLJp(x) + c2Np(X) (p integer or arbitrary positive) where el, c2 denote arbitrary (real or complex) constants. Thus Zp(x) can apply to Jp(x) by letting.e 0, to Np(x) for c = 0, and to Ip(x) by other constants All formulae on pages following use this definition of Z(x)O 2. The function Ip(x) is not listed as such, but is found as i J (ix) on pages 224-229. 35 The function Kp(x) is not listed, but 2K (X) in 1) (ix) -n+1 (2) = -i ( (ix) The functions iHI ) (ix) = -iHo (-ix) and -H1 (ix) = -H1 (-ix) are tabulated on pages 236-24-30 4. This reference is full of extremely interesting, beautiful and helpful pictures of many functions, almost suitable for hanging in the living-room. References o 1. G.o o Watson; "A Treatise on the Theory of Bessel Functions ", 2nd edition, Cambridge University Press, 1944, (exhaustive treatment) 2. E Jahnke and Fo Eide "tTables of Functions" (extensive tabulation of equations leading to Bessel Functions and of the related cylindrical functionso Also has further properties, see "Comment on notation of Jahnke and Emide ) 3. R, Vo Churchill "1Fourier Series and Boundary Value Problems', McGraw Hill, 1941 (good elementary discussion) 4. IO No Sneddon; "Special Functions of Mathematical Physics and Chemistry" University Math. Texts, 1956 (thorough discussion for practical use. His use of subscript is not always clear or general) 5~ D. Jackson; t"Fourier Series and Orthogonal Polynomials1" Carus Math. Monographs, Noo 6, 1941 (good brief discussion) 60 Whittaker and Watson;'Todern Analysis" 4th edition, Cambridge University Press, 1958 (more rigorous development)

-317. N. W. McLachlan; "Bessel Functions for Engineer"T, 2nd edition, Oxford, Clarendon Press, 1955. 8. H. and B. So Jeffreys; "Mathematical Physicst" 3rd edition, Cambridge University Press, 1955, (compact, but rigorous presentation. They use a different notation, but it is clearly defined.)

-52T1E LAPLACE RANSFORMATION I. Introduction A. Description The Laplace trarsformation permits many relatively complicated operations upon a function, such as differentiation and integration for instance, to be replaced by simpler algebraic operations such as multiplication or division~ upon the transform. It is analogous to the way in which such operations as multiplication and division are replaced by simpler processes of addition and subtraction when we work not with the numbers themselves but with their loarithms. B. Definition The Laplace transformation applied to a function f(t) associates a function of a new variable with f(t). This function of s is denoted by, f(t) or where no confusion will result, simply by ((f) or F(s); and the transform is defined by: j(f) = f (t) e-st dt Jo C. Existance Conditions For a Laplace transformation of f(t) to exist and for f(t) to be recoverable from its transform it is sufficient that f(t) be of exponential order, i.e. that there should exist a constant, a, such that the product; e-at (f(t)( is bounded for all values of t greater than some finite number T; and that f(t) should be at least piecewise continuous over every finite interval 0 Lt _-T, T any finite number. These conditions are usually met by functions occuring in physical problems. The number a is called the exponential order of f(t). If a number a exists such that e-at Jf(t)/ is bounded, f is said to be of exponential order.

-33D. Analyticity of F(s). If f(t) is pie-ceWise continuous and of exponential order a, the transform of f(t), i.e.r F(s), is an analytic function of s for Re(s)?a. Also, it is true that for Re(s) ) a, lim F(s) = 0 and lim F(s) = O wher s = x + iy. X -o o y E. Theorems. Theorem I {Linearity) The Laplace transform of a sum of functions is the sum of the transforms of the individual functionso jd(f + g) i ~ (f)+ f(g) Theorem II (Linearity) The Laplace transform of a constant times a function is the constant times the transform of the function;(ef) = e (f) Theorem III iBasc cOperational_ Propert) If f(t) is a function of exponential order which is continuous and whose derivative is at least piecewise continuous over every finite interval 0! t t2, and if f(t) approaches the value f(0+) as t approaches zero from the right, then the Laplace transform of the derivative of f(t) is given by (f') s (r(f) - f(+) and (tf") s2 ~(f) _ Sf(0+) f (O+) the latter, of course, requires an extension of the continuity of f(t) and its derivatives to include f"(t), and may be formally shown by partial integrations. More generally, if f(t) and its first n-1 derivatives are continuous and dnf is piecewise continuous, then dtn 4lnl|= iff) - s11' f(04) _ s-2 f(O+) ___ _ f(nt) (O) [UnfT] snf(f) _ sn_ f(0+) nf(0 +)

-34Theorem IV (Transforms of Integrals) If f(t) is of exponential order and at least piecewise continuous, the t transform of j f(t) dt iS given by t[i ] ft)dtJ =' (f) + i1 f(t) dt F. Further Properties Below, let us assume all functions of the variable t are piecewise continuous, 0 -. t ^ T, and of exponential order as t-,o. Then Theorem V [eat f(t)] F(s - a). Theorem VI If (f(t-b), t _b f(t) - f (t t < b, then [fb(ti - eb- (s) Theorem VII fConvolution) r[ f(t - T) g(T) dT = F(s)*G(s) -=J g(t - T) f(T) d Theorem VIII (Derivatives of Transforms) [tf(t)] -3dr(s) ds and, in general, lt f(t)] (1)n dnF dsn II.o xaZples. A. Solving Simultaneous Equations Solve for y from the simultaneous equations Y' + y + /3 zdt cos t + 3 sin t 2Sy + 5z' + 6z = 0 Yo = -3, zo = 2

The transform of each equation is: Jt (y') + 3(y) t+ zdt + (cos t) + 5(sin t) 2J~) + 35(z) + 6 (z) = 0 orj Csf(y) +5J +3(y) +5(z) = s + 3 LJS 2 82.2. ~~~$cy, + a + 1 +1 2L[S(Y) + +5 [s (z)- 2 + 6 (z) = 0 collecting terms and transposing (s + l)(y) + 3 (z) = s + 3 $2+ 1s s 1 2s (y) + 5(s + -2)z (z) = The two original integro-differential equations are now reduced to two linear algebraic equations inf(y) and (Z). Applying Cramer's rule and solving for (y) since it is y which we want; |ccgS (+)_ )3 s + 3) 3: ) (s + l) f^ - 0 5(sa) (s +2)1 s2 4- 1 (s + 1) 3 2s 3(s + 2 or, (y) = s + 2 -+ 2'(.s (s +' )1)J Applying the method of partial fractions; J(y) = / + -s + - 1 + 2 s l 2s2 + +3 -2 / s \ + 1 - 1s+ [ s2 ^-2 f s7 +

-36And, finding the inverses in the table of transforms, which are tables relating functions of s to the corresponding functions of t, and will be found in section IV of this paper, y = 2 -2:os t + sin t - e-3t (for t> O) It should be noted that one of the inherent characteristics of solving differential equations by the use of Laplace transforms is that the initial conditions are included in the solution. B. Electric Circuit Example Since Laplace transforms are widely used in the determination of the transient response of electric circuits, a simple circuit example is given below. Given circuit below; R I R L Find, equation of current flow after switch is closed, a. Circuit equation; E = iR + L di dt b. TRANSFORMING; E = I(s)R + LsI(s) - Lf(O0+) at t = 0, i = so; f(0+) = 0 c.'solving for I(s) I(s) E s(R + Ls) d. TAKING INVERSE i(t) = E (1 - eRt/L) R C. Transfer Func tions For certain control functions, and for representing the dynamic behavior of various devices such as reactors, heat exchangers) etc., it is advantageous to use a "'transfer function" because of the convenience in manipula

-57tion which obtains. The transfer functions of many elements of a systemwhen strung together in a block diagram, represent a convenient way of writing complicated system equations. The transfer function of a system may be defined as the ratio of the output to the input of the system in transform (s) space. Conditions for using transfer functions 1. Initial condition operator 0. 2.No.loading between transfer functions. 3. Transfer function satisfies existence conditions for Laplace transformations. 4, Linear system,. Exap le of Transfer Function; * —- i -h --— r., -, i'.. I ~, v t Find Eo (s) of -- I I a, Equations ei Ri + L di dt eo L di dt b. Transforms Ei (s) - RI(s) + -sLI(s) Eo (s);sLI(s) co Solving Eo (s) = sLI(s) = sL st _T (R+sL) I () R + sL 1 + st where t L= L/R = circuit time constant. d. Block diagram Ei(s) st E (s) 1.~ okSZ

-38III. Inverse Transformations A. Heavsiside Methods. When solving equations by the Laplace transform technique, it is frequently the most difficult part of the procedure to inyert the transformed solution for F(s) into the desired function f(t). A simple way of making this inversionl, but unfortunately a method only applicable to special cases, is to reduce the answer to a number of simple expressions by using partial fractions, and then apply the Heaviside theorems as outlined below: Theorem I If y(t) = rPl[ } >, where p(s) and q(s) are polynomials, and the order of q(s) is greater than the order of p(s), then the term in y(t) corre~spoding t-o an unrepeated linear factor (s-a) of q(s) is pa) eat or a) at qI(a) a(a where Q(s) is the product of all factors of q(s) except (s-a) Example: If (f (t)) = s2 +2 what is f(t)? s (s4i) (s+2) a. Roots of denominator are s =- 0 - s = - 2 b. p(s) = 2 + 2 c. q(s) s3 + 35s2 + 2s; q' (s) = 3s2 + 6s ++ 2 p(o) - 2, p(-1) 3, p (-2) = 6 d. qt(o) = 2, q (-l) -1, q=(-2) = 2 e. f(t) = e~t + 3 e 6 e2t 1 - 3et + 3et 2 -1 2 Theorem II If y(t) = l[p(s) where p(s) and q(s) are polynomials and the order of q(s) is greater than the order of p(s), then the terms

in y(t) corresponding to the repeated linear factor (s-a) of q.(s) are; (r-2)' I 1!' r (r-2) where 0(s) is the quotient of p(s) and all the factors of q(s) except (s-a) Example t(f(ft)) = s + 3 what is f(t)T (s2)2 (s+l) a. p(s) s= S3 J'(s) (s+1) - (s+3) = L- 2 8~.s+1 (s+()2 (s+1)2 b. (-2a) = -1;,'(-2) = -2 so8 terms in f(t) corresponding to (s+2)2 are e-2t r2 + -1t -e2t'(2+t) 1L! 0t1' then, as in the example of Theorem I; p(s) s= +3 q(s);= s3 + 5s +8s + 4 qt (s)= 3s + i0s + 8 p(-l)= 2 qt (-1) sol f(t) -e 2t (2+t) + 2e-t Theorem III If y(t) = pl(s) where p(s) and q(s) are polynomials and the order of q(s) is greater than the order of p(s), then the terms in y(t) corresponding to an unrepeated quadratic factor (s+a)2 + b2] of q(s) are e-at (0i cos bt +; sin bt) where b r r and i are respectively, the real and imaginary parts of S(-a+ib), and p(s) is the quotient of p(s) and all the factors of q(s) except (sha)2 + b2. Example s (f(t)) = ---- s(s+2)2 (s22s+2)

-40:a. Considering the linear factor as in the example of Theorem II <s);=; ) i (s) =- -s + 2 (s2 + 82s +2) (s2 2s + 2)2,0(-2) = -1 S'(-Q2) 1 so, the terms in f(t) corresponding to the linear factor arej e-'2t (-1/2 - t) = - (1+ 2t) e-2t 2 b. eonsidering, the quadratic factor s2+ 2s + 2 (s + +12 (s)-.-.. (S + 2)2 * 0(-a + ih) = 1(-1 + i) = -1 + i -1 + i = -1 + i = 1 + (1 + i)2 i 2 2 so; 0r = i = 1/2 so; the terms in f(t) corresponding to (s2 + -2s + 2) are e-t(cos t + sin t) 2 Now, adding the two partial inverses, we get f(t) - (1 + 2t) e-2t + et(cos t + sin t) 2 2 B. The Inversion Integral When the function cannot be reduced to a form ammenable to inversion by tables of transforms or Heaviside methods, there remains a most powerful method for the evaluation of inverse transformations. The inversion is given by an integral in the complex s-plane, +f(t) 1 t F(s) d f(t) = 1 e est F(s) ds

_41-L where is some real number so chosen that F(s) is analytic (see Appendix A) for Re(s)?, and the Cauchy principle value of the integral is to be taken i.e. f(t) = 1 aim ( eStF(s) ds. Let us illustrate the formal origin of the inversion integral in the following way. In the complex plane let S(z) be a function of z, analytic on the line x = X v and in the entire half plane R to the right of this line. Moreover, let |I (z) approach zero uniformly as z becomes infinite through this half plane. Then if so is any point in the half plane R, we can choose a semi-circular contour c, composed of cl and C2, as shown below, and apply Cauchy's integral formula, (see Appendix B) (^;i - 5 ^^ ^- di c Here, 4S)is analytic within and on the boundary c of a simply connected region R and 5 is any point in the interior of R (Integration around C in positive sense). A s-planc C,,Cz Thus, Cauchy'ts integral formula yields 0{(sa) _=1 = 1 (z)dz + 1 z)dz 2; J z-s 2i z-s si 0 -s ^^ Of,~~Ci

Now, for values of z on the path of integration, c2, and for b sufficiently large, )2 - |' b - s - Z| b sIs hencJe f I ( Al Z- I - I Z- C2 C2 b M s (dz C2 b -ISI where M is the maximum value of [q (z)| on c2. As b-30, the fraction b approaches 1, and at the same time M approaches zero. Hence b- Isl him f J (z)d.z 0 b"0 - J z - s c2 and the contour integral reduce4 to ~ Z~ /y-Lb / A(s) = lim 1 ( 0(Z)dz = 1 (z)d1z b 2t1f z - s 2 s - z now, taking the inverse transform of the above equation, -1 f d s (- z 1l_ (z)dz bt-1^^: = f(t) I T.sr' 1 fZz) dz since from our table of transforms eS-ct o

Our final equation after switching fronm z to s as duammy variable in the last integral + X (s)} _ f(t) = J (S) est ds, is just the inversion integral which we were establishing. At this pointJ it would be advantageous to know how to evaluate the integral on the right. According to the residue theorem, the integral of estz(s) around a path enclosing the isolated singular points sl, s2 — sn cs' e0s(s) has the value 2riEPl(t) +Og(t) + /( (t)t where (t) = the residue of est0(5) atS s. For discussion of residues, see Appendix C; for singular points, Appendix D. Let the path of integration be ade up of the line segment - ib, + ib,, and c3, then + ab ~ Snpianl / s_ plane 3/ 12i est 6 (s) ds ~+ 1) eS (s) ds = p (t ) -ib c3 n=l If the second integral around c vanishes for b — ^30, as often happens, we are led to the immediate result that i]-l )(sl = f(t)= 2 p n(t) w> < J ~~~~n=l

Note that in the formal derivation of the inversion formula, we assumed that p(s) (and therefore e5st (s)) is analytic for s g and that limI (s)/ - 0 in that plane. In our discussion of the residue form of the inversion, we work in the left half-planeo This is because Laplace transforms have the property that they are analytic in a right half-plane, and that in that plane, lim I4(s)) O. s -3ar Questions of the validity of the above procedures, alterations of contour, and applications to problems are not dealt with here, as they are presented in detail in the references. IV. Table of Transforms. F(s) f(t) 1 Unit impulse at t 0=, (t) -s Unit doublet impulse at t = O, 2(t) 1 Unit step at t - O0 u(.) -at (s + a) 1 e-at e-bt (S+a)(s) b s.ct (c-a) e-at -(c-b) ebt (s+a) (sb) b - a -at -bt s+-e + c - a e~ + c - b ebt S(8+a) (s+b) ab a(a-b b(bs 2- 1 e, + eb e-c (s+a) s+b) (s+c) (c-a) (b-a (a-b) (c-b) (a-c )(b-) ( s4i _ (d a ) e-t + (d.-b) et + (d-c) e ct (s+a) (sI)(s+) (b-aU( e-Yc ) (a-b)(c-b) (a-c) (b-c) 2 es + d (a2 - ea + d)e-at + (c2-ec + d)e-ct + (b2-eb+d)e-bt -sta)(s+b )+c (b-a) (c-a) (a-c) (b-c) (a-b) (c-b) 1 1 sin bt s2 +b2 b

-45s + d 1 (d2 + b2)1/2 sin (bt +^) arc tan b 2 2b d s -cos bt s2 + b1 e-at sin bt (sa)2 + b2 b s +'a e-at cos bt (S-a)2 +~ s + d a 3.(d-a)2 + b e-at sin (bt +L) (s+a)2 + b2 a rb e ta/ b \ d-a j1_~~ ~ 1 + 1 e+-at sin (bt -i) 5s (sa)2 + b 2 2 b2 s )= arc tanb b = a + b as +d d + 1 a(d - a)2 + b2 1/2 eat sin (bt+_) s F(s8a) + bj2 L J - L arc tan/b )- arc tan b = a + 1 L s inm bt s2 - b2 b s cosh bt 2 -. b2 1 1 tn-1 (n is an integer O0) sn (n-1)! Ts —- 1 t' (I 0) () may be non integer) r(lv)~ 1 e-'at + at - 1 s2 (s + a) a2 s + d d - a e"at + d t + a - d (s + a) s2 a2 a a2

s 4- d1 (d - a)t + 1'et (s + a)-2 J 1 1 - (at +1) e-at s(s + a)2 a + dd + *a d t - d + e"at 8(s + a)2, a s2 + es + dd ea - - d t + ad - d e_-at s(s + a)~ a J a2 1 1 t - 1 sin bt.2 (S2 b2) b2 b3 1 1 (cos bt - 1) + 1 t2 3 ('2 + b2)4' b2 1 1 sinh bt - 1 t s2 (2 - b2) b3 b2 1 1 (cosh bt- 1) - 1 t2 s3 (s2 _ b2)' 2b2 1 1 (sin bt b- cs bt) (s2+ b)2 23 s __1 t sin bt (s + b2 2 22b s2 1 (sin bt + bt Cos bt) (s2 + b2)22b s2 + b2 t Cos bt (s2 + b2)2 1 1 e-at (sin bt - bt cos bt) s +:a 1 te"at sin bt B 2 a)2b2

-47(s+a) b te-at cos bt [(s+a)2 + b22 1 t 1 (t - tl) u (t - tl) 1_ (tls + 1) e-tl tu(t - tl) 2 s 1 (tl s + 2tls + 2) e-tls t2u(t tl) Appendix A Analyticity Let w be a single valued complex function of z, 0.= f(z) = u(x,y) + iv(x,y) -where u and v are real functions. The definition of the limit of f(z) as z approaches z and the theorems on limits of sums products and quotients correspond to those in the theory of functions of a real variable. The neighborhoods involved are now twodimensional; however; and the condition lim z zo f(z) = uo + ivo is satisfied if and only if the two-dimensional limits of the real functions u(x,y) and v(x,y) as x — x, y y have the values uo and v respectively. Also, f(z) is continuous when z = z if and only if u(x,y) and v(x,y) are both continuous at (xoY o) The derivative of o at a point z is dco = rf'() = lim ao = lim f(z +A z) - f(Z) dz z z — 0 z Z O - z provided this limit existso (it must be independent of direction)o Suppose one chooses a path on which Ay = 0 so that Lz = Ax. Then,

-48since LO = u a + iAv^ d lis / u + i Zv\= - u + i ds xA-O (4AX AX x ax or ifxf x -O so that z i A y, then dk) lia. /u -i + v yO iAy i ay3 y Ay E-uating real and imaginary parts of the above equations, since we insist that the derivative must be independent of direction, we get U v, cv = -_U ~x )y Dx y These are known as the - Cauchy-Riemann conditions". Now, the definition of analyticity is that Ma function f(z) is said to be anytic at a point zo if its derivative f' (z) exists at every point of some neighbrhood -of zo l* And, it is necessary and sufficient that f(z) =u + iv satisfy the CaucMhy-Riemnn conditions in order for the function to have a derivative at point Zo Appendix B Cauchy' s Integral Formula Theorem Io If f(z) is analytic at all points within and on a closed curve, c, then e f(z) dz = 0 Proofp f(Z)dz = c(u +iv) (dx+idy) _ (udx-vdy) +(udx i+i vdx-udy Applying Green's lemma to each integral,,f (z) dz ( | Z - ) dXdy + iif fu V) dx dy but, because of -analyticity the integra-nds on the right vanish identically, giving yf(z) d = 0 c

Theorem II If f(z) is analytic within and on the boundary c of a simply connected region B and if is ay point in the interior of R, then f(zo) i f(z) dz ^24 ri Z Z- 0.. where the integration around c is in the positive sense. Prbof: Let c be a circle with center at z whose radius,0 is sufficiently small that c0 lies entirely within R (see Figure below) f (z) is analytic everywhere within RB hence f(z) is analytic z - everywhere within R except at z = z. By the "principle of defrmation of contours", (see any complex variable book) we get fZ(K d d.z t f(? ) f f (z) z dz: o z - z c cO = z + f()z + f( zo) dz o - ZI - zO Co C 0 Consider the first integral, dz Co Z -'o and. let z - zo =r e, dz = r i e, getting r e d = i f d 1 2ii o r e o

Aand observe that IJ -f( z'-' d j1 f |f(Z) - f( )l 1dz 0o |I On c, z Zo =/ Also, f(z) - (zo)Zeprovided Iz - zOj,~s Choosingp to be less than, we write t f(z) - f(zo) |dzL d = = 2o = 2 Since the integral on the left is independent of 6, yet cannot exceed 2 r-E which can be aade arbitrarily small, it follows that the absolute value of the integral is zeroo. * We have 6 f(Z) ( f(ZO) 2Xi + 0 or, oz 0 f(zo) 1 f() dz chh which is "Cauchy's integral fornmularo Appendix C Calculation of Residues Io Laurent series. Theorem I:* If (z) is analytic throughout the closed region, R,- bounded by two concentric circles, c1 and c2, then at any point in the annular ring bounded by the crcles f(z) can be represented by the series f(z) - an (z -)n n= -dO where a is the commbn center of the circles, and a = f(W) dU

- 51each integral being taken in the counter-elockwise sense around any curve c, lying within the annulus and encircling its inner boundary (for proof see any complex variable book) This series is called the Laurent series. II. Residues The coefficient, al'of the term (z - )1 in Mthe Laurent expansion of a function f(b) is related to the integral of the function through the formula 1'a 1 1 f f(z) z, In particular, the coefficient of (z - a)- in the expansion of f(z) around an isolated singular point-" is called the "residue'" of f(z) at that point. If'we consider a simply closed curve c containing in its interior a number of isolated singularities of a function f(z), then it can be shown that'f(z) dz = 21i[ + rr2 + n3.c where rnsS are residues of f(z) at the singular points within c. IIIo Determination of Residues. The determination of residues by the use of series expansions is often quite tedious. An alternative procedure for a simple-or first order pole at z = a can be obtained by writing f(z) a - + a Q + a(z -a) + and multiplying this by (z - a), to get )f(z- ) = + a%(z - a) + al(z - and letting z -ja, we get al = lim ((z - a) f(z) z -a A general formula for the residue at a pole.of order m is (m - 1)I al = lim dm- 1 E(z - a~ f(z)] z - a dmnl L

-52For polynomials, the method of residues reduces to the Heaviside method for finding inverse Laplace transforms. Appendix D Regular and Singular Points IfW - f(z) possesses a derivative at z - z0 and at.every point in some neighborhood of z, then f(Z) is said to be "analytic" at z = z and Z is called a "regular point" of the functiono If a function is analytic at some point in every neighborhood of a point Zo, but not at zo0 then zo is called a "Singular point" of the function. If a function is analytic at all points except zo, in some neighborhood of zo, then Zo is an "isolated singular point". About an isolated singular point zo a function always has a Laurent series representation f() = A -1 + A-z + -- A A (z - zO) + -— (O <~ Z - ZoLo) z - (Z - zo) where ro is the radius of the neighborhood in which f(z) is analytic except at Zo. This series of negative powers of (z - zO) is called the "principle part" of f(z) about the isolated singular point zo. The point tZ is an "essential singular point" of f (z) if the principle part has an A-j A m and An 0 = when n11 m. It is called a'"simple pole" when m = 1. References 1. Churchill; "Operational Mathematics' - A Complete discourse on operational methods, well written and presented, 2. Wylie; "Advanced Engineering Mathematics" - A concise review of the highlights of transformation calculus both Fourier and Laplace transforms. 35 Murphy; "Basic Automatic Control Theory!! - An exposition of Laplace transforms with many good electrical engineering examples, A fairly complete table of Laplace transforms.

-554e Erdelyi, A. et al; "'Tables of Integral Transform"n, Vols I and II, McGraw HEill New York (1954). Very extensive coeipilation of transforms of bmany kindso %5 Gardner and arnes; "Transients in Line.ar Systems" Extensive tables of transforms of ratios of polynomials.

-54FOURIER TRANSFORMS I. Definitions A. Basic Definitions In addition to the Laplace transform there exists another commonly-used transform, or set of transforms, the Fourier transforms. At least five different Fourier transforms may be distinguished. Their definitions follow: Finite Range Cosine transform H[n] = C [f] _ xf(x) cos nx d(n = 0,1,2 —-) Finite Range Sine transform = s [f f(x) sin nx dx (n = 1,2, —-) Infinite Range Cosine transform c [r] = Cr If] Y f(x) cos rx dx (Ot r ro) Infinite Range Sine transform J-s [r] = Sr[f] / | jf(x) sin rx dx (O rz:7) Infinite Range Exponential transform 5e[r] = r[f = f(x) eir dx (-e r- o) B. Range of Definition In the infinite range transforms, the irange sform variable is continuous; in the case of the finite range transforms, the variable takes only positive integer values or zero. Considering the range of integration used in the definition of each transform, we see that the finite range transforms apply to functions defined on a finite interval, the infinite range sine and cosine transforms to functions defined on a semi-infinite interval, while the exponential transform applies to functions defined on the infinite interval.

-55C. Existence conditions. As an existence condition for all these transforms it is customarily required that the function be absolutely integrable over the range, i.e. rgf(x) dx exists range Note that although for the derivations to follow, the more stringent conditions of continuity or sectional continuity are imposed upon the function, absolute integrability is all that is required in the general case. II. Some Fundamental Properties A. Transforms of Derivatives of Functions Consider the finite range cosine transform of the derivative f' of the function f, CnfJ = Jf (x) cos nx dx Integrating by parts, Cn f] = f(x) cos nx + n f(x) sin nx dx 0 O = f(x) cos nr - f(O ) + n Sn[f and since n is an integer, Cn, f [] ()n f() - f(O)]+ + nSn[f] Consider also, Sn [ f (x) sin nx dx = f(x) sin nx - n f(x) cos nx dx = -n [] Now take for f, f = g'; we get by iteration Cn [f] = Cn[g' = [(l)n g',() -g' (0+) +n Sn[gj] = [(-l)n g() - g( 0+1) - n2 Cn[g]

-56Similarly, Sn[g"t = -n Cn[g' = n[(-l)" g(1C) - g(O -n2 Sn[g = [g(O) - (-1) g(t)| - n2 n [g Now consider the infinite range cosine transform C~r[f] = Uf7Jf'(x) cos rx dx and again integrating by parts, and assuming lim f(x) = 0, which is a consequence of our condition of absolute integrability, we get Crf'] = -f(1 -f r'ff(x) sin rx dx = -2 f(0) + r Sr[ and also Srf] = f2f f(x) sin rx dx = -r 2 fx) coS rx dx = -r Cr fj Iterating once, we find Cr [f' = _ 47 fv(0) + rSr[f] = - fv(0) - r2 Cr f] I t Similarly, Sr [f] = -r Cr [1f = - rT f(0) - r2 Sr [fJ Finally, consider 7ri irx f = 1 ff(x) e dx and assuming lim f(x)O0 r CfT - r f(x) eir dx r ir"""' i Iterating, ~r [f']= - r2grf]

-57In each case we have assumed continuity for f' and f" in order to perform the indicated parts integrations. One may proceed with the iterations, obtaining relations involving transforms of higher derivatives. Further properties are derivable with similar ease, the procedure usually involving an integration by parts* B. Relations among Infinite Range Transforms. It is interesting to note some relations among the infinite range transforms. Recalling the identity eir =cos rx + i sin rx we find that 6r[f] = 1 jf(x) cos rx dx +' f(x) sin rx dx = 1 If(x) cos rx dx + 1 f(x) cos rx dx + i f(x) sin rx dx + i _ f(x) sin rx dx = 1 |f(-x) cos rx dx + 1 |f(x) cos rx dx +ac i o fxX)snrx' o + i If(x) sin rx dx - i f(-x) sin rx dx or Er )f) 1 f r [f(-x)J + r [f(x)] + i Sr f(x) - i Sr [f(-x which is not very interesting except when f(x) is either even or odd on the infinite interval; if even, i.e. if f(x) = f(-x), then the exponential transform reduces to the cosine transform; if odd, i.e., if f(x) = -f(-x), then the exponential transform reduces to the sine transform, with a factor? C. Transforms of Functions of Two Variables. The transforms may also be used with functions of two or more variables; for example, if f is a function of x and y, defined for 0 ^ x_, 0 y then,

-58Sm 1111 = f(x,y) sin mx dx o Sn [] = f(x,y) sin ny dy Smn[f = Sn f sin my y = f Sm[f] sin n dx f(x,y) sin mx sin ny dx dy J 00 Furthermore, SmnL = m Sn f (Oy) - ( -1) S [fM (y)7J - m2 Sn [f] so that if f(O,y) = f(iT,y) = f(x,O) = f(x,rt) then, Smn - m Smn2a + 2 -(m2 + n2) Sm [ LX2 j J Similar formulae may be derived for Cm n and extensions can be worked out in analogy to the single-variable properties. These transforms of more than one variable amount to transforms of transforms, obtained by taking the transform of the function with respect to a single variable, and subsequently taking the transform of this transformed function with respect to another variable. In fact, if the boundary conditions in the various dimensions are not all of the same type, more than one type of transformation may be usedo (one fairly common combination is the Fourier plus the Laplace transformation). D. Fourier Exponential Transforms of Functions of Three Variables. Consider a function of three variables, f(xl, x2, x3), piecewise continuous and absolutely integrable over the infinite range with respect to

-59each variableo We may apply the exponential transform with respect to each variable, defining the three-times transformed function. E jk J] = g(klk2,k3) = 1 I(klxl+k2x2+k3x3)f(x 2x dxdx2dx Using vector notation, this may be written iki x 5 k = g(k) = 1 f e -- f(x) d3x (2)/3 2 I.X where k has components kl, k2, k3 x has components xl,x2,x3, and d3x = dxldxpdx3, and the integration is to be taken over the full range, -, to CO of each variable. The inverse transformation gives back f(x), -i k -x f() = 1 g(k) d3k (2-)5/2 ) Properties~ say k g(k); ] = G (k) then 1. k ] i k g (k) 2. Ek[bF] = ik G (k) 35 - kIVx - = -i k x G(k) 4. ~kkC J = -k2f (From a glance at formulae 1 to 4, we see that under this transformation, the vector operator V operating on a function transforms into the vector ik times the transformed function).

-60III. Summary of Fourier Transform Formulas A. Finite Transforms (Functions defined on any finite range can be transformed into functions defined on the interval 0 x x C r) 1. Definitions a. Sn [Y] tj y(x) sin nx dx n =,2,.. b. 0n [yj =2 y(x) cos nx dx n = 0,1,. Cn o 2. Inversions (0 t x ~ x ) a. y(x) 21 Sn Iy] sin nx n=l ~1 b. y(x) i? I Cn Y] cos nx CO [y] n=l 3. Transforms of Derivatives a. Sn [ = -n Cn [y n =1,2 b, Cn [y' = n Sn [y - y(0) + (-1) y() n =0,1,2,..(note that functions must be known on boundaries) c. Sn [ ] = -n Sn []+ n y(0) +(-1)y(G n=l12,...(note that functions must be known on boundaries) d. Cn [y = -n Cn [y] - y(0) + (-1)n y(r) n=0,1,2,...(note that derivative must be known on the boundaries) 4. Transforms of Integrals a. Sn f y(()dj = 1 Cn n C [ n=,2,... O n - n b. Cn )j y() dj= -1 Sn [ Y] C, [f(x ) d - ICon Y 3 COrxy

5. Convolution Properties a. Define convolution of f(x), g(x) (-lr_ x'-r) P * J P(x - ) q(~) d = *p b. Transforms of Convolutions Define extension of f(x), where f(x) defined an range 0 x L x Odd extension: fl(-x) = -fl(x) fl(x + 2g) = fi(x) Even extension~ f2(-x) = f2(x); f2(x + 2c) = f2(x) 1. 2 Sh [f] Sn[g= -Cn [f*gl] 2 2 Sn [f] Cn[g] = Sn [f 2 3. 2 Cn[f] Cn [g = Cn [f 2 6. Derivatives of Transforms a. d n [Y] = Cn [xy] dn b. d Cn = -Sn [xy dnn (Here the differentiated transforms must be in a form valid for n a continuous variable instead of only for integral n). B. Transforms on Infinite Intervals. It must be true that Jy(x) dx or Jy(x) dx exists. 1. Definitions: a. Sr [y =2'jy(x) sin rx dx r7O b. Cr[ Y =/y(x) cos rx dx r30 c. Er [j = < Iy (x) eirx C. Fy. = -_ Y (x) eY dx -_ OO oO 2 it, L

-622. Inversions co a. y(x) sin rx dx x > = ( x S, [y]] ijx sr b. y(x) =\ |Cr[Y] cos rx dx x 70 \ x r c. y(x) e=1 [y] e dx = -xi.[yi [ -QD (The Cauchy principal value of the integral is to be taken). 3. Transforms of derivatives a. Sr yl = -r Cr [yl b. Cr [y' = r ry] - y(O) c. [y'] =-ir r[Yj d. Sr [yjt = -r2 S [y]+ ry(O) e. r [y ] =-r2 Cr [] -'(0) f~ & [ytJ = -r2 r [yj g dny (-ir)n r [Cy 4. Transforms of Integrals b. [ y(f)o ( =- 1 Sr [y] x (In a and b, require 5o y(r)dTbe sect. cont. and-gO as x-) ). c. y(') dl = c is any lower limit r r x (in c. require ( y(~) dpto be sect. cont. and-oas x-co). -eo~

e. Jf oy(T) d J= -1 rlJ I Y(5) td~g 12 r 2 X X rd; I r2 5. Some Relations Between Transforms for real y(x), ao Cr [] + i Sr[Y = r[yj or JCr [Y = Re (4[Y] ), Sr EY3 -J (CrJ ) b. For y(x) = y(-x), y(x) Q(e-t ) 6 eo Cr[ y = 2 L y where Laplace transform variable is taken as ir. 6. Convolution Properties a. 2 Sr []Sr[g] = Cr [g(T) (f(x+7) - fl(x-T)) dj b. 2S[fr Cr[g] Sr[ g((T) (f(x+T) + fi(x-T)) d] = Cr o(T) (g2(x-~) - g(x+V)) dfI c. 2 cr.[f Cr[gJ = Cr[Ji g( ) (f (x-)) f( d where extensions defined yj(-x) = -y(x); Y2 (-x) = y(x), all x. d. Er fr3 I [r [f (r) g(x-T) d 7. Derivatives of Transforms a. d Sr [y = Cr [xy dr b. d Cr [yj = -Sr [xy dr E[ ir:xy C. d Er [Y~: i rx

IV. Types of Problems to which Fourier Transforms are Applicable. A. General Discussiono It is to our great advantage to have some inkling as to just which transform to use where. We have noted that finite-range transforms are useful on functions defined over a finite range, Ac [rand s [r] are useful on functions defined over semi-infinite intervals, and eCrjon functions defined over the infinite range. Still more can be said. First, it goes almost without saying that if it can be avoided, it is undesirable to introduce an unknown quantity into an equation. Now, if an equation in f, which is defined 0 ~ x r contains a differential operator which one wishes to reduce, say d2f and f(g) and f(O) are known, while f' () and f'(O) are not, dx2 then clearly Sn is used, for in so doing we introduce f(g) and f(O) and need not know the value of fV at any point. We would not use Cn for f'(O) and f' () are unknown and would enter the transformed equations as unknowns, which would not be solved for until later in the work. On the other hand, if f'(0) and fV (x) are known, one uses Cn for the same reason. The situation is similar with respect to the infinite range transforms; use is r7 reduce d2f when f(0) dx2 is known, pcr]when f (O) is known. No such question arises with respect to leCr)r We have noted at the start that the functions to which the Fourier transforms are applicable are usually required to be absolutely integrableo This kind of knowledge of a function is usually evident from the physical meaning of the function, before the function itself is known. The Laplace transform, on the other hand merely requires that the function be of exponential order, i.e., If(x)I Mea M If any real numbers B. An Example of the Use of Fourier Transforms Consider the following steady-state heat conduction problem in a medium with no internal heat generationo

-65Face x=0 has a heat flux q, 0 L y. a, and is insulated, y > a, and face x=-I is insulated for all y. Face y=O is held at temperature <, all x, 0 ( x LC C. The slab extends 0 _ y _aM. The equation to be solved is Laplace's equation with boundary conditions. ~ ~= 0< Xxx=O 0 y a al1l =0 ax x=: -$(x,O) = m We propose to do the problem by the method of Fourier transforms, but intuitively we know lim -, 0 and, therefore, the transform of does not exist. Iowever, the function - pEO is such that lim l = lim r- =- ~Oand the transform may (in fact, does) exist. Let us, therefore, substitute in the above problem 69+ )+ to obtain 2 W72 = O -k _. J (o 0 Oy a x x x=O 0 ~ y a a6 g = o x X=0 9(x,o) +: = 0; (xo) = _- ( _ 9o

-66The structure of the problem is not essentially changed, except that now (9 is not known since * is not known. We must reduce the operators a and i2n. In x, we known )6| x2 a y2 x x=O and ~~= ~,. Thus, a finite range Fourier cosine transform is indicated (see section IIA) In y, we known k(x,o) =t and that lim = O. Therefore, an infinite range Fourier sine transform is indicated. Denote x-transformed functions by superscript fn, y-transformed functions by superscript F. Recall aAnd f =-n26+f(-l)n -x x tax2)' X x x=O and (/ ) = -r2 F + rG(%^o) It is irrelevant in which order the transformations are applied or inverted, although one order may prove nicer than another. Let us transform first with respect to x. -n2 ~ fn + (l)n __ - as + d2fn = 0 X lxE=.T x x=O dy2 1/k 0 ( y C a ax x=O = - 0 y-a g l = 0 x x-= 0 j n=O fn (0) =( 0c O n=1)2,., then with respect to y: (Churchill pag. 300, formula 3) F F _n2 )fnF ( Fa + (- r (0) t0 )x x|= x xFO ae F = -q/k (l-cos ar) Yx x=O r (See Erdelye, p. 63, formula 1)

)x x=f= fn"(0) = 0 n = 1,2,5,.. Making substitutions, this yields the single algebraic equation. (-n2 - r2) ^fnF + q (1 - cos ar) = 0; (n=l,2,...) k r -r2f ~F + + (1 - cos ar) + ir = 0; (n = 0) k r Solve for nF foF - q (1- cos ar) + r c; (n=0) k r3 r fnF =q (1 - cos ar); (n=l,2,...) i k r(r2 + n2) We propose to invert first with respect to r, but we would run into difficulties for n=0. Let us, therefore, integrate the x-transformed equations directly for n=0 to get fo. We have d26f~ + q 0 0 yL a dy2 k d2fo = 0 y a dy2 ~o(y0o) =- fo(y=O) lim fo = 0 integrating, efo = q y2 + Cy + C2 0 < y a 2k 0fo = 3 + C4 y > a Now lim f = 0, so C3 = C4 = 0. Also fo~(0) = Co =C2 It is necessary to cook up another condition to get C1. Ln a problem of this type, we must require,@ and 9 to be continuous, therefore afo a" ay

-68and fo are continuous. Apply these conditions at y=a. d ~ / - d dy a- dy a+ - qa + C1 = 0; C = qa kj k fo (a-) = &fo (a+) -qa + qa + I t= 0. 2k k Somewhat surprisingly, applying this last condition yields ~ 2<k Thus ofo 2= (y - ya a 2k (O y_ a We have 0fo. Let us invert efnF to get efn fnF = q (1 - cos ar) (n=,2, ) k r(n2 + r2) The inverse of 1 is 1 (1 - e-ny) (See Erdelye, p.65, formula 20). r(n2 + r) Also, a property of the Fourier sine and cosine transforms is F1 [g(r) cos ar] = 1 [G(y + a) + G(y - a and it is also true that for the sine transform, if F-l[gf= G(y), then G(-y) = -G(y). Therefore, F-1 1 (1 - cos ar) = r(r2 + n2) r2 2 1 (1 - e-ny) - 1 e-n(y+a) + n1 - e (_ n2 2n2 j 11_ (1 - e-nY) -_ 1 1 - e-n(y+a) _ 1 + e-n(a-Y)

-69f1 1 - e-ny -1 + e-ny (en na) )1 f - e-ny + in(6ya) _ 2-n(a-y) y yCa n2 Now lacking a known inversion to invert with respect to n, we use the series form fn2'+ 2 CO nx n= 2q e"ny (cosh na - ) cosnx a jik ni and recall that @= 0 +' 2nrk <$e-n - a so n) 2n2 = 9 +&e+4a V. Inversion of Fourier Transfor ms. A. Inversion of Finite Range Transforms Inversions of the finite range transforms are easily seen to be a consequence of the completeness and orthogonality of the cosines in the case of the n=n cosine trand of the sine in the case of the sine transform on the interval of integration. Indeed, one sees that the integrals defining C

-70and S are just the Fourier coefficients for expansions of f in a cosine series or a sine series. Thus their inversions are given by f(x) = Cnl [Cn] = 1 CO [f] + 2 Cn f cos nx (0 x 4 ) t it n=l f(x) = Sn S 2s Sn [f sin nx (0 - x 4- ) n=l Two facts} though obvious) should be noted with regard to these transforms. If the function is defined over some range other than 0 _ x S a, say 0 4 x L there arises no difficulty since one can define a new variable, say = x L such that when x L4,= jt, and f(x) = g(?) = f(L A) and proceed. If the function is extended out of the range 0 L x g, the inversion of the cosine transform is the even extension, i.e., Cn [Cn (x)] = [Cn (-x) while the inversion of the sine transform is the odd extension, i.e. sn1 [Sn (x)] = - S1 Sn (-x) B. Inversion of Infinite Range Transforms. Inversions of the infinite range transforms follow from the Fourier integral theorem in various forms. The inversion of the cosine transform, for examplearises from the formula f(x) =2 cos rx f(y) cos ry dy dr = cos rx 2 f (Y) cos ry dy dr The interior integral is just what we above defined as Cr f] thus f(x) = Cr1 [Cr] = r Cr f] cos rx dr The other inversions follow immediately in the same way from other forms of the Fourier integral theorem. It is to our great advantage to note that, with the normalization factory or 1 inserted as above, the inversion integral is just the transform of the transform, i.e.,

-71f(x) Cjr1 [r] = Cx [Cr [fi] Similar formulae for the sine and exponential transforms are f(x) = S1 [s S [S r [] f(x) = Er1 [ Ex[Erf)J Knowing this fact doubles the utility of a table of transforms.since it can be used backwards as well as forwards. That is, given a transform one wishes to invert, one may first look for it among tabulated transformed functions; not finding an inversion there, one may equally well look for his transform among the tabulated functions, if it is found there, the inversion of the given transform is the transform of the tabulated function. There are tables of both the finite range transforms and the infinite range transforms, useful for the purpose of inverting these transforms. However, this is just one way of obtaining an inversion (the easiest, of course). In the case of the finite range transforms, where the inversion is a Fourier series, and one does not know how to sum it, that is, get the inversion in closed form, then the truncated series in a useful approximation to the inverse. In the case of the infinite range transforms, the inversion integral is subject to evaluation by the methods of complex integration and residue theory. C. Inversion of Fourier Exponential Transforms. We have seen that if the transform of a function F(x) is f(r) = 1 Fx) eirx then (under proper conditions on F) the function can be recovered from its transform through the inversion formula, F(x) 1 (r) e dr. Note that in the inversion formula, r is a real variable. Let us change the variable r to a new (complex) variable, ir = s, and say 0(s) = f(r). Then p(s) = 1 F(x) e dx l[FLx)

-72and the inversio is (sine idr = ds) F(x); - i f6(s) e-s ds c where 1 the urve e i the s-plane belows and. the Cauchy Principjl value is to be taken, - s-plane h.,C that is ia F(x) = - i limn 1 ( (s) e"SX ds Suppose p(s) is analytic in the left half plane (as) < O, except at a finite number.of isolated singular points sno Let us close the-path cB ~ - p _ Ct ~_~ with a semicircle in the left half-plane, choosing p. 6i' * C s-plane - 53 so large as to include all finite singular points in the plane ((.s) O. By Cauchy's reisidue theorem, then we have

-.475 J e-^ ~s + J e-^ d(S) ds - 2i g pa ote the reside Where eB the residue of esX(s) at the singolar point 8, and We have astunied that there are k -such singular points..Sine we have hpothetsized that 5 be so large that c' ilude al fi-te s lr pi in. the eft half plae in the limit as >0, the right side remains c t ant d we have lim J e9'(s) d+ s l+ r efe'^X(s) da - 2(j u J lim ex (s) das = i ei x (~ 5 -e ( or ilp k ( k -4X3 F(x) = - ii.m ( ex (s) ds P^ Qnlip 5 - + i lim js ei < (s) ds J f P Jc K Many times., the limit on the right hand side is zerot or is easy to evaluate, so' that the above formula is a very useful device for inverting the transform-. Reasoning in similar fashion, but'oanpleting the path with a semicircle in the right half plane (j(s) ) 0> we obtain a similar formula, F(x) = -f;2 P _ i liim f e-sX(s) ds whreth cu where the curve c"i is shown below, P -^/ " —^s-lne __^ y ^~~~~~r/ 0/^ —

-74. One smay find. that for x ( 0, the limit e.X (s) ds 0 so that the first fonrmula be'acomes w(x) (l (recall that ft. are residues at singular points in the left half-plane (s)L0 )4 Again) o'ae may find. that for x? 0, the limit lira eX -(s) d = 0 that thhe second. formla becomes F(x): f Z 0^ At x 0) F(o) [ F() + where:F(0 ) lim F(),, and.F(0) = l. F( -6) Appendix VI TrjA'form. Table' (Y) 5 y.(x) sin nx dx n a pO itive integer.o a' ~a~a Ch(Y) o y(x) cO's nj.c dx n a on-negative integer y (x) Asn(y) 1- (^i )~flj.a_ (-1)n n2 + x^ at (-)^ - 2/^ pL - a

75 r1. - co s h + C..t yWx) slk)x toX a n J os x On asih ex 1' (-1) Inh J a 22 221 c-oDS \ sh1 eh- (-1) enosh caa Xfa - x) a sin x (a. - x) 3 [ - ] sinca i-a' -a V / si c(a x) nta sinh ca 2 y(xO) C (x)) 1 a1 (n=O) 0 (nrc, &,..) x 1 a2 (n=0)

y(x) _Ca(x) x~2~,,3 (n=o) 3 eCX 1 II+a2l i) ea - si, k x 0 a BLW Wo an i (nk ) cos W x la (n 4Sa cosh cx a2c (.1 )n snh ca u=,:2 + c2a2 (x - a)2 4 a3 (n=o) 2 n___ cEos't(a - x) a2W cosh c(a - x) a2c ^inh c a n2 + ca2 Fbr a few additionai tfransfoxs of this type, see Churchill or Sneddon (Refer'en'ce 2 and 3) For transforis of the form (x) einx dx, x) sin nx dx, and. (x) o nx dx, see.rdelyi (reference i).

-77 - References lo Sneddon, Ian N "Fourier Tr rms McGraw Hill, 1951. 2. Churchill, Ruel V., "Operational Mathematics", McGraw Hill, 1958o 5. Erdelyi, Volume I, "Bateman Mathematical Tables. Mis C I1 i ID1.ENTITE, DiES 3N ITIQfO, FUCTION AND NOTATIOS IL ILeibnitz's Rule b(x) If f(x) = g(x,y) dy a(X) Then b(x) d f(x) = g [x,b(x)1 b'(x) - g [xa(x)] a'(x) +,x 2 g (xy) dY a(x) 2x II. General aolution of first order linear differential equation. dy + a(x) y f(x) dx with boundary condition y(xo) = YO The procedure is to find an integrating factor. Define h such that dh = a(x). Thus dx x h / a(xt) dxt Yc The integrating factor will be eh, since d(ehy) =eh dy + ey dh= eh dy + eh a(x)y =eh:f(x) dx dx dx dx Then,yeh x eh d(ehy) - eh' f(xt) dxx where ho = a (xx ) dx xo h' = ( a(x") adX 0 0~~"

x ho hi yel - yOehQ eh' f(x') dx' 0 Hence x y eh/-h + y eh'-h f(x') dx' Recalling that x h = / a(x') dx' -c x'x x h1 - h = a(x') dx" - / a(x") dx' - a(x") dx x a.x(' )dx y = yo eho-h + f(x7) e4- dx' o (Note that the constant c appearing as lower limit in the integral of the integrating factor is not a boundary condition: it disappears in the final solution). III.Identities in Vector Analysis Below, underscored quantities are vectors, and V is the vector A A\ ^\ A\ AAA differential operator v = + 4- and i,j,k are unit vectors in x, y, z directions respectively.

-791) a b b c b x a = * a x b 2) x (b x -) b(a * ) - c (a' b) 3) (a x ) b (c x ) = a b x (c x d) a (b d - b cd) (a c) (b d) - (a d) (b c _ ) 4) (a x b) x ( x d) (a x b d) - (a x b ) d 5) V(A +Y) = + vL6) (y) - i7t + d0 7) V(a b ) = (a_ ) b + (b * 7) a + ax(x b) + b x (v x a) 8) (a + ) =V_- a + +V b 9) V x(a + b) =Vx a Vx b lo) __(O a) = _a + - a 11) Vx (0 a) = (7V) x a + x a 12) V. (a x b) =b 7 xa- a x 13) 7 x(a x b) = a V - b - b V a + (b -V) a - (a *7) b 14) 7V Vx a V x( a ( ) V 15) VX-V -= 0 16) V7.V Vx a - If r i x +j y + k z 17) V r = 3,Vxr = 0 If V represents a volume bounded by a closed surface S with unit vector normal to S and directed positively outwards, then. 18) JV0 dv = f d a 20) r d v = da (Ggaus Ig Thorem) 20 g^ f

-8o21) g Vf d v = g n d s - f - g d v 22) x a) d v = x a d ad 23) j,(<PV- V7Z ) dv =,~, ) n d s (Green's Theorem) If S is an unclosed surface bounded by contour C, and ds is an incrementof length Ialong. 24) f xVO d a 0 d s 25),fx a * n d a = a *f d s (Stokes' Theorem) IV, Cartesian, Cylindrical^ and Spherical Coordinate Systems Cartesian. 7: -- - Cylindrical ya' / ^ ~ ~~ - r-^ q4 VN I 7 I.~'^~aG/'6.!~ 4r p$~

Spherical - - - -^J /.:g / r_ V. Index Notation A short note will be given on this notation which greatly simplifies certain mathematical problems (to mention one advantage). What is involved is essentially just the adoption of a convention. The convention used here suffices for work in rectangular Cartesian coordinates. For more general coordinate systems a more elaborate convention is needed; it is explained in reference works, see for example 1) 2, 3. Consider a simple example which illustrates the utility and application of the index notation. Suppose:one has a set of three equations u -= ax + ayy + azz v = bxx + byy + bxz w = e Cx y + yy z By defining u u1,v u2 = u3; x = x1, y = x2, z =x3 a = all a a xb a11ay at2a Za) a x a by = a22 bz =a23 C'= a 1- ay = 2 C - = a;

-82These may be written u1 axl +x + al2x2 + al u2 = a2lxl + a2x2 + a23X3 u3 a3xl + + a32x2 + a33x3 or 3 oUl 5 u =3 a3cx or 53 U >. " "aid (i = 12)53) To this point we have effected considerable simplification of the original equations. By the intrQduction of the:summation convention", we can go still further. Notice that there are two kinds of indices on the right, ig which occurs once in the product, and a, which occurs twice. Index i is called a "single-occuring" index$ a is called a "doubly-occuring" index. The convention to be introduced is: a. Doubly-occuring indices are to be given all possible values and the results summed within the equation b. Singly-occuring indices are to be assigned one value in an equation) but as many equations are to be generated as there are available values for the index. Thus by Part a we may drop the sum symbol, and by Part b we may drop the parenthesis denoting values for i. With the summation convention in force, we have u. -= aica i1~

85which unambiguously represents the original e'quations) if a and i have the same range, which we shall assume. A nice but unnecessary finishing touch can be put ohz the convention which seems to make things clearer in the work: for all singly occuring indices use lowercase Roman letters; for all doubly-occuring indices, use lower-case Greek letters. We remark that it is possible to have any number of indices on a quantity. VI. Examples of the Use of Index Notation A. Some Handy Symbols 1.,. = j1 = 0 i A This is the Kronecker delta. 2. E. -=l, 1i j k, j k, ij,k in cyclic order ijk =-1,i j i k, j / k, i>j,k in anticyclic order = =0 i = j or i k, or j = k This is called the Levy-Civita Tensor densityo B. Some Relationships Expressed in Index Notation 1. Dot product (a scalar) a b a= aab 2. Cross-product (a vector; consider ith component) (a x b)i = 6 abC = - Ei ba 3 Triple product (a scaler) a bxc = a6ob 4. Gradient ( Q is a scalar) (s_)i = ax axi

-845. Divergence (V is a vector) ty = a\V a 6. Curl (V is a vector) (~x V). = ia OV~ 7. Laplacian v27 = a. xacxa x xj C. Some Identities in ij and iJk in 3-D Space 1. 5 3 2. Eapc? y = 6 3. 6 iQc~jiC = 26ij 4. ijj6k3 = iikjkl - il:jk 5- 6ijk:n iljmkn i15j + 8imjn51 inJl-km- iljnkm r imSjl1kn - 5in:jmkl VII. The Dirac Delta "Function"'The Dirac delta "function" symbolizes an integration operation and in this sense is not strictly, in the interpretation of Professor R. V. Churchill) a function. Thus the quotation marks around the word "function". It cannot be the end result of a calculation, but is meaningful only if an integration is to be carried out over its argument. We define the Dirac &-"function" as follows: 8(x) - o, x J:(x) d7. =1, 6?0 J~f(x) b(x) dx = f(O) -6

Very often it is convenient to think of the 2-functiont" as a function zero everywhere except where its argument is zerop, but which is so large at that point where its argument vanishes that its integral over any region of which that point is an interior point, is equal to unity. Mathematicians shudder at the idea of the -"tfunction!', but physicists have used them for years (carefully), finding them of great utility. Schiffts'tQuantum Mechanics" lists some properties of the Dirac 5e:(x) = 8(-x) s (x) -8 (-x) ( 1(x) d5(x) ) dx x-.(x) = X&: (x) - ~(x) ~(ax) - 1 B(x) a> 0 a b(x2 a2) = 2 (x- a) + (x+ a) a70 a L,, j 5(a - x) (x b) dx -= (a - b) f(x) &(x - ) = f(a) (x - a) Professor Churchill uses as a S-tfutnction" the operation (f(x) U(h,x) dx, 70 h-,>o ~E where U(hx) is the function 1( h ~ h h U(h,x) = h 3i 2 2 h h 0; x < -2 x A'?CL.

-86Here, note that the limit is taken after integration. Other such representations are common, like that given by Schiff; 6(x) = lim sin gx - *o ICX which means not that the limit is to be taken exactly as shown, but rather that it is taken after integration, i.e., with this representation, f6(x) dx lim (sin gx dx _ ^ 6-o ) i xx fx) (x) (x)dx = lim f(x) sin gx dx. -. )c _ ax Schiff gives still another representation, in terms of an integral; 6(x) x.d- l 1 g lim 1 2 sin gx g -a0 2r x = lim sin gx g ->ao Ax thus / ff(x x) (x)dx = e f(x) ddx. -e —. VIII. Gamma Functions. A. Definitions F'(x) = J( et tx-l dt W e

-87B. Properties a. r(x + ) = x (x) b. () = ( - 1) ( l 1) (n positive integer) c. r(x)r(l-x)= rt sin nx d. r(x) x r (x + 1/2) 2x-2x 1/2 e. r (1 b) = 1 - a 2 - a 3 - a... (1 -a) 1 -b a - b 3 - b f. f(l/2): = f Since by (a) one may reduce P(x) to a. product involving r of some number between 1 and 2, a handy table for calculations is one like that found at the end of Chemical Rubber Integral Tables forr (x) 1 X _ 2. References 1. H. Margenau and G. M. Murphy; TThe Mathematics of Physics nd Chemistry" Do Van Nostrand Company, Inc. New York, 1956, pp. 93-98. 2. Whittaker and Watson, "Modern Analysis", 4th Edition,, Cambridge University Press (1927), Chapter VII IX. Error Func'tion A. x 2 erf(x) = 2 e d X B. Jo.~x = -erfxco B. erfc(x) =1 - erf(x) = 2 e"-2 d X x

-8-8 NOTES AND CONVERSION FACTORS I Electrical Units A. The Electrostatic CGS System 1. The electrostatic cgs unit of charge, sometimes called the escoulomb or statcoulomb, is that "point" charge which repels an equal "point" charge at a distance of 1 cm. in a vacuum, with a force of 1 dyne. 2. The electrostatic cgs unit of field strength is that field in which 1 escoulomb experiences a force of 1 dyne. It, therefore, is 1 dynq/escoulomb. 3. The electrostatic cgs unit of potential difference (or esvolt) is the difference of potential between two points such that 1 erg of work is done in carrying 1 escoulomb from one point to the other. It is 1 erg/escoulomb. B. The Electromagnetic CGS System 1. The unit magnetic pole is a "point" pole which repels an equal pole at a distance of 1 cm, in a vacuum, with a force of 1 dyne. 2. The unit magnetic field strength, the oersted, is that field in which a unit pole experiences a force of 1 dyne. It therefore is 1 dyne/unit pole. 3. The absolute unit of current (or abampere) is that current which in a circular wire of 1-cm radius, produces a magnetic field of strength 2,t dynes/(unit pole) at the center of the circle. One abampere approximately equals 3 x 1010 esamperes or 10 amp. 4. The electromagnetic cgs unit of charge (or abcoulomb) is the quantity of electricity passing in 1 sec. through any cross section of a conductor carrying a steady current of 1 abampere. One abcoulomb equals 10 coulombs.

5. The electromagnetic cgs unit of potential difference (or abvolt) is a potential difference between two points, such that 1 erg of work is done in transferring abcoulomb from one point to the other. One abvolt 10-8 volt = approximately l/(3 x 1010) esvolto C. Practical Electrical Units and Their Equivalents in the Absolute System Practical Electrostatic Electromagnet cgs cgs Quantity 1 coulomb 3x109 escoulombs 1/10 abcoulomb Current 1 ampere 5x109 esamperes 1/10 abcoulomb Potential Difference 1 volt 1/300 esvolt 10 abvolts Electrical field 1 volt/cm 1/300 dyne/escoulomb 108 abvolts/cm strength D. Some Energy Relationships 1 esvolt x 1 escoulomb = 1 erg 1 abvolt x 1 abcoulomb = 1 erg 1 volt x 1 coulomb = 107 ergs 1 joule The electron volt equals the work done when an electron is moved from one point to another differing in potential by 1 volt. 1 electron volt = 4.80 x 1010 escoulomb x 1/300 esvolt = 1.60 x 10-12 erg.

II Physical Constants and Conversion Factors; Dimensio Anal ysis Numerical Constants e(base of natural logarithm) 2.718 loge 10 25303:t;2.a^~~ ~9. 870 Lengths, Areas Micron 10- 1 cm Angstrm unit A 10-8 cm X-unit XU l 1o11m Wavelength of 1-volt photon 12,396 A Calcite grating space at 200C 3d 5.036 A Separation of electron and proton in groun state of H a 0.591 x 10 cm -co 2.426 x 10.8 cm Compton velegth h/me 2.426 x l010 em "Conventional electron radiust e2/m 2.8175 x 10'13 cm De Brigl e eVae of 1-volt electron h/r 12.26 A Bam /b 10- cm Masses and Mass lEquivalents'28 Electrons mo 9.107 x 10 28 16-.~~ ~5.488 x 10o AMU 1/16 mss of 01 AMU 1.6595 x 10-24 = Atomic mas unit Proton Mn 1.6722 x 10-24 gm.00758 AMU, Neutron MN 1.6744 x 10-24 g 1.00894 AMU Deuteron D 3.343 x 10-24 gm 2.01417 AMU Alpha particle Mo 6.642 x 10 gm 4.oo079 AMU H1 atom 1.00812 AMU H2 atom 2.01472 AMU He4atom 4.005389 AMU Proton masa over electron mass R/mo 1836.1 Energies and Speeds Electron volt 4.80 escoulomb x 1/300 esu ev 1.602 x 10-12 erg Million electron volts Mev 1.074 x 10 3 AMU Energy equivalent of electron mass moc2 0.5110 Mev Ionization energy of H atom 13_.60 ev Speed of 1-volt electron 5.931 x 10 cm/sec Speed of light c 2.9979 x 1010 cm/sec 2 8.9874 x 1020 (cm/sec)

-91Other Electronic and Atomic Constants Electronic charge e 4.802 x 10 esu 1.602 x 10-20 emu Charge/mass for electron e/m 5.273 x 1017 esu/gn P c cott 1-759 x 107 emu/gm Planck's constant h 6.624 x 1027 erg sec, or 4.155 x 10-15 ev sec Unit of angular momentum h/2i 1.054 x 1027 erg sec Duane s constant h/e 1.579 x 10-17 erg sec/esu tydberg constant For H1 atom 109,678 cm1l For infinite 109>737 cm Bohr magneton ^B 9.271 x 1020 erg/oersted Fine-structure constant 7.297 x 10"3 = 1/137.04 Constants Needed in Kinetics and Radiation Theory Gas constant R 8.314 x 107 erg/(mole OC) Boltzmann' s constant (gas constant for 1 molecule) k R/N 1.380 x 10-16 erg/(~K molecule) Molar volume of perfect gas at 0OC and 760 mm of Hg Vm 22415 cm3/mole Faraday?* F 9652.2 emu/equivalent Avogadro's number N = F/e 6.025 x 10 3 molecules/mole Number of molecules in 1 cm3 of perfect gas at 00C and 760 mm of Hg F/eV 2.687 x 1019 molecules/cm3 Average kinetic enery of a molecule at 0~C and 760*w.m - of Hg (To - 273-16~K) 3/2 kT 5.655 x 10- erg of H~g (T= 273.160K) g25k4 5.669 x 105 erg/(cm2deg4sec) Stefan-Boltzmann tonstant a 15 h35c First radiation constant c 4.99 x 10-5 erg cm Second radiation constant c2 hc/k 1.439 cm ~C * Based on the "physical" scale of atomics weights, in which 016 = 16 exactly. On the chemical scale the value 16 refers to the natural mixture of oxygen isotopes. The two differ by 0.018 per cent

-92(C I I CQ 00 0 0 1)) *rd;4i~ r^dH'H 5C0 H (1 0)H I 4 Q 03 | h g |h H| |k h0 | g |H 0 0)0 0 $-43 B 4* ( *H) 0^2D fr rTJ H P?.> l d S a bO o a rdrd'H ~2.Q 0), > w. H O' o 9 U o o o QC, 0k ~ ~ ~ ~ ~ ~ 40 A7 g g I SV X A p WIg X W w n m N N N X i_ Q td vl F~1 l1 I* *r1i Q2 d) U S ~ O a)H @ H PL1 t Q o O O t b U-a tO9 > pt. O rd a) H O o 103 41-^ ) ( 1 1 H d rd} S 0 |HO * H CH H ~C 2 CH ~2*: 4 3 -,'- FH 0) 0-R ^ I3 0 0 O I 03 I0 I -P -P -P 4t-. OX HOr-~4Ot +k 43> 0d0~ 0) Nc | 0d %( @ 3 03 E F(1 V H3 co P4 l co c 0 ~ <D 0 IHI=03~~ I0) ~ a H -. 0^4 a) IL, 4P> IH~ ^ Ttg, c3 0H'?- CUg g I 1 CP Ql 0r * rJ (D 6 4?- N r t *1 T i 3 4 l H a5 eP. a) ~2 l 3 p( { r ( 0 ( 0 0 C (4H 00+~ 0 3 0' l + w mm 4> b ) ~2H10)rP 0 ) 03 0 ) Q 0 r (~2 (3 ~l03 Ca _0'u M'S ". ~CHr l m-{ ^ QH ^ rQ 0 ^ fCH HU 0 0'HSri~~~oi 0, Q'H H 0) 0 P, yH 03H 0 ~2H'H.'H'H'H-H-F ~2'H'C a) g bbgo~kk 00 ooo^ ^k g g?-( P P 0 03-( 4 g3 ~ 0)0) Q H r4 0P0))r~j r 4 4> o3 i) -(D (2UO (L>J)r r< d H rQ -P f d a) EQ 4-l> r -P ro.a; * > p3~ t? U U 0 -P 03 Q0 ~Q ^ ^ +> d 4-^ I<

Table 2 Conversion Factors To obtain Qutanti ty Symbol Multiply Number of By Nmber of Dynes. 10-5 Force F Pounds 4,448 Netons OQunees 02780o Centimeters 10-2 Length A Feet 0.3048 Meters Inche3s 2.540 x -10 Mls 2o540 x 10-5 Centimeter2 104 Feet 0.0 20. Area A Ia.'hes"2 6.^j452 x 10- 4 Meter.Mils2 6.452 x l10 10 Circular mils 5.67 xlO1 Iyne —centimeters 10-7 Torque T Po.und-feet 1.356 Newton:meters Oumice-inches 7.062 x 10-3 Ergs 10-7 Energy w Foot-poxids 1.356 Joules British thermal uunits 1.055 x 103 Kil-owatt hours 3.600 x 106 Charge Q Statc oumb 3.b5335 x 10-10 Coulombs Abcouombs 10 Eleetric potential D:Statvolts 299 8 Volts Abvolts 10-8 Statvolts/centimeter 2.998 x 104 Abvolts/centimeter 10 Electric field Volts/cent imeer 100 Volts/meter inteasity Volts/mil.937 x 10

Table 2 Q1uantity Symbol Multiply Number of y To obtain Number of Tjlnrationalized NKS units of electric flux 7.958 x 102 Electric flux 9 Esu of electric flux 26,54 x 10"12 Co-ulmbs Eiu of electric flux 0.7985 UxInrationalized MS units of electric flux density 7.958 x 102 ELectric flux density D Esu of electric flux 2 density 26.54 x 10-8 Coulbs/meter kEmi rf electric flux density 7.958 x 10-3 Capacitance C Statfarads 1.112 x l012 Abfarads 109 Current I Stataperes 5 5 x 1 peres Abamperes 10 Pragilberts 7,958 x Magnetic potential Esu of jmagtetie potential 26.54 x 10-12 Ampere-turn Gilberts 0.7958 Praoersteds 7.958 x.10 aFgneti~c field t of magmetic field inteity su f ati field intensity 26.54 x 100 Ampere-t rs/eter Oe tteds 79-958 Mazetia flux density B Esu of magnetic flux density 2.998 x 106 ebers/eter Gauss 10-4 ndatc taine L Stathenrys 8.988 x 10 Henrys Abhenrys 10-9

-95Table 3 Dimensional Analya&is, Us-ing, F L, T> abdt Q Mechanical Qatity_ Syibol1 Uit Dimension Force f newton F Length x meter L Time d aeond' Velcity v meter/?second LT-1 AcCeleration a meter/seo dLT.2 Mass M kilogram M -1 Spring Cn:t'.'a~ntio KTrnaf meter'/newt:on FL' Damping acwntant (translation) R= f newto'-e.-seond/meter ^Torquie T newton —,meter FL A'ngle.0 radian An.gular veloeity' radian/se ond TAnguilar a-ccelerati.on radia"'/e:O nd2 T'Moent of inertia I kilo.gram-meter 2 FLT2 T Spring.constant (rotation) KR ne'wto-meter FL Damping constant (rotatian) RR = newton-meter-secnd FLT Energy W joule FL P:oer P wsatt FLTElectric or -aNeti.atiiety Symb Unit Dibolmesion'Charge Q.coulomb Q Pe'mittivity E farad/meter F-1 L"El-etric field intensity volt/meter Fl Electric flux density D coulomb/meter'F L-'Q Electric flux co lomb 1 Capacitance C farad FQ-"1 t Q Current, ampere TMagaetiC flux density B'veber/metei2 germe abt'ity ~-..~~TIC"'-: P'rrmeabllity 4. henry'/meter LPQ Magnetic field intensity H ampere/meter Magneti potent ial ampere MNagetic flux (or flux inkage). (or x) reber FLT QI Inductance L henry'FT2 Magnetic pole (a mathematieal on cept) p w'eber FLTQ-! Resistance o hm FLT 2

Table 4 Dimensional Analysi, Uaing M, L. 1T and Mechahicc'al Qantity Symbol Un it Dimension Maass i kilogram M Length x mieter L Tpime seaond. T Velocity meter/second LT-i AceleratiL a e~telr/seod nC2 F:ore f newton MOLT; Spring c'-otant (tr anlatioi) KT f netonf/meter MT' Damping constant (tr atin) I e e.on sond/meter "1 To3re T neo.t-eme./t.err ME2T An-gle 0 radian. Angular "relocity radlian/.Second T-1 Angul ar.acelerati a radi~aa/seC-ond T-2 Momet of inte.rtia I k1logram -meter2 ML2 T Spring Ornstant (rotation) newtn-meter ML T Damping c.nstasnt (rOt ation) newton-ieter-second E Ene.rgy W joule ML2T2 wPer P watt MLRT -3 Electri or Magufletc Qu^antity_ S 0_____g'uynit Dimensi'on Pe.raeability he nry/eter Current i ampere l2l AC agnetic ~flux de- ity B weber/meter 2 T Maget i field.denity H a.pere/iaeter M1A2L-/'2yT -l/ 2 Magnetoi potential ampere M1/2L/2 "!/2 Magnetic fiux (or flux linkage) ~ (orx) weber Ml/2L5/2T-l/2 Charge eC-.ul, omb Pemnittivity farad/eter 2 Electric field intens ity Eo t/meter 2 P1l21/2 Elecbteric* potei2 T-al L/ Electrlic!L-ux density D'coomb/meter/ / Electrie flux q "mcoul.omb 21/P"' tcaacitance C farad i1^P241 Magetic p.iel (a mathematical concept) p weber J /T' / 2 Resistance R ohm LT~'i.

Contents 1 Introduction to the Leader's Guide Judith Wylie-Rosett, EdD, RD 2 Addressing Motivational Issues 5 Nicole Schaffer, PhD 3 Goal Setting 11 Lee-Ann Klein, MS, RD, and Brenda Axelrod, MS 4 Using the Workbook in a Group Setting 19 Lee-Ann Klein, MS, RD, and Brenda Axelrod, MS 5 Emotions and Roadblocks 23 Charles Swencionis, PhD, and Arlene Caban, MS 6 Health Issues 31 CJ Segal-Isaacson, EdD, RD 7 Six-Session Curriculum for Relapse Prevention 37 Nicole Schaffer, PhD

APPENDICES A General Obstacles to Changing Habits Audit (GOTCHA) 49 B Perceived Obstacles to Physical Activity (POPA) 51 C Book List 53 D Hunger-Fullness Weekly Food Record 57 iv LEADER'S GUIDE: THE COMPLETE WEIGHT LOSS WORKBOOK

Errata - II NOTES ON MATHEMATICS (AMERICAN NUCLEAR SOCIETY PROJECT) Pg. 8 -Next to last line: Orthagonal should be orthogonal Pg. ll-c: F(x) = 2+., not ao + anot 2 Z Pg. 12-g: F(x) = 2+ Z... not a 2 Z Pg. 13: II: Legendre Polynomials H(x,y) = ( ) = ( 2=o Previously A omitted Also the line below this, should read P, = - (the factorial was unclear). Pg. 16: should read (absolute signs were missing) 1. p (x) = (1 - x2) ~ d P,(x) Iml < ) myimi A ~ P~ 1m1dx [not (0 <m < )] 2. P2 (x) = [ ] (O <m< m ) Pg. 24: c:. should read: Z (-_1)r (x)2r + n Jn (x) = 2r + n r=o 2 r r lrl + n + r) [was missing] Pg. 25: No. 4-c: lim Jn(x) not lim Jn(x) x -0 x -*o

-2Pg. 28-e: I = x, Z (1/4 x ) v 2vr(v + l) r=0 ro F(v + 1 + r) Pg. 51: II - line 20 f(z) = a-1 not -+ z -a z -a Line 22: (z - a)f(z) = a"- + * not a1 + Pg. 52: line 14 A- A-2 f(z) = z + zo was missing Page 56: 4th line from bottom: should be | X -oo, not X - oo. Page 59: line 6, k, not k Line 7: x, not x Page 60: should read 1, Definitions a) Sn[ y] = y(x) sin n x d x 0 b) Cn [ Y ] = f y(x) cos n x d x 2 V a) y(x) = Z ( ( ) n=l b) y(x) = 2 ( ) Jc n=l

-3Pg4 64: line 17. after J [r] insert the word to S Pg, 66: line 3. We know 2x Ix = not we known 2x Ix = 0 Line 7, should read fn, not f Line 8, all f should be fn Line 17, fn ^ eo0 ef (p) - not f 0 O Pg. 68. Last two equations should have after them y > a y < a y>a yea Pg. 69: line 5, should read Now, lacking a known inversion in closed form (this part missing) Pg. 77: I: Leibnitz Rule, —rx):( )+ fx ay, fo 2/' d b(x) dg(x,y) b(x) 2y(xty) f ~(X) a+ f dy, not Jf' dy a(x) a(x) Pg: 79': No. 19 V a dv = f a d s (not da) Pg. 80: No. 22 f (V x a)dv = f n x a ds (not da) v S — - Pgo 91: Gas constant R = 8.314 x 107 erg/1 ot /o, 3 x/ mole rL not / o /mole

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