ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN TECHNICAL REPORT NO. 2 GENERALIZED FOURIER INTEGRAL FORMULAS 2 '::. R. V.CURCHILL, Professor of Mathematics CONTRACT NO. DA-20-O18-ORD-1;2916 DEPARTMENT OF ARMY PROJECT NO. 599-01-004 ORDNANCE R AND D PROJECT NO. TB2-0001 OOR PROJECT NO. 867 August, 1954

,c27 r( r~~~~~~~,.I" ti

- ENGINEERING RESEARCH INSTITUTE a UNIVERSITY OF MICHIGAN - ABSTRACT Let tan a(t) = (h-mt2) / (kt) where h, k, and m denote non-negative constants. The functions y(x,t) = cos[xt-a(t)], where the parameter t assumes all positive values, are the characteristic functions of the boundary-value problem y" + ty = (x > O), hy- ky' + -iy" = 0 at x = O, y(x,t) bounded for x > 0, Integral formulas representing an arbitrary function f(x)(x > 0) in terms of the functions y(x,t) or cos[xt-a(x)], analogous to the Fourier cosine-integral formula, are derived and established assuming that f(x) satisfies conditions under which it is represented by the classical Fourier integral formula. ii

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN - GENERALIZED FOURIER INTEGRAL FORMULAS* 1. Introduction The solution of certain types of boundary value problems in partial differential equations depends on a representation of a function f(x) prescribed on the semi-infinite interval x > O, in terms of the functions that satisfy the system y"(x,X) + X2 y(x,\) = 0 (X > 0) (1) hy(O,X) - ky'(OX) + my" (0,X) = 0 where y(x,\) is bounded for all x (x > O), Here h, k, and m denote nonnegative constants at least one of which is positive, and x is a paramenter, Since y"(O,X) = k2 y(O-)), the above boundary condition involves the parameter explicitly when m O0. The boundary-value problem (1) with m 4 0 arises, for example, in problems of diffusion of gases within electrodes of vacuum tubes to the surface where they enter the vacuum chamber. It also arises in problems of transverse displacements of membranes with certain forms of elastic support along an edge. Applications in which m = 0 while h and k are positive are treated in references [l], [2], and [3]. The characteristic functions of the system (1) are (2) y(x,) = cos[Xx-a()], * Sponsored by the Office of Ordnance Research, U.S. Army, under Contract DA-2o-018-ORD-12916. 1

- ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN where x is real and non-negative and, when k > 0, a(X) is defined by the conditions,h -mX2 / < < x (3) tan a(X) = k- (- = (X) = ). In case k = 0 the characteristic functions are y = sin Xx and the representation of the function f(x) is the classical Fourier sineintegral formula. When h = m = 0, y = cos Xx and the representation is the Fourier cosine-integral formula. In these two special cases the generalized formulas found in this paper reduce to those classical formulas. In establishing the generalized representation formula, then, the cases k = 0 and h = m = 0 will be excluded from consideration. 2. The Trigonometric Functions Certain properties of functions associated with the characteristic functions (2) will be needed in the following sections. Their introduction here simplifies the presentation of the principal results. The identities cos(Xx-a) cos(Xa.-a) cos[X(x+4))-a] cos a + sin Xx sin Xi (4) = sin[X(x+4)-aJ sin a + cos Xx cos X[ can be verified by elementary trigonometry. When m > 0 and k > 0, consider the function ( P(t,r) cost co (X) = y(t,)y( (5) P(t,r) = f cos[%t-a(%)] cos a(%) - ry(t)y(oX)d io Let H (6) = h/m and P(t,r) dI-b ~ ~ ~ ~ ~,Jo 0 1 2K = k/m; then in view of equation (3) r 2 2 = 2K 2KX cos Nt + X(H-X) sin t dX 0o 4K2X2 + (H-X2)2 2

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN It is to be proved that P(tr) + 0 as r > o, uniformly with respect to t for t > x, where x is any positive constant. An alternate form of the integral (6), found by formally writing the Laplace transform of that integral with respect to t, leadsto the formula (7) P(t,r) = K E(T-t) sin rT d_ T where the function E is defined by the equations -Kt.-Bt Bt BE(t) = Kt [(K+B)eB -(K-B)eB ] B = (K2H)1/2 when H f K2. When H = K2 -Kt E(t) = 2e (1-Kt). To verify that formulas (6) and (7) represent the same function P(t,r) it is only necessary to show that both give the same function aP/ r and the initial value P(t,0) = 0. If K2 > H then B is real and 0 < B < K; if K2 < H then B is a pure imaginary constant. Thus it is seen from the above two formulas for E(t) that, whether B is zero or not, the function E(t) is either a function of the form -(a+-ib)t F(t) = A e (t+C), where A, a, b, and C are constants, a > 0 and real, and b is real, or else E(t) is the sum of two such functions. Now by a partial integration, F(T-t) sin rd cos rt + =cos rT — IIl d T r T r t +t sr T

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN > When t x > 0 the absolute value of the first term inside the braces does not exceed the constant IF(0) x It is easy to see that the absolute value of the integrandof the integral inside the braces does not exceed a constant multiple of exp [-a(T-t)] where the constant is independent of tO since T/t < 1 and l/t < l/x. Thus the quantity in the braces is bounded uniformly with respect to t and it follows that (8) jP(t,r) | < rX r (t > x > 0) for some constant X1 independent of t. When m = 0 and k > 0 it will now be shown that the function r Q(tr) = o sin [\t-a(X)] sin a(X) dX (9) h o rkX sin Xt - h cos Xt tendo zero uy h k tends to zero as r - co y uniformly with respect to t for t By the same procedure used above to obtain formula (7), the alternate form of the function (9) is found: x > 0. following (10) CO Q(t,r) = - exp - (T-t) sin rT -- k t Since the integral here has the same form as the integral in equation (7) when E(t) is replaced by a special case of the function F(t), it follows at once that a constant X2 exists, independent of t, such that (11) |Q(t,r)| < 'r (t = x > 0).

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN 3. Formal Development The linear representation of an arbitrary function f(x) prescribed on the semi-infinite interval, in terms of the characteristic functions y(xA) where the values of the parameter X range over all positive numbers, may be written in the form (12) f(x) = y(x,x) g(X) dX o a formal solution of this integral equation for the function g(X) will give the expansion formula that is to be established. The method of solution used here is analogous to the procedure with orthogonal functions when the eigenvalues X are discrete. It has some advantages over the formal method used in the special case m = 0 by Karush [1]. Let both members of equation (12) be multiplied by y(x,t) and integrated with respect to x from 0 to r, then interchange the order of integration in the resulting iterated integral. This leads to the equation 00 lim r00 (13) f(x) y(,t) x = i g(\) y(x,X) y(x,t) dx d. raw From either the Green's formula for the boundary-value problem (1) or the direct integration of the product y(x,X) y(x,t) of trigonometric functions (2) it follows that assuming k > 0 2m (14) 2 y(x,\) y(x,t) dx = - y(O,t) y(0,2) + Sin (t)r-( )-(tsin [] -t)r-a(X)+-(t) Xk +t X - t 5

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN According to the Riemann-Lebesgue theorem in the theory of Fourier integrals, co lirm G(X) sin [(Xkt)r] dX = im r G(X) cos. [(Xtt)r]dX = 0 r+OJ rI+~Jo when the function G(X) is sectionally continuous on each finite interval 0 < ~ _ X and absolutely integrable on the semi-infinite interval [5]. O The function (X-t) sin [a(x)-a(t)j that arises in equation (14) has a limit as -+t. Thus if g(X) satisfies the conditions just cited on G(X) it follows from equation (12) that f(O) = y(O,) g() d, 0o and from equations (13) and (14) and the Riemann-Lebesgue theorem, that 00 2m 2J f()y(xt) dx = f(Q) y(0,t) + (15) 00 (15)m 0 g(X) cos [a(X)-c(t)] sin[(X-t)r] dX 0 t Now suppose that in addition to satisfying the conditions of sectional continuity and absolute integrability assumed above, the function g(X) has one-sided derivatives of the first order from the right and from the left at the point X = t (t > 0). If g(X) is discontinuous at the point X = t let g(t) represent the mean value of the limits g(t+O) and g(t-O). Then the limit that appears in equation (15) represents 7g(t), according to the primitive form of the classical Fourier integral formula or the extension of the Dirichlet integral formula to the unbounded interval r4]. Thus (16) g(t) = 2 f(x) y(x,t) dx + k f(O) y(O,t) (t > 0) Jo 6

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN When m > 0 the representation (12) now becomes 00 00 00 f(x) = y(x,x) / f(L) y([,X) di dX + 2 f(0) y(x,X) y(O,X) d. f0x) rJo 0o ixk J The last integral here vanishes when x > 0 since it represents the limit, as r oo, of the function P(x,r) defined by equation (5) and satisfying condition (8). When m = 0 the final term in equation (16) vanishes. Thus for m 0 the expansion formula becomes (17) f(x) = 2_ y(x,x) f(k) y(,QX) dr dX (x > 0) tr Jo o0 When the trigonometric form (2) of the function y is substituted here the formula becomes 00 00 (18) f(x) o= 2 os Tfx-xa(x)J f(i) cos [X\-a(x)] dC dX Oq O0 (x > 0). This is the principal generalized Fourier integral formula that is to be established. When h = m = 0 and k > 0 it follows from equation (3) that a(X) = 0; then formula (18) reduces to the Fourier cosine-integral formula, When k = 0 then 0a = + /2 and the formula reduces to the Fourier sine-integral formula. 4. Theorem 1 Let f(x) denote a function that is sectionally continuous on each finite interval 0 < x < xo, defined at each point of discontinuity.X as the mean value of the two limits f (X+O) and f (X-0) a nd absolutely integrable on the semi-infinite interval x > O. Then at each point x (x > 0) at which the right- and left-hand derivatives of f(x) exist, the generalized Fourier integral formula (18) represents the function f(x). I -. L 7

- ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN Under the conditions stated in the theorem the function is represented by its Fourier sine- and cosine-integral formulas, at the points specified in the theorem. In view of the identities (4) it follows that formula (18) represents f(x) at those points, provided that either 00 00 (19) / f(GIt) cos [X(x+ii)-ca() cos Ca(X) d. dX = 0 0 o or 00 00 (20) 4 o sf (()I) sin [C(x+)-( sin d() d. = 0 Let each of the outer integrals here, with respect to X, be written as the limit as r + o of the integral from 0 to r. The conditions on f(G) insure the uniform convergence, with respect to X, of each of the inner integrals and the validity of interchanging the order of integration of the definite and improper integrals. Thus the two conditions can be written 00 (21) lim f(G) P(x+I,r) d = 0 (x > 0).r- o o (22) 00 r-oo O =0 (x > 0), where the functions P and Q Theorem 1 is true then if satisfied. When m > 0 and k Hence for each fixed x(x>O) I 0o Jo are those either of defined by equations (5) and (9). the conditions (21) or (22) is > 0 the function P satisfies condition (8). a constant X1 exists such that P(x+p, r) d(L| r< r f r J 8

r- ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN - r L and from the absolute integrability of f([p) it follows that condition (21) is satisfied. When m = 0 and k > 0 it follows from the condition (11) on Q(t,r), in the same manner, that condition (22) is satisfied. As noted earlier, the case k = 0 is included in the classical theory. This completes the proof of Theorem 1. 5. Remarks The preceding proof is clearly valid under conditions other than those stated in the theorem. It depends on f(x) being representable by its Fourier sine- and cosine-integral formulas and satisfying either of the conditions (19) or (20). In the above proof that one of those conditions is satisfied, it is sufficient that f(x) be bounded and integrable on each finite interval and absolutely integrable on the infinite interval. The formal development in section 3 above involved conditions on the function g(t). Those conditions are the same as the conditions imposed on f(x) in the theorem. Under those conditions, the improper integral in equation (12) converges uniformly with respect to x (x=0) and equation (13) is true provided that the limit on the right-hand side exists. But for each positive t at which the one-sided derivatives of g(t) exist, it was shown that the limit does exist and that g(t) is represented by formula (16). When the expression (12) for f(x) is substituted into (16) it follows that* when k > 0, 00 00 00 = 2 y(x, t) d +2m (23) g(t) = y(x,t). g(X)y(x,) d dx + m y(O,t) t/g(X)y(O,\) dX *"* ^o ^dnik O for each value of t just described. The following expansion is therefore established. 9

I ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN Theorem 2: Let f(x) satisfy the conditions stated in Theorem 1. Then when k > 0 for each positive value of x for which f(x) has a derivative from the right and from the left, the function is repres.ented.by the formula 2m (24) f(x) = 2 / cos [x-a(x)] A f(p) cos [x4-a] do d\ + km cos a(x) X f(4) cos a(j) da ~o When h = m = 0 then a(x) = 0 and this formula reduces to the Fourier cosine-integral formula. Thus, formula (24) is another generalization of the Fourier integral formula. It is a representation of.f(x) in terms of the function cos Rx-.(x)] (X = 0), not in terms of the characteristic functions of the boundary-value problem (1). The special case of formula (18) when m = 0 was arrived at formally in references [1] and [2]. The general case of that formula and the representation (24) together with the two expansion theorems are believed to represent new results. L 10

L - ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN BIBLIOGRAPHY 1. 2. 5. 4. Karush, W., "'A Steady-state Heat Flow Problem for a Quarter Infinite Solid", J. Appl. Physics, 23, (1952), pp. 492-494. Churchill. R. V., Modern Operational Mathematics, New York, 1944, pp. 206-210. Amer. "Integral Transforms and Boundary Value Problems" Math. Monthly, 59, (1952), pp. 149-155., Fourier Series and Boundary Value Problems, New 1941, p. 90. York, 5. Wiener, N., The Fourier Integral, Cambridge, 1933, PP. 14-15. J 11

UNIVERSITY OF MICHIGAN 3 9015 02828 547911111 3 9015 02828 5479