Dirichlet and Fresnel Integrals via Iterated Integration

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1 VOL. 78, NO. 1, FEBRUARY 5 63 While not ll of the ssumptions of the model will be stisfied in every cse, nevertheless, this nlysis might be used to justify limiting the size of committees. Now, if we could only come up with n nlysis to justify limiting the number of committees we re ssigned to! Acknowledgment. The uthors would like to express their pprecition to the reviewers for mny helpful suggestions. REFERENCES 1. W. Feller, An Introduction to Probbility Theory nd Its Applictions, Vol. I, John Wiley & Sons Inc., New York, S. Krlin, nd H. M. Tylor, A First Course in Stochstic Processes, nd ed., Acdemic Press, New York, E. Przen, Modern Probbility Theory nd Its Applictions, John Wiley & Sons Inc., New York, 196. Dirichlet nd Fresnel Integrls vi Iterted Integrtion PAUL LOYA Binghmton University Binghmton, NY pul@mth.binghmton.edu Mny rticles [, 3, 4, 6] hve been devoted to estblishing the vlues of some importnt improper integrls: sin x dx = π cos x sin x π nd dx = dx = x x x. The first integrl is clled the Dirichlet integrl nd the other two re clled Fresnel integrls. One wy to estblish these formuls is to consider the iterted integrls of the functions f (x, y = e xy sin x nd g(x, y = y 1/ e xy+ix, respectively, over [, [,. For instnce, if only we could justify switching the order of integrtion, we would evlute the Dirichlet integrl like this: ( e xy sin xdy dx = ( e xy sin xdx dy. (1 Since e xy sin xdy= sin x/x, the left-hnd integrl is (sin x/x dx.inviewof the integrtion formul e xy sin xdx= e xy (y sin x + cos x + C, ( 1 + y which is proved using integrtion by prts, the right-hnd integrl in (1 is 1 dy = lim 1 + y t tn 1 (y y=t = π y=. Hence, we hve computed the vlue of the Dirichlet integrl: (sin x/xdx= π/. Unfortuntely, justifiction for these steps is not t ll obvious! The reson is tht the

2 64 MATHEMATICS MAGAZINE stndrd hypotheses justifying iterting improper integrls, nmely Fubini s theorem, which requires bsolute integrbility, do not pply to the bove mentioned f nd g. After ll, f (x, is sin(x, which is certinly not integrble over the whole line, nd g(x, is not even defined. However, in this pper we hve just the right theorem to justify the desired steps. This theorem pplies to the functions f nd g nd is useful nd pproprite in n undergrdute nlysis course for two resons: (1 The hypotheses re strightforwrd to verify nd they pply to mny importnt exmples (see Exmples 1 4; ( The proof is very short (given certin well-known results. THEOREM. Let F(x, y be continuous function on (, (α,, where nd α re rel numbers, nd suppose tht the improper integrls G(x = α + F(x, y dy nd H(y = + F(x, y dx (3 exist nd converge uniformly for x nd y restricted to compct subintervls of (, nd (α,, respectively. In ddition, suppose tht for ll b, c >, c F(x, y dx M(y, (4 where α + M(y dy exists. Then the improper integrls b exist nd re equl. + G(x dx nd α + H(y dy (5 The integrls in this theorem re Riemnn integrls nd they re improper t, α, nd ; hence the plus signs on nd α. Since the integrls in (3 re uniformly convergent, G(x nd H(y re continuous on their respective domins [1, Th. 33.6], which gurntees they re Riemnn integrble over compct subintervls of their respective domins. If ll integrls re understood s Kurzweil-Henstock integrls or (improper Lebesgue integrls, or if more knowledge concerning Riemnn integrls is ssumed, then the hypotheses cn be wekened considerbly. We invite those reders fmilir with more dvnced theories to formulte such generliztions. Before proving our theorem, we need the following stndrd results (the reder not interested in the proof cn skip to Exmple 1 below: LEMMA 1. ( If f (x, y is continuous on finite rectngle [, b] [α, β], then β ( b b ( β f (x, y dx dy = f (x, y dy dx, α nd the inner integrls re continuous functions of y nd x, respectively. (b If { f n } is sequence of continuous functions on finite intervl [, b] tht converges uniformly on [, b] to limit function f, then f is continuous nd α b b fdx= lim n f n dx.

3 VOL. 78, NO. 1, FEBRUARY 5 65 (c DOMINATED CONVERGENCE THEOREM. Suppose tht f (x = lim f n (x for ll x > where f nd f n,n N, re continuous on (,. Suppose tht f n (x M(x for x > nd n N where M(x dx exists. Then f hs n integrl over [, nd + + fdx= lim n + f n dx. Sttements ( nd (b re found in most elementry nlysis books, see, for instnce, Brtle [1]. The Dominted Convergence Theorem cn be found there s Theorem 33.1 nd it follows directly from the usul one on compct intervls, simple proof of which is given by Lewin [5]. (Techniclly, Brtle s Theorem 33.1 is stted for integrls tht re improper only t infinity, but n nlogous proof works for integrls improper t both limits of integrtion. Now to the proof of the theorem: Let < n < b n be sequences with n nd b n,ndletα<α n <β n be sequences with α n α nd β n.sincef is continuous on the rectngle [ m, b m ] [α n,β n ], by ( of Lemm 1, βn ( bm α n m F(x, y dx dy = bm ( βn m α n F(x, y dy dx, nd the inner integrls re continuous functions of y nd x, respectively. As n, the inner integrl on the right converges uniformly to G(x, so by (b of Lemm 1, the limit s n of the right-hnd integrl exists nd equls b m m G(xdx. Thus, the improper integrl of the inner integrl on the left exists, nd ( bm bm F(x, y dx dy = G(x dx. (6 α + m m As m, the continuous function on (α, given by the inner integrl on the left in (6 converges to the continuous function H(y nd by (4, the inner integrl is dominted by function tht hs n integrl over [α,. Thus, (c of Lemm 1 implies tht s m the limit of the left-hnd side in (6 exists nd equls H(ydy.It α + follows tht the improper integrl G(xdx exists nd equls H(y dy.this + α completes the proof. + We now demonstrte how esy it is to use this theorem. Henceforth we drop the plus signs on the lower limits of integrtion to simplify nottion. EXAMPLE 1. For our first exmple, consider once gin f (x, y = e xy sin x on [, [,. One cn check tht f is not bsolutely integrble over [, [,, so the usul Fubini s theorem does not imply the existence or equlity of the iterted integrls of f over this qudrnt. However, we cn pply our theorem, s we now show. First, becuse of the exponentilly decying fctor, it follows tht e xy sin xdynd e xy sin xdxre uniformly convergent for x nd y restricted to compct subintervls of (,. Second, using the formul (, one cn verify tht for ll b, c >, c f (x, y dx K, for some K >, 1 + y b which hs n integrl on [,. Thus, the conditions of the theorem re stisfied, nd so the formul (1 t the beginning of this pper is indeed true! We cn now proceed exctly s we did before to derive the vlue of the Dirichlet integrl: sin x x dx = π.

4 66 MATHEMATICS MAGAZINE EXAMPLE. Now let g(x, y = y 1/ e xy+ix on [, [,. As with the previous exmple, the usul Fubini s theorem does not pply to this function, but we shll see tht our theorem does pply. Becuse of the exponentilly decying fctor, it follows tht y 1/ e xy+ix dy nd y 1/ e xy+ix dx re uniformly convergent for x nd y restricted to compct subintervls of (,. Moreover, one cn esily check tht for ny b, c >, c b y 1/ e xy+ix dx = y 1/ y + i e cy+ic e by+ib 1 y 1 + y (e cy + e by y 1 + y, which hs n integrl over [,. Thus, the conditions of the theorem re met, nd so ( ( y 1/ e xy+ix dy dx = y 1/ e xy+ix dx dy. Since y 1/ e xy+ix dy = e ix y 1/ e xy dy = π x 1/ e ix, where we mde the chnge of vribles y x 1 y nd used the Gussin integrl e y dy = π/, the left-hnd integrl is π x 1/ e ix dx = π x 1/ cos xdx+ i π on the other hnd, chnging vribles y y, the right-hnd integrl is y 1/ 1 y + i dy = y i dy = y 1 + y 4 dy + i x 1/ sin xdx; 1 + y 4 dy. Ech integrl on the right hs the vlue π/, which cn be found using the method of prtil frctions s in [4]. Thus, we hve computed the Fresnel integrls: cos x sin x π dx = dx = x x. EXAMPLE 3. Let < < 1. Then, rguing s in Exmple, we cn pply our theorem to the function y e xy+ix on [, [,. Working out the iterted integrls in the sme spirit s we did in Exmple nd using some elementry properties of the Gmm nd Bet functions, we rrive t the following generlized Fresnel integrls: ( π ( π x 1 cos xdx= Ɣ( cos nd x 1 sin xdx= Ɣ( sin, where Ɣ( is the Gmm function evluted t. EXAMPLE 4. We remrk tht our theorem immeditely implies the celebrted Weierstrss M-Test for iterted integrls [1, Th ]: Let F(x, y be continuous function on (, (α, nd suppose tht F(x, y L(x M(y,

5 VOL. 78, NO. 1, FEBRUARY 5 67 where L(x nd M(y hve improper integrls over [, nd [α,, respectively. Then the iterted integrls in (5 exist nd re equl. Finlly, we end with brief outline of how the Dirichlet nd Fresnel integrls cn be derived from the stndrd Fubini s theorem. As we lredy mentioned, both f (x, y = e xy sin x nd g(x, y = y 1/ e xy+ix re not bsolutely integrble on [, [,, so, without using the theorem, some ingenious trick is usully required to justify the itertion of integrls. For instnce, Brtle [1] integrtes the function f (x, y over [s, [t, with s, t >, where Fubini s theorem is vlid, nd fter integrtion is performed, one tkes the limits s s, t to estblish Dirichlet s integrl. Leonrd [4] pplies Fubini s theorem to e tx g(x, y with t >, which is bsolutely integrble on [, [,, nd, fter integrting, tkes the limit s t to estblish the Fresnel integrls. Acknowledgment. The uthor thnks the referees for helpful comments. The uthor ws supported in prt by Ford Foundtion Fellowship. REFERENCES 1. Robert G. Brtle, The elements of rel nlysis, second ed., John Wiley & Sons, New York-London-Sydney, Hrley Flnders, On the Fresnel integrls, Amer. Mth. Monthly 89:4 (198, Wclw Kozkiewicz, A simple evlution of n improper integrl, Amer. Mth. Monthly 58:3 (1951, I. E. Leonrd, More on Fresnel integrls, Amer. Mth. Monthly 95:5 (1988, Jonthn W. Lewin, A truly elementry pproch to the bounded convergence theorem, Amer. Mth. Monthly 93:5 (1986, J. Vn Yzeren, de Moivre s nd Fresnel s integrls by simple integrtion, Amer. Mth. Monthly 86:8 (1979, A Publishing Prdox Alert reder Jck C. Abd nd his brother Victor Abd send the following: A recent Birkhäuser-Verlg book list included Unpublished Philosophicl Essys/Kurt Gödel, edited by Frncisco A. Rodriquez-Consuegr, If the title is ccurte, it might mke pproprite brbershop reding in tht town where the brber shves everyone who doesn t shve himself. A quick serch of Amzon.com turns up welth of similr mteril unpublished recordings by Elizbeth Schwrtzkopf nd Mrin Anderson, unpublished letters from Generl Robert E. Lee to Jefferson Dvis, nd unpublished opinions of the Wrren Court. Author Michel McMullen is more scrupulously logicl: The title of his book is The Blessing of God: Previously Unpublished Sermons of Jonthn Edwrds.