FUNCTIONS DEFINED BY IMPROPER INTEGRALS

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1 FUNCTIONS DEFINED BY IMPROPER INTEGRALS Willim F. Tench Andew G. Cowles Distinguished Pofesso Emeitus Deptment of Mthemtics Tinity Univesity Sn Antonio, Tes, USA This is supplement to the utho s Intoduction to Rel Anlysis. It hs been judged to meet the evlution citei set by the Editoil Bod of the Ameicn Institute of Mthemtics in connection with the Institute s Open Tetbook Inititive. It my be copied, modified, edistibuted, tnslted, nd built upon subject to the Cetive Commons Attibution-NonCommecil-SheAlike 3. Unpoted License. A complete instucto s solution mnul is vilble by emil to wtench@tinity.edu, subject to veifiction of the equesto s fculty sttus.

2 Foewod This is evised vesion of Section 7.5 of my Advnced Clculus (Hpe & Row, 978). It is supplement to my tetbook Intoduction to Rel Anlysis, which is efeenced sevel times hee. You should eview Section 3.4 (Impope Integls) of tht book befoe eding this document. 2 Intoduction In Section 7.2 (pp ) we consideed functions of the fom ; c y d: We sw tht if f is continuous on Œ; b Œc; d, then F is continuous on Œc; d (Eecise 7.2.3, p. 48) nd tht we cn evese the ode of integtion in! to evlute it s Z d c Z d c F.y/ dy D F.y/ dy D Z d c (Coolly 7.2.3, p. 466). Hee is nothe impotnt popety of F. Z d c f.; y/ dy Theoem If f nd f y e continuous on Œ; b Œc; d ; then! dy d ; c y d; () is continuously diffeentible on Œc; d nd F.y/ cn be obtined by diffeentiting () unde the integl sign with espect to yi tht is, F.y/ D f y.; y/ d; c y d: (2) Hee F./ nd f y.; / e deivtives fom the ight nd F.b/ nd f y.; b/ e deivtives fom the left: Poof If y nd y C y e in Œc; d nd y, then F.y C y/ y F.y/ D f.; y C y/ f.; y/ y d: (3) Fom the men vlue theoem (Theoem 2.3., p. 83), if 2 Œ; b nd y, y C y 2 Œc; d, thee is y./ between y nd y C y such tht f.; ycy/ f.; y/ D f y.; y/y D f y.; y.//yc.f y.; y./ f y.; y//y: 2

3 Fom this nd (3), F.y C y/ F.y/ y f y.; y/ d jf y.; y.// f y.; y/j d: (4) Now suppose >. Since f y is unifomly continuous on the compct set Œ; b Œc; d (Coolly 5.2.4, p. 34) nd y./ is between y nd y C y, thee is ı > such tht if jj < ı then jf y.; y/ f y.; y.//j < ;.; y/ 2 Œ; b Œc; d : This nd (4) imply tht F.y C y y F.y// f y.; y/ d <.b / if y nd y C y e in Œc; d nd < jyj < ı. This implies (2). Since the integl in (2) is continuous on Œc; d (Eecise 7.2.3, p. 48, with f eplced by f y ), F is continuous on Œc; d. Emple Since f.; y/ D cos y nd f y.; y/ D sin y e continuous fo ll.; y/, Theoem implies tht if then F.y/ D Z Z cos y d; < y < ; (5) sin y d; < y < : (6) (In pplying Theoem fo specific vlue of y, we tke R D Œ; Œ ;, whee > jyj.) This povides convenient wy to evlute the integl in (6): integting the ight side of (5) with espect to yields sin y y Diffeentiting this nd using (6) yields Z sin y d D To veify this, use integtion by pts. D sin y y 2 D sin y ; y : y cos y ; y : y We will study the continuity, diffeentibility, nd integbility of ; y 2 S; 3

4 whee S is n intevl o union of intevls, nd F is convegent impope integl fo ech y 2 S. If the domin of f is Œ; b/ S whee < < b, we sy tht F is pointwise convegent on S o simply convegent on S, nd wite D lim!b Z (7) if, fo ech y 2 S nd evey >, thee is n D.y/ (which lso depends on ) such tht Z F.y/ D < ;.y/ y < b: (8) If the domin of f is.; b S whee < b <, we eplce (7) by nd (8) by F.y/ D lim!c D Z < ; <.y/: In genel, pointwise convegence of F fo ll y 2 S does not imply tht F is continuous o integble on Œc; d, nd the dditionl ssumptions tht f y is continuous nd R b f y.; y/ d conveges do not imply (2). Emple 2 The function is continuous on Œ; /. ; / nd conveges fo ll y, with f.; y/ D ye jyj D 8 ˆ< ˆ: theefoe, F is discontinuous t y D. y < ; y D ; y > I Emple 3 The function f.; y/ D y 3 e y2 is continuous on Œ; /. ; /. Let D ye jyj d y 3 e y2 d D y; < y < : 4

5 e y2 / d D F.y/ D ; < y < :.3y 2 2y 4 /e y2 d D ( ; y ; ; y D ; so d if y D : 3 Peption We begin with two useful convegence citei fo impope integls tht do not involve pmete. Consistent with the definition on p. 52, we sy tht f is loclly integble on n intevl I if it is integble on evey finite closed subintevl of I. Theoem 2 ( Cuchy Citeion fo Convegence of n Impope Integl I) Suppose g is loclly integble on Œ; b/ nd denote G./ D Z g./ d; < b: Then the impope integl R b g./ d conveges if nd only if; fo ech > ; thee is n 2 Œ; b/ such tht jg./ G. /j < ; ; < b: (9) Poof Fo necessity, suppose R b g./ d D L. By definition, this mens tht fo ech > thee is n 2 Œ; b/ such tht jg./ Lj < 2 nd jg. / Lj < 2 ; ; < b: Theefoe jg./ G. /j D j.g./ L/.G. / L/j Fo sufficiency, (9) implies tht jg./ Lj C jg. / Lj < ; ; < b: jg./j D jg. / C.G./ G. //j < jg. /j C jg./ G. /j jg. /j C ; < b. Since G is lso bounded on the compct set Œ; (Theoem 5.2., p. 33), G is bounded on Œ; b/. Theefoe the monotonic functions G./ D sup G. / < b nd G./ D inf G. / < b 5

6 e well defined on Œ; b/, nd lim!b G./ D L nd lim!b G./ D L both eist nd e finite (Theoem 2.., p. 47). Fom (9), so jg./ G. /j D j.g./ G. //.G. / G. //j jg./ G. /j C jg. / G. /j < 2; G./ G./ 2; ; < b: Since is n bity positive numbe, this implies tht lim!b so L D L. Let L D L D L. Since.G./ G.// D ; G./ G./ G./; it follows tht lim!b G./ D L. We leve the poof of the following theoem to you (Eecise 2). Theoem 3 (Cuchy Citeion fo Convegence of n Impope Integl II) Suppose g is loclly integble on.; b nd denote G./ D g./ d; < b: Then the impope integl R b g./ d conveges if nd only if; fo ech > ; thee is n 2.; b such tht jg./ G. /j < ; < ; : To see why we ssocite Theoems 2 nd 3 with Cuchy, compe them with Theoem (p. 24) 4 Unifom convegence of impope integls Hencefoth we del with functions f D f.; y/ with domins I S, whee S is n intevl o union of intevls nd I is of one of the following foms: Œ; b/ with < < b ;.; b with < b < ;.; b/ with b. 6

7 In ll cses it is to be undestood tht f is loclly integble with espect to on I. When we sy tht the impope integl R b hs stted popety on S we men tht it hs the popety fo evey y 2 S. Definition If the impope integl D lim!b Z () conveges on S; it is sid to convege unifomly.o be unifomly convegent/ on S if; fo ech > ; thee is n 2 Œ; b/ such tht Z < ; y 2 S; < b; o; equivlently; < ; y 2 S; < b: () The cucil diffeence between pointwise nd unifom convegence is tht.y/ in (8) my depend upon the pticul vlue of y, while the in () does not: one choice must wok fo ll y 2 S. Thus, unifom convegence implies pointwise convegence, but pointwise convegence does not imply unifom convegence. Theoem 4.Cuchy Citeion fo Unifom Convegence I/ The impope integl in () conveges unifomly on S if nd only if; fo ech > ; thee is n 2 Œ; b/ such tht Z < ; y 2 S; ; < b: (2) Poof Suppose R b conveges unifomly on S nd >. Fom Definition, thee is n 2 Œ; b/ such tht < 2 nd < 2 ; y 2 S; ; < b: (3) Since Z D (3) nd the tingle inequlity imply (2). Fo the convese, denote ; Z : 7

8 Since (2) implies tht jf.; y/ F. ; y/j < ; y 2 S; ; < b; (4) Theoem 2 with G./ D F.; y/ (y fied but bity in S) implies tht R b conveges pointwise fo y 2 S. Theefoe, if > then, fo ech y 2 S, thee is n.y/ 2 Œ; b/ such tht < ; y 2 S;.y/ < b: (5) Fo ech y 2 S, choose.y/ mœ.y/;. (Recll (4)). Then D Z.y/ C.y/ ; so (2), (5), nd the tingle inequlity imply tht < 2; y 2 S; < b: In pctice, we don t eplicitly ehibit fo ech given. It suffices to obtin estimtes tht clely imply its eistence. Emple 4 Fo the impope integl of Emple 2, D jyje jyj D e jyj ; y : If jyj, then e ; so R conveges unifomly on. ; [ Œ; / if > ; howeve, it does not convege unifomly on ny neighbohood of y D, since, fo ny >, e jyj > 2 if jyj is sufficiently smll. Definition 2 If the impope integl D lim!c conveges on S; it is sid to convege unifomly.o be unifomly convegent/ on S if; fo ech > ; thee is n 2.; b such tht < ; y 2 S; < ; o; equivlently; Z < ; y 2 S; < : 8

9 We leve poof of the following theoem to you (Eecise 3). Theoem 5.Cuchy Citeion fo Unifom Convegence II/ The impope integl D lim!c conveges unifomly on S if nd only if; fo ech > ; thee is n 2.; b such tht Z < ; y 2 S; < ; : We need one moe definition, s follows. Definition 3 Let f D f.; y/ be defined on.; b/ S; whee < b : Suppose f is loclly integble on.; b/ fo ll y 2 S nd let c be n bity point in.; b/: Then R b is sid to convege unifomly on S if R c nd both convege unifomly on S: R b c We leve it to you (Eecise 4) to show tht this definition is independent of c; tht is, if R c nd R b c both convege unifomly on S fo some c 2.; b/, then they both convege unifomly on S fo evey c 2.; b/. We lso leve it you (Eecise 5) to show tht if f is bounded on Œ; b Œc; d nd R b eists s pope integl fo ech y 2 Œc; d, then it conveges unifomly on Œc; d ccoding to ll thee Definitions 3. Emple 5 Conside the impope integl =2 e y d; which diveges if y (veify). Definition 3 pplies if y >, so we conside the impope integls F.y/ D =2 e y d nd F 2.y/ D septely. Moeove, we could just s well define F.y/ D Z c =2 e y d nd F 2.y/ D whee c is ny positive numbe. Definition 2 pplies to F. If < < nd y, then Z Z =2 e y d < =2 d < 2 =2 ; so F.y/ conveges fo unifomly on Œ; /. c =2 e y d =2 e y d; (6) 9

10 Definition pplies to F 2. Since Z =2 e y d < =2 e y d D e y y =2 ; F 2.y/ conveges unifomly on Œ; / if >. It does not convege unifomly on.; /, since the chnge of vible u D y yields Z Z =2 e y d D y =2 y u =2 e u du; which, fo ny fied >, cn be mde bitily lge by tking y sufficiently smll nd D =y. Theefoe we conclude tht F.y/ conveges unifomly on Œ; / if > : Note tht tht the constnt c in (6) plys no ole in this gument. Emple 6 Suppose we tke y sin u u du D 2 s given (Eecise 3(b)). Substituting u D y with y > yields (7) sin y d D ; y > : (8) 2 Wht bout unifom convegence? Since.sin y/= is continuous t D, Definition nd Theoem 4 pply hee. If < < nd y >, then Z sin y d D cos y Z Z cos y y C d ; so sin y 2 d < 3 y : Theefoe (8) conveges unifomly on Œ; / if >. On the othe hnd, fom (7), thee is ı > such tht u This nd (8) imply tht sin u u du > 4 ; u < ı: sin y d D y sin u u du > 4 fo ny > if < y < ı=. Hence, (8) does not convege unifomly on ny intevl.; with >.

11 5 Absolutely Unifomly Convegent Impope Integls Definition 4.Absolute Unifom Convegence I/ The impope integl D lim!b Z is sid to convege bsolutely unifomly on S if the impope integl jf.; y/j d D lim!b Z jf.; y/j d conveges unifomly on S; tht is, if, fo ech >, thee is n 2 Œ; b/ such tht Z jf.; y/j d jf.; y/j d < ; y 2 S; < < b: To see tht this definition mkes sense, ecll tht if f is loclly integble on Œ; b/ fo ll y in S, then so is jf j (Theoem 3.4.9, p. 6). Theoem 4 with f eplced by jf j implies tht R b conveges bsolutely unifomly on S if nd only if, fo ech >, thee is n 2 Œ; b/ such tht Since Z jf.; y/j d < ; y 2 S; < < b: Z Z jf.; y/j d; Theoem 4 implies tht if R b conveges bsolutely unifomly on S then it conveges unifomly on S. Theoem 6. Weiestss s Test fo Absolute Unifom Convegence I/ Suppose M D M./ is nonnegtive on Œ; b/; R b M./ d < ; nd jf.; y/j M./; y 2 S; < b: (9) Then R b conveges bsolutely unifomly on S: Poof Denote R b M./ d D L <. By definition, fo ech > thee is n 2 Œ; b/ such tht L < Theefoe, if < ; then Z M./ d D Z Z M./ d L; < < b: M./ d L Z M./ d L <

12 This nd (9) imply tht Z jf.; y/j d Z Now Theoem 4 implies the stted conclusion. M./ d < ; y 2 S; < < < b: Emple 7 Suppose g D g.; y/ is loclly integble on Œ; / fo ll y 2 S nd, fo some, thee e constnts K nd p such tht If p > p nd, then e p jg.; y/j d jg.; y/j Ke p ; y 2 S; : D K e.p p / e p jg.; y/j d e.p p / d D Ke.p p / p p ; so R e p g.; y/ d conveges bsolutely on S. Fo emple, since j sin yj < e p nd j cos yj < e p fo sufficiently lge if p >, Theoem 4 implies tht R e p sin y d nd R e p cos y d convege bsolutely unifomly on. ; / if p > nd. As mtte of fct, R e p sin y d conveges bsolutely on. ; / if p > nd >. (Why?) Definition 5.Absolute Unifom Convegence II/ The impope integl D lim!c is sid to convege bsolutely unifomly on S if the impope integl jf.; y/j d D lim!c jf.; y/j d conveges unifomly on S; tht is, if, fo ech >, thee is n 2.; b such tht jf.; y/j d jf.; y/j d < ; y 2 S; < < b: We leve it to you (Eecise 7) to pove the following theoem. Theoem 7.Weiestss s Test fo Absolute Unifom Convegence II/ Suppose M D M./ is nonnegtive on.; b ; R b M./ d < ; nd jf.; y/j M./; y 2 S; 2.; b : Then R b conveges bsolutely unifomly on S. 2

13 Emple 8 If g D g.; y/ is loclly integble on.; fo ll y 2 S nd fo ech y 2 S, then jg.; y/j A ; < ; g.; y/ d conveges bsolutely unifomly on S if >. To see this, note tht if < <, then Z Z jg.; y/j d A A C d D C < A C C : Applying this with D shows tht cos y d conveges bsolutely unifomly on. ; / if > G.y/ D sin y d conveges bsolutely unifomly on. ; / if > 2. nd By eclling Theoem (p. 246), you cn see why we ssocite Theoems 6 nd 7 with Weiestss. 6 Diichlet s Tests Weiestss s test is useful nd impotnt, but it hs bsic shotcoming: it pplies only to bsolutely unifomly convegent impope integls. The net theoem pplies in some cses whee R b conveges unifomly on S, but R b jf.; y/j d does not. Theoem 8.Diichlet s Test fo Unifom Convegence I/ If g; g ; nd h e continuous on Œ; b/ S; then g.; y/h.; y/ d conveges unifomly ( on S if ) the following conditions e stisfiedw () (b) lim!b sup jg.; y/j y2s D I Thee is constnt M such tht Z sup h.u; y/ du < M; y2s < bi 3

14 (c) R b jg.; y/j d conveges unifomly on S: Poof If then integtion by pts yields Z g.; y/h.; y/ d H.; y/ D D Z Z h.u; y/ du; (2) g.; y/h.; y/ d D g. ; y/h. ; y/ g.; y/h.; y/ (2) Z g.; y/h.; y/ d: Since ssumption (b) nd (2) imply tht jh.; y/j M;.; y/ 2.; b S, Eqn. (2) implies tht Z Z g.; y/h.; y/ d 2 < M sup jg.; y/j C jg.; y/j d (22) on Œ; S. Now suppose >. Fom ssumption (), thee is n 2 Œ; b/ such tht jg.; y/j < on S if < b. Fom ssumption (c) nd Theoem 6, thee is n s 2 Œ; b/ such tht Z jg.; y/j d < ; y 2 S; s < < < b: Theefoe (22) implies tht Z g.; y/h.; y/ < 3M; y 2 S; m. ; s / < < < b: Now Theoem 4 implies the stted conclusion. The sttement of this theoem is complicted, but pplying it isn t; just look fo fctoiztion f D gh, whee h hs bounded ntdeivtive on Œ; b/ nd g is smll ne b. Then integte by pts nd hope tht something nice hppens. A simil comment pplies to Theoem 9, which follows. Emple 9 Let The obvious inequlity I.y/ D cos y d; y > : C y cos y C y C y is useless hee, since d C y D : 4

15 Howeve, integtion by pts yields Z cos y C y d D sin y y. C y/ sin y D y. C y/ Theefoe, if < <, then Z cos y C y d < 2 y C y C Z sin y C y. C y/ d 2 Z sin y y. C y/ C sin y y. C y/ d: 2. C y/ 2 3 y. C y/ 3 2. C / if y >. Now Theoem 4 implies tht I.y/ conveges unifomly on Œ; / if >. We leve the poof of the following theoem to you (Eecise ). Theoem 9.Diichlet s Test fo Unifom Convegence II/ If g; g ; nd h e continuous on.; b S; then g.; y/h.; y/ d conveges unifomly ( on S if ) the following conditions e stisfiedw () (b) lim!c sup jg.; y/j y2s D I Thee is constnt M such tht sup h.u; y/ du M; y2s < bi (c) R b jg.; y/j d conveges unifomly on S. By eclling Theoems 3.4. (p. 63), (p. 27), nd (p. 248), you cn see why we ssocite Theoems 8 nd 9 with Diichlet. 7 Consequences of unifom convegence Theoem If f D f.; y/ is continuous on eithe Œ; b/ Œc; d o.; b Œc; d nd (23) conveges unifomly on Œc; d ; then F is continuous on Œc; d : Moeove;!! Z d dy D Z d c c f.; y/ dy d: (24) 5

16 Poof We will ssume tht f is continuous on.; b Œc; d. You cn conside the othe cse (Eecise 4). We will fist show tht F in (23) is continuous on Œc; d. Since F conveges unifomly on Œc; d, Definition (specificlly, ()) implies tht if >, thee is n 2 Œ; b/ such tht < ; c y d: Theefoe, if c y; y d, then jf.y/ F.y /j D f.; y / d Z Œf.; y/ f.; y / d C C f.; y / d ; so jf.y/ F.y /j Z jf.; y/ f.; y /j d C 2: (25) Since f is unifomly continuous on the compct set Œ; Œc; d (Coolly 5.2.4, p. 34), thee is ı > such tht jf.; y/ f.; y /j < if.; y/ nd.; y / e in Œ; Œc; d nd jy y j < ı. This nd (25) imply tht jf.y/ F.y /j <. / C 2 <.b C 2/ if y nd y e in Œc; d nd jy y j < ı. Theefoe F is continuous on Œc; d, so the integl on left side of (24) eists. Denote! I D Z d c dy: (26) We will show tht the impope integl on the ight side of (24) conveges to I. To this end, denote! I./ D Z Z d c f.; y/ dy Since we cn evese the ode of integtion of the continuous function f ove the ectngle Œ; Œc; d (Coolly 7.2.2, p. 466), I./ D Z d c Z d: dy: 6

17 Fom this nd (26), I I./ D Z d c! dy: Now suppose >. Since R b conveges unifomly on Œc; d, thee is n 2.; b such tht < ; < < b; so ji I./j <.d c/ if < < b. Hence,! lim!b Z Z d c f.; y/ dy d D Z d c! dy; which completes the poof of (24). Emple It is stightfowd to veify tht e y d D y ; y > ; nd the convegence is unifom on Œ; / if >. Theefoe Theoem implies tht if < y < y 2, then Z y2 Z dy y2 Z y2 D e y d dy D e y dy dy y y y y Z e y e y 2 D d: Since it follows tht Z y2 y e y e y 2 Emple Fom Emple 6, dy y D log y 2 y ; y 2 y > ; sin y d D log y 2 y ; y 2 y > : d D 2 ; y > ; nd the convegence is unifom on Œ; / if >. Theefoe, Theoem implies tht if < y < y 2, then Z y2 Z 2.y sin y Z y2 sin y 2 y / D d dy D dy d y y cos y cos y 2 D d: (27) 2 7

18 The lst integl conveges unifomly on. ; / (Eecise (h)), nd is theefoe continuous with espect to y on. ; /, by Theoem ; in pticul, we cn let y! C in (27) nd eplce y 2 by y to obtin cos y 2 d D y 2 ; y : The net theoem is nlogous to Theoem (p. 252). Theoem Let f nd f y be continuous on eithe Œ; b/ Œc; d o.; b Œc; d : Suppose tht the impope integl conveges fo some y 2 Œc; d nd G.y/ D f y.; y/ d conveges unifomly on Œc; d : Then F conveges unifomly on Œc; d nd is given eplicitly by Z y F.y / C G.t/ dt; c y d: y Moeove, F is continuously diffeentible on Œc; d ; specificlly, F.y/ D G.y/; c y d; (28) whee F.c/ nd f y.; c/ e deivtives fom the ight, nd F.d/ nd f y.; d/ e deivtives fom the left: Poof We will ssume tht f nd f y e continuous on Œ; b/ Œc; d. You cn conside the othe cse (Eecise 5). Let F.y/ D Z ; < b; c y d: Since f nd f y e continuous on Œ; Œc; d, Theoem implies tht Then F.y/ D F.y / C D F.y / C F.y/ D Z Z y y Z y Z y G.t/ dt C.F.y / F.y // f y.; y/ d; c y d: f y.; t/ d dt Z y y f y.; t/ d! dt; c y d: 8

19 Theefoe, F.y/ F.y / G.t/ dt jf.y / F.y /j y Z y C f y.; t/ d dt: (29) Z y y Now suppose >. Since we hve ssumed tht lim!b thee is n in.; b/ such tht jf.y / F.y /j < ; < < b: F.y / D F.y / eists, Since we hve ssumed tht G.y/ conveges fo y 2 Œc; d, thee is n 2 Œ; b/ such tht f y.; t/ d < ; t 2 Œc; d ; < b: Theefoe, (29) yields F.y/ F.y / G.t/ dt <. C jy y j/. C d c/ y Z y if m. ; / < b nd t 2 Œc; d. Theefoe F.y/ conveges unifomly on Œc; d nd Z y F.y / C G.t/ dt; c y d: y Since G is continuous on Œc; d by Theoem, (28) follows fom diffeentiting this (Theoem 3.3., p. 4). Emple 2 Let Since it follows tht Z I.y/ D e y2 d; y > : e y2 d D p Z p y y I.y/ D p e t2 dt; y e t2 dt; nd the convegence is unifom on Œ; / if > (Eecise 8(i)). To evlute the lst integl, denote J./ D R e t2 dt; then Z Z Z Z J 2./ D e u2 du e v2 dv D e.u2 Cv 2/ du dv: Tnsfoming to pol coodintes D cos, v D sin yields J 2./ D Z =2 Z e 2 d d D. e 2 / ; so J./ D 4 9 q. e 2 / : 2

20 Theefoe e t2 dt D lim! J./ D p 2 nd e y2 d D 2 y ; y > : Diffeentiting this n times with espect to y yields 2n e y2 d D 3.2n /p y > ; n D ; 2; 3; : : :; 2 n y nc=2 whee Theoem justifies the diffeentition fo evey n, since ll these integls convege unifomly on Œ; / if > (Eecise 8(i)). Some dvice fo pplying this theoem: Be sue to check fist tht F.y / D R b f.; y / d conveges fo t lest one vlue of y. If so, diffeentite R b fomlly to obtin R b f y.; y/ d. Then F.y/ D R b f y.; y/ d if y is in some intevl on which this impope integl conveges unifomly. 8 Applictions to Lplce tnsfoms The Lplce tnsfom of function f loclly integble on Œ; / is F.s/ D e s f./ d fo ll s such tht integl conveges. Lplce tnsfoms e widely pplied in mthemtics, pticully in solving diffeentil equtions. We leve it to you to pove the following theoem (Eecise 26). Theoem 2 Suppose f is loclly integble on Œ; / nd jf./j Me s fo sufficiently lge. Then the Lplce tnsfom of F conveges unifomly on Œs ; / if s > s. Theoem 3 If f is continuous on Œ; / nd H./ D R e s u f.u/ du is bounded on Œ; /; then the Lplce tnsfom of f conveges unifomly on Œs ; / if s > s : Poof If, Z e s f./ d D Z Integtion by pts yields Z e s f./ dt D e.s s/ H./ e.s s / e s f./ dt D C.s s / Z Z e.s s /t H./ dt: e.s s / H./ d: 2

21 Theefoe, if jh./j M, then Z e s f./ d M e.s s / C e.s s/ C.s s / Z 3Me.s s / 3Me.s s / ; s s : e.s s/ d Now Theoem 4 implies tht F.s/ conveges unifomly on Œs ; /. The following theoem dws considebly stonge conclusion fom the sme ssumptions. Theoem 4 If f is continuous on Œ; / nd H./ D Z e s u f.u/ du is bounded on Œ; /; then the Lplce tnsfom of f is infinitely diffeentible on.s ; /; with F.n/.s/ D. / n e s n f./ di (3) tht is, the n-th deivtive of the Lplce tnsfom of f./ is the Lplce tnsfom of. / n n f./. Poof Fist we will show tht the integls I n.s/ D e s n f./ d; n D ; ; 2; : : : ll convege unifomly on Œs ; / if s > s. If < <, then Z e s n f./ d D Integting by pts yields Z Z e.s s / e s n f./ d D Z e.s s / n H./ d: e s n f./ d D n e.s s / H./ n e.s s / H./ Z H./ e.s s/ n d; whee indictes diffeentition with espect to. Theefoe, if jh./j M on Œ; /, then Z e s n f./ d e M.s s/ n C e.s s/ n C j.e.s s/ / n / j d : Theefoe, since e.s s/ n deceses monotoniclly on.n; / if s > s (check!), Z e s n f./ d < 3Me.s s/ n ; n < < ; so Theoem 4 implies tht I n.s/ conveges unifomly Œs ; / if s > s. Now Theoem implies tht F nc D Fn, nd n esy induction poof yields (3) (Eecise 25). 2

22 Emple 3 Hee we pply Theoem 2 with f./ D cos ( ) nd s D. Since Z sin cos u du D is bounded on.; /, Theoem 2 implies tht conveges nd F.s/ D e s cos d F.n/.s/ D. / n e s n cos d; s > : (3) (Note tht this is lso tue if D.) Elementy integtion yields Hence, fom (3), F.s/ D e s n cos D. / n d n s s 2 C 2 : ds n s ; n D ; ; : : :: s 2 C 2 22

23 9 Eecises. Suppose g nd h e diffeentible on Œ; b, with g.y/ b nd h.y/ b; c y d: Let f nd f y be continuous on Œ; b Œc; d. Deive Liebniz s ule: d dy Z h.y/ g.y/ D f.h.y/; y/h.y/ f.g.y/; y/g.y/ C Z h.y/ g.y/ f y.; y/ d: (Hint: Define H.y; u; v/ D R v u nd use the chin ule.) 2. Adpt the poof of Theoem 2 to pove Theoem Adpt the poof of Theoem 4 to pove Theoem Show tht Definition 3 is independent of c; tht is, if R c nd R b c both convege unifomly on S fo some c 2.; b/, then they both convege unifomly on S nd evey c 2.; b/. 5. () Show tht if f is bounded on Œ; b Œc; d nd R b eists s pope integl fo ech y 2 Œc; d, then it conveges unifomly on Œc; d ccoding to ll of Definition 3. (b) Give n emple to show tht the boundedness of f is essentil in (). 6. Woking diectly fom Definition, discuss unifom convegence of the following integls: Z d () d (b) e y 2 d C y 2 2 (c) 2n e y2 d (d) sin y 2 d (e).3y 2 2y/e y2 d (f).2y y 2 2 /e y d 7. Adpt the poof of Theoem 6 to pove Theoem Use Weiestss s test to show tht the integl conveges unifomly on S W () (b) Z e y sin d, S D Œ; /, > sin d, S D Œc; d, < c < d < 2 y 23

24 (c) (d) (e) (f) (g) (h) (i) e p sin y d, p >, S D. ; / e y d, S D. ; b/, b <. / y cos y d, S D. ; [ Œ; /, >. C 2 y2 e =y d, S D Œ; /, > e y e 2 d, S D Œ ;, > cos y cos d, S D. ; / 2 2n e y2 d, S D Œ; /, >, n D,, 2, () Show tht.y/ D y e d conveges if y >, nd unifomly on Œc; d if < c < d <. (b) Use integtion by pts to show tht.y/ D.y C / ; y ; y nd then show by induction tht.y/ D.y C n/ ; y > ; n D ; 2; 3; : : :: y.y C /.y C n / How cn this be used to define.y/ in ntul wy fo ll y,, 2,...? (This function is clled the gmm function.) (c) Show tht.n C / D nš if n is positive intege. (d) Show tht e st t dt D s. C /; > ; s > :. Show tht Theoem 8 emins vlid with ssumption (c) eplced by the ssumption tht jg.; y/j is monotonic with espect to fo ll y 2 S.. Adpt the poof of Theoem 8 to pove Theoem Use Diichlet s test to show tht the following integls convege unifomly on S D Œ; / if > : 24

25 Z sin y sin y () d (b) y 2 log d cos y (c) C y d 2 (d) sin y C y d 3. Suppose R g; g nd h e continuous on Œ; b/ S; nd denote H.; y/ D h.u; y/ du; < b: Suppose lso tht ( ) lim!b sup jg.; y/h.; y/j y2s D nd g.; y/h.; y/ d conveges unifomly on S: Show tht R b g.; y/h.; y/ d conveges unifomly on S. 4. Pove Theoem fo the cse whee f D f.; y/ is continuous on.; b Œc; d. 5. Pove Theoem fo the cse whee f D f.; y/ is continuous on.; b Œc; d. 6. Show tht C.y/ D e continuous on. ; / if f./ cos y d nd S.y/ D jf./j d < : f./ sin y d 7. Suppose f is continuously diffeentible on Œ; /, lim! f./ D, nd Show tht the functions C.y/ D jf./j d < : f./ cos y d nd S.y/ D f./ sin y d e continuous fo ll y. Give n emple showing tht they need not be continuous t y D. 8. Evlute F.y/ nd use Theoem to evlute I : (), b > d C y 2 2, y ; I D tn tn b d, 25

26 (b) (c) (d) (e) (f) I D Z e I D e Z y d, y > ; I D e y cos d, y > e b cos d,, b > e y sin d, y > e b e sin y d; I D sin d,, b > e cos y d; I D Z b log e cos d sin e d d,, b > 9. Use Theoem to evlute: () (b) (c) (d) Z Z Z.log / n y d, y >, n D,, 2,.... d d, y >, n D,, 2, C y/ nc 2nC e y2 d, y >, n D,, 2,.... y d, < y <. 2. () Use Theoem nd integtion by pts to show tht (b) stisfies Use pt () to show tht e 2 cos 2y d F C 2yF D : p 2 e y2 : 2. Show tht Z y e 2 sin 2y d D e y2 e u2 du: (Hint: See Eecise 2.) 22. Stte condition implying tht C.y/ D f./ cos y d nd S.y/ D 26 f./ sin y d

27 e n times diffeentible on fo ll y. (You condition should imply the hypotheses of Eecise 6.) 23. Suppose f is continuously diffeentible on Œ; /, nd lim! n f./ D. Show tht if then C.y/ D C.k/.y/ D k n. 24. Diffeentiting unde the integl sign yields j. k f.// j d < ; k n; f./ cos y d nd S.y/ D k f./ cos y d nd S.k/.y/ D cos y d sin y d; f./ sin y d; k f./ sin y d; which conveges unifomly on ny finite intevl. (Why?) Does this imply tht F is diffeentible fo ll y? 25. Show tht Theoem nd induction imply Eq. (3). 26. Pove Theoem Show tht if F.s/ D R e s f./ d conveges fo s D s, then it conveges unifomly on Œs ; /. (Wht s the diffeence between this nd Theoem 3?) 28. Pove: If f is continuous on Œ; / nd R e s f./ d conveges, then lim s!s C e s f./ d D (Hint: See the poof of Theoem 4.5.2, p. 273.) 29. Unde the ssumptions of Eecise 28, show tht lim s!s C e s f./ d D e s f./ d: e s f./ d; > : 27

28 3. Suppose f is continuous on Œ; / nd F.s/ D e s f./ d conveges fo s D s. Show tht lim s! F.s/ D. (Hint: Integte by pts.) 3. () Stting fom the esult of Eecise 8(d), let b! nd invoke Eecise 3 to evlute e sin d; > : (b) Use () nd Eecise 28 to show tht sin d D 2 : 32. () Suppose f is continuously diffeentible on Œ; / nd jf./j Me s ; : (b) Show tht G.s/ D e s f./ d conveges unifomly on Œs ; / if s > s. (Hint: Integte by pts.) Show fom pt () tht G.s/ D e s e 2 sin e 2 d conveges unifomly on Œ; / if >. (Notice tht this does not follow fom Theoem 6 o 8.) 33. Suppose f is continuous on Œ; /, eists, nd F.s/ D conveges fo s D s. Show tht s f./ lim!c F.u/ du D e s f./ d e s f./ d: 28

29 Answes to selected eecises 5. (b) If f.; y/ D =y fo y nd f.; / D, then R b does not convege unifomly on Œ; d fo ny d >. 6. (), (d), nd (e) convege unifomly on. ; [ Œ; / if > ; (b), (c), nd (f) convege unifomly on Œ; / if >. cos y 7. Let C.y/ D d nd S.y/ D S./ D, while S.y/ D =2 if y. sin y d. Then C./ D nd 8. () 2jyj ; I D 2 log b (b) y C ; I D log C b C (c) y y 2 C ; I D b 2 C 2 2 C (d) y 2 C ; I D tn b tn (e) y y 2 C ; I D 2 log. C 2 / (f) y 2 C ; I D tn 9. (). / n nš.y C / n (b) nš (c) 2y.log y/ 2 (d) nc.log / 2 j n f./j d < 24. No; the integl defining F diveges fo ll y.! 2n 2n y n =2 n 3. () 2 tn 29