OTHER TITLES IN THE SERIES ON PURE AND APPLIEMDA THEMATICS IntrodutcotA ilgoenb raTiocp ology Vol1.. by A.H .W ALLACE Circles Vol2.. by D.P EDOE AnalytCiocnailc s Vol3.. by B.S PAIN IntegErqaula tions Vol. 4. by S. MIKHLIN ProbleimnEs u clidSepaanc e: Vol. 5. ApplicatoifCo onn vexity by H.G .E GGLESTON HomoloTghye oroynA lgebraViacr ieties Vol6.. by A. HW.A LLACE METHODSB ASED OTNH E WIENER-HTOEPCFH NIQUE fotrh seo luotpfi aornt ial differeeqnutaitailo ns by B. NOBLE SeniLoerc tuirnMe art hematics TheR oyaClo lloefgS ec ieanncdTe e chnology Glasgow PERGAMON PRESS LONDON · NEW YORK ,P ARIS, LOS ANGELES 1958 PERGAMON PRESS LTD. 5F itzrSoqyu arLeo,n don 4 & W.l PERGAMON PRESS, INC. Eas5t5 tSht reeNte,w York 122 22,N .Y. P.O.B ox LosA ngeleCsa,l if. 47715, PERGAMON PRESS S.A.R.L. Rued esEc olesP,a ris 24 Ve Copyright © 1958 B.N OBLE LibraorfCy o ngreCsasr dN o. 58-12676 PrintIenNd o rtIhreerlnaa tTn hdUe n ivePrrseiBstesil.ef sa st CONTENTS PAGE Preface vii Hombea snioct atainodrn e suflrtoCsmh apIt er x I.C OMPLEXV ARIABLAEN D FOURIETRR ANSFORMS Introduction 1 1.1 Complex variable theory 5 1.2 Analyftuincc tdieofinnsbe ydi ntegrals 11 J.3T heF ouriinetre gral 21 I. 4 Thew aveeq uation 27 1.1) Contionutre gorfaa c lesr ttayipne 31 l.aT heW ienHeorp-pfr ocedure 36 1.7 Miscelleanxeaomupsal nerdse suIl ts 38 IIB.A SIPCR OCEDUR:E HSALF-PLAPNREO LBEMS Introduction 48 2.1 22. Jon'emsse thod 52 A duailn teegqruaalt mieotnh od 58 2.3I ntegral feoqrumautliaotni ons 61 2.4 2.1iS olutoifto hnie n teegqruaalt ions 67 26. Discusosfti hoseno lution 72 Comparoifms eotnh ods 76 2.7 Boundcaornyd istpieocnisbfi yge edn efruanlc tions 77 2.HR adiatiobno-utnydcpaoern yd itions 83 2.9M iscelleaxnaemopualsne rds e suIlIt s 86 IIIF.U RTHERW AVEP ROBLEMS Introduction 98 :1.1 A plawnaev ien cidoentn wtos emi-ipnafirnaipltlleae nle s1 00 :1.2 :Eadiaftriootmnw op araslelmeil- ipnlfiantietes 105 :I.:J :1.l�4a diaftriooamn cylindrical pipe 110 Sem-niifinistter piaprsa tlolt ehwlea lolfas d uct 181 :1.[; A straicpr oss a duct 122 :I.Ma iscelleaxnaemopualsne drs e suIlItIs 125 lV.E XTENSIOANNSD LIMITATIOOFNT SH EM ETHOD Introduction 141 ·1.T1 heH ilbperrotb lem 141 4·.2 Genecroanls iderations 147 4·,3 SimultaWnieeonuesr -eHqoupaft ions 153 4.4 Approxifmaactteo rization 160 4.5 Laplea'ecsq uatiinpo onl caor- doirnates 146 4. (} Miscelleaxnaemopualsne drs e suIlVt s 167 v VI UONTEN'rs APPHOXIMATEM E'l'HODFl V. SOMJ<: 5.1I ntroduction 178 5.2S omep roblewmhsic cha nnboets olveexda ctly 180 5.3G enertahle oorfya s peceiqaula tion 184 5.4D iffracbtyia o tnh iscekm i-insfitnriitpe 187 5.5G enertahle oorfya nothsepre ceiqaualt ion 196 5.6Di rffaticnob ys triapnsds litosffin itwei dth 203 Miscellaenxeamopulsea sn dr esults 207 V THE GENERALS OLUTIONO F THE BASIC VI. WIENER-HOPFP ROBLEM 6.1I ntroduction 220 6.2T hee xascotl uotfic oenr tdauiainln e gtrale quations2 22 Miscelleaxnaemopualsne drs e suVlIt s 228 Bibliography 237 Inde 243 x PREFA CE Them ethoddess criinbt ehdib so osko lcveer tbaoiunn dary-value probleomfsp ractiicmaplo rtainncveo lvpianrgt idailff erential equatiAo ntsy.p ipcraolb lreemq uisroelsu toifto hnes teady-state wavee quatiionfn r esep acweh ens emi-inbfionuintdea ries are preseEnxta.m plaergsei vferno eml ectromatghneeoatrciyoc,u stics, hydrodynamics,a ndepl oatsetntithcieiaotlry y . Thet win aimosft hibso oakr et:ot akteh set udefnrtoo mr dinary dogrseteu diinetsto h er eseafireclhcd o verbeytd h Wei ener-Hopf techniaqnudet ,op rovitdheer eseawrocrhk weirt ah r easonably comprehesnusmimvaero yfw hacta na ndw hacta nnboetd onaet them omenbty ttheec hniqTuheer. e adeart'tse ntiisdo rna wn particutloa rtlvhyae r iomuest hofdosra prpxoiamtseo lutoifo n probleOmnseo. ft hree markfaebalteui rste hsre a ngofea pparently unrelattoepdic cosv erbeydr amificaotfit ohnets e chniqIutie s. hopetdh asotm eo ft hceo mmenitnts h e taenxdit ne xampmleasy suggseusitt albilnfeeo sfr u rtrheesre arch. TheW iener-Htoepcfh niwqausie n venatbeodu1 t9 3t1os olavne integerqaula toifao s np ectiyapleD .u ritnhgwe a ri tw asn otebdy J.S chwingeri nd(eapnedn dbeynE t.Tl .yC opsotnh)ap tr oblems involving dbiysff ermaic-tinfiniitoen pcloaulndbe esf ormuliant ed termosfi ntegerqaula tiwohniscc ho ulbdes olvbeydt heWi ener HoptfechniTqhuese o.l utdieopne nodnts h ues oef F ouriinetr egrals too btaai cno mplveaxr iaebqluea twihoinc ihss olvbeyda nalytic continuaMtoisootfnt .hi bso oiksb aseodna d ifferbeunettq uivalent approdauceth o D . JoneFso.u rtirearn sfaorrem sa pdpilrieecd.t ly S. tot hpea rtdiiaffle reenqtuiaatlai nodtn h ceo mplveaxr iaebqluea tion iso btaidnierde cwtiltyh otuhten ecessfiotrfy o rmulaotfia onn integerqaula tiForno.m t hipso inotf v iewth eW iener-Hopf technipqruoev iads eisg nifiacnadnn att uerxatl ensoifto hner ange ofp robltehmasct a nb es olvbeydt hues eo fF ourieLra,p laacned Melltirna nsfoIrs mtsa.rt theibdso owki tthh ien tenotfir ounn ning thien tegerqaula tainodJn o nemse'tsh oadl ongesaicdohet hebru,t as twrhiet inpgr ogreistss eede mepdo intlaensdcs o nfustion g elabotrwaoet qeu ivamleetnhto dJso.n emse'tsh osde emssi mptloe r mea ndh avien cluodnleysd u fficideentta oiftl hsie n tegerqaula tion I methotdoe nabtlhere e adteofr o lltohwle i terature. Them ateriinta hli bso oskh oubleda ccesstioab nlyeo nweh oi s familwiiatrh Ltahpel atcrea nsfiotrcsmo ,m plinevxe rsfioornm ula, andi ntegriantt ihoceno mplpelxa nTeh.efi rscth aptiesinr t ended tos upplemtehnuets uauln dergracdouurastieenc omplveaxr iable vii PREFACE Vlll theoarnytd, of amiltiharere iazdweei rtt hh ues oeft hFeo urier transifnto hrcemo mpplleaxnA est. h bioso hka bse ewnr itftoern workwehrosis net earrepesr tism airnai plpyl icoaftt hiteoh neso ry rathtehrai nnt hteh eoirtyst ehlseft ,a ndard of rigour may not sattishpfeuy rm ea thematthiocuiigstahh n o suludffif coor practical purops.e s It iimpso rttoea mnpth atshiafztre ot mhp eo ionfvt i eawd opted int hbioso tkhe es seonfc Weti heen- eHrotpefncihqiutseh aitct an beu setdoo btaniunm erviaclaufleo psrh ysqiucaanilte .is Ftor variroeuassI oh nasvde e citdooe mdin tu mertiacbaallle tsh aousg h faars p ossriebsluaelr gteis v ienna f orsmu itfaobnrlu em erical computaantdrie orfnee ncaeres gtiov eftneh we exsiesttsi nogf tabulvaatleuPder sa.c tniocd ailslcyui sss igointv hepenh yosfi cal implicoaft iroenFssou erll tesc.t rotmhaegotnrhegyita sisp ch ould befil lebdyt hveo luimne stehribsiy De s. JSo.n es. Int heex amptlreesai tnte htdee xhta vcea rrtihaeend a lfyasri s I enouigneh a ccha tsoeo btaatil ne aosntree soufpl hty ssiicganli fi canicnse i mfprolmeT. h iisps a rttole yn coutrhabege eg iwnhnoe r migthhti tnhkac to mplifcroamutlecadabe ne i nteropnlryew tietdh thaei d oefl eacntc roomnpiuct er. Thset imtuowrl iutste h bioso cka moer igifonrmaal c loyu orfs e post-glreacdtusuautrgeeg seb syPt reodDf .C. .P acAkm.o nogt her thingasmg ratteofP urloP fa.c fko r orgiadnewiaozlri knign g I condiitnhi iodsne sp artamtte hnReto yCaoll loefSg cei eanncde TechnoGlloagsyg,oa wma. l gsroa tteoPf ruolf .S nedfdoorn I I.N . askmient gow ritev otlhufimosher i sse rainefdso ,hr e lllIp sfugges tioinncs o nnewxiitsohhn o rtaem nainnugs ctrhiawptat sm uch longero rtihgaiennn avlgilesydaI .ti sp erhwaoprsmt ehn tioning thath avIhe a dt oo mitc haa pdteearl wiintaghp plications of Besfsnueclt diuoanil n teegqruaaltt iodo inpssrk o ebmlsA.r eferee fotrh Per oCacm.bP .h l.iS o(cw.h onsaemc ea nnotb etn roawc ed) suggetsomt ete hda t thesaerb epe trsaotcb klbleytem hdsWe i ener Hopmfe thHoadv.i wrnigt ttehnbi oso akm n oetn ticroenlvyi nced I thatths eu ggeissct oirornbe ucittna , s entshere e f'escr oemement provotkhbeeod o Ik !a imn detbotP erdo f.J oDn.fe osvSr a. r ious rerfeencaensdc orrespaonndtd oeD nrcW.e. E .W illiwahmos carefcuhlelcCykh eadp Vt.eM ryt hanakrdseu teo ptrhien ftoerr s acrcautwoer okna d ifficmualntu script. B. N. 12.57. 31. SOME BASICN OTATIONA ND RESULTS FROM CHAPTER 1 Thef ollomwaiypn rgo uvsee furle ffeeorr.Ae ntcimfeca tor ex(p- wi)ti uss etdh routghhbioousot k . oc + iT k kl+ ik'2 (1k 0,k 2 0). = (f : = > > (2oc- k2)1/2= -iP( -OC)21/2. [(.11]4 ) Y= (k OC)1-/2 i(-ock) 1/2 (-k oc-1)/2 -i(oc+ 1k2./) [(1.]1 2) = : = -iki foc locii foc i rse aanlld a r.g e Y= = ° : y!":::i -00 where Wewr itfeo,ir n stance,- 00 whearneyo ft hefsroem sma yb eu seadc rcdoitnogc onvenience provtidheatdth eirse r nioos fkc ounsfion. IfI (c/>)\x ex(pT _aXsX) � + +(ocir)se guilnTa> r T _ < A } If\c/> (x )\ ex(pT X)a sX� - _(ocir)se guilnTa rT < B + < + [§ 1.3] Ifc/> x() ""a sXx �'1 + 0t,h en+( oc"") - oc'11-a soc � T_} Ifc/> x() x""'1a s X � -0,t hen_( oc"" )-oc '1-1a soc �

Methods Based on the Wiener-Hopf Technique for the Solution of Partial Differential Equations

Similar Methods Based on the Wiener-Hopf Technique for the Solution of Partial Differential Equations