Multidimensional Laplace Transforms over Quaternions, Octonions and Cayley-Dickson Algebras, Their Applications to PDE

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1 dac i Pu Mathatic Pubihd Oi Mach ( Mutidiioa Lapac Tafo o uatio Octoio ad Cayy-Dico gba Thi ppicatio to PDE Sgy Victo Ludoy Dpatt of ppid Mathatic Moco Stat Tchica Uiity Moco uia Eai: udoi@aiu cid Juy 8 ; id Nob ; accptd Nob BSTCT Mutidiioa ocoutati Lapac tafo o octoio a tudid Tho about dict ad i tafo ad oth popti of th Lapac tafo o th Cayy-Dico agba a pod ppicatio to patia difftia quatio icudig that of iptic paaboic ad hypboic typ a itigatd Moo patia difftia quatio of high od ith a ad copx cofficit ad ith aiab cofficit ith o ithout bouday coditio a coidd Kyod: Lapac Tafo; uatio S Fid; Octoio gba; Cayy-Dico gba; Patia Difftia Equatio; No-Coutati Itgatio Itoductio Th Lapac tafo o th copx fid i aady caica ad pay y ipotat o i athatic icudig copx aayi ad difftia quatio [- 3] Th caica Lapac tafo i ud fquty fo odiay difftia quatio ad ao fo patia difftia quatio ufficity ip to b od fo xap of to aiab But it t ubtatia difficuti o do ot o fo ga patia difftia quatio ith cotat cofficit pciay fo that of hypboic typ To oco th dabac of th caica Lapac tafo i th pt pap o ga ocoutati utipaat tafo o Cayy-Dico agba a itigatd I th pcdig pap a ocoutati aaog of th caica Lapac tafo o th Cayy-Dico agba a dfid ad itigatd [4] Thi pap i dotd to it gaizatio fo a a paat ad ao aiab i th Cayy-Dico agba Fo thi th pcdig ut of th autho o hooophic that i (up) difftiab fuctio ad oophic fuctio of th Cayy-Dico ub a ud [56] Th up-difftiabiity of fuctio of Cayy-Dico aiab i tog tha th Fécht' difftiabiity I tho o ao a ocoutati i itgatio a itigatd W id that quatio ad opatio o th had b fit dfid ad itigatd by W Haito i 843 [7] Sa ya at o Cayy ad Dico had itoducd gaizatio of quatio o o a th Cayy-Dico agba [8-] Th agba pciay quatio ad octoio ha foud appicatio i phyic Thy ud by Max Yag ad Mi hi diatio of thi quatio hich thy th ha itt i th a fo bcau of th iufficit dopt of athatica aayi o uch agba i thi ti [-4] Thi i ipotat bcau ocoutati gaug fid a idy ud i thotica phyic [5] Each Cayy-Dico agba ha gato i i o th a fid i uch that i ii i fo ach ii fo y h Th agba i fod fo th pcdig agba ith th hp of th o-cad doubig pocdu by gato i I paticua C coicid ith th fid of copx ub H i th fid of quatio 3 i th agba of octoio 4 i th agba of dio Thi a that a quc of bddig xit Gato of th Cayy-Dico agba ha a atua phyica aig a gatig opato of fio Th fid of quatio i aociati ad th agba of octoio i atati Th Cayy- Dico agba i po aociati that i z z z fo ach N ad z It i oaociati ad o-atati fo ach 4 Copyight Sci

2 64 S V LUDKOVSKY cougatio z * z of Cayy-Dico ub z i aociatd ith th o * * z zz z z Th octoio agba ha th utipicati o ad i th diiio agba Cayy-Dico agba ith 4 a ot diiio agba ad ha ot utipicati o Th cougat of ay Cayy-Dico ub z i gi by th foua: * * (M) z : Th utipicatio i i dfid by th fooig quatio: (M) fo ach z: : t th bgiig of thi atic a utipaat ocoutati tafo i dfid Th typ of th dict ad i ocoutati utipaat tafo o th ga Cayy-Dico agba a itigatd paticuay ao o th quatio fid ad th agba of octoio Th tafo a coidd i phica ad Catia coodiat t th a ti pcific fatu of th ocoutati utipaat tafo a ucidatd fo xap atd ith th fact that i th Cayy-Dico agba th a iagiay gato i i apat fo o i th fid of copx u- b uch that th iagiay pac i ha th diio Tho about popti of iag ad oigia i couctio ith th opatio of ia cobiatio difftiatio itgatio hift ad hoothty a pod xtio of th ocoutati utipaat tafo fo gaizd fuctio i gi Foua fo ocoutati tafo of poduct ad cooutio of fuctio a dducd Thu thi o th pob of o-coutati athatica aayi to dop th utipaat Lapac tafo o th Cayy-Dico agba Moo a appicatio of th ocoutati itga tafo fo outio of patia difftia quatio i dcibd It ca a a ffcti a (too) to o patia difftia quatio ith a o copx cofficit ith o ithout bouday coditio ad thi yt of difft typ ( ao [6]) agoith i dcibd hich pit to it fudata outio ad fuctio of G typ oig bouday pob ad patia difftia quatio ith dicotiuou cofficit a ao tudid ith th u of th ocoutati tafo Fquty fc ithi th a ubctio a gi ithout ub of th ubctio apat fo fc h ubctio a difft ut of thi pap a obtaid fo th fit ti Mutidiioa Nocoutati Itga Tafo Dfiitio Tafo i Catia Coodiat Dot by th Cayy-Dico agba hich ay b i paticua H th quatio fid o O 3 th octoio agba Fo uificatio of th otatio put C fuctio f : ca a fuctio-oigia h N if it fufi th fooig coditio (-5) ) Th fuctio f t i aot yh coti uou o ati to th Lbgu au o ) O ach fiit ita i ach fuctio g t f t t by t ith ad a oth aiab ay ha oy a fiit ub of poit of di cotiuity of th fit id h t t t t ca that a poit u i cad a poit of dicotiuity of th fit typ if th xit fiit ft ad ight iit g u : g u ad : iu u u< u iuu u> u g u g u 3) Ey patia fuctio g t f t t atifi th Höd coditio: g t h g t h fo ach h < h < cot > > a cotat fo a gi t t t yh o ay b bid poit of dicotiuity of th fit typ 4) Th fuctio f t ica ot fat tha th xpotia fuctio that i th xit cotat C cot > a a h fo y uch that f t < C xp q t fo ach t ith t fo ach q a a ; h 5) x y : x y dot th tadad caa po duct i Ctaiy fo a boudd oigia f it i poib to ta a a Each Cayy-Dico ub p it i th fo 6) p p i h i i i i th tadad bai of gato of o that i i ad ii i ii fo ach > ii ii fo ach > ad > ith p fo ach If th xit a itga pt 7) F p: F p; : f t dt th F p i cad th ocoutati utipaat (Lapac) tafo at a poit p of th fuctio-oigia f t h Copyight Sci

3 S V LUDKOVSKY 65 i i i th paat of a iitia pha fo ach d t dt 8) pt p t t pti ao put u p; t p t 8) Fo cto ha coid a patia odig 9) if ad oy if fo ach ad a xit o that < Tafo i Sphica Coodiat No coid ao th o-ia fuctio u u p t; taig ito accout o coutatiity of th Cayy-Dico agba Put u p t : u p t; : p M p t h ) ) M p t M p t; p i cop i i p co p i i p p p i p p p i co i i i fo th ga Cayy-Dico agba ith < ) : ; t: t t fo ach o that t t t Mo gay t 3) up t up t; pp t h p t i a ocay aaytic fuctio p t fo ach p ad t z: z z * z z dot th cougatd ub fo z Th th o ga o-coutati utipaat tafo o i dfid by th foua: F p; : f t xp u p t; dt 4) u fo ach Cayy-Dico ub p h thi itga xit a th picipa au of ith i a o Lbgu itga Thi o-coutati utipaat tafo i i phica coodiat h up; t i gi by Foua () t th a ti th copot p of th ub p ad fo i up; t it i th p - ad -ptatio pctiy uch that * 5) h hi i h i hi fo ach ) xpip xpi3p xpip33 3 * 6) h h h i hi h N h h i h i h fo ach i * i i fo ach > i h Dot Fu p; i o dtai by F f u; p; Hcfoth th fuctio up; t gi by (88) o () a ud if aoth fo (3) i ot pcifid If fo u p; t coct foua a ot tiod it i b udid that th fuctio up; t i gi i phica coodiat by Expio ) If i Foua (7) o (4) th itga i ot by a but oy by t() t( ) aiab h < < < th dot a ocot ; utati tafo by () t ( F ) u p; o t ; () t( ) F f u; p; If ( ) th dot it hoty by Fu p; o F f u; p; Hcfoth ta ad t ad p fo ach if othig oth i ot pcifid 3 a Th phica coodiat appa atuay fo th fooig coidatio of itatd xpot: p i p i p p 3 33i3 p p xp co i co i i Coid i th gato of th doubig pocdu of th Cayy-Dico agba fo th Cayy- Dico agba uch that ii i fo ach ) W dot o th fuctio M pt ; fo Dfiitio o i o dtai by M Th by iductio it: M p t M ipi p t t t i i i p M i p i p t t i i xp ; xp ; xp xp ; Copyight Sci

4 66 S V LUDKOVSKY h t t t ; t fo ach ; t t t( ) ; t ; t ic fo ach iag fuctio ca b itt i th fo 3) ; : Fu p i ; Fu p h a fuctio f i dcopod i th fo 3) f t i f t f : fo ach Fu p; d-ot th iag of th fuctio-oigia f If a autoophi of th Cayy-Dico agba i ta ad itad of th tadad gato i i gato N N a ud thi poid ao M pt ; MN pt ; ati to baic gato h N I thi o ga ca dot by NFu p; a iag fo a oigia f t o i o dtai dot it by F f u N ; p ; Foua (7) ad (4) dfi th ight utipaat tafo Syticay i dfid a ft utipaat tafo Thy a atd by cougatio ad up to a ig of baic gato Fo a aud oigia thy ctaiy coicid Hcfoad oy th ight utipaat tafo i itigatd ) Paticuay if p p p ad t t th th utipaat o-coutati Lapac tafo (7) ad (4) duc to th copx ca ith paat a a Thu th gi abo dfiitio o quatio octoio ad ga Cayy-Dico agba a utifid 4 Tho If a oigia f t atifi Coditio (-4) ad a < a th it iag F f u; p; i -hooophic (that i ocay aaytic) by p i th doai z : a < z < a a a by h N th fuctio u p; t i gi by (88) o () Poof t fit coid th chaactitic fuctio U t h t U fo ach t U hi t U fo y t \ U U : t : t i th doai i th Eucida pac fo ay fo Thfo F p; : ) u : U f t xp u p t; d t ic U U fo ach f txp u p t; dt U ach p ith th a pat Each itga i abouty cogt fo a < p < a ic it i aoizd by th cogig itga f t xp u p t ; d t C xp a y a y d y d y U z h p ic C a xp z fo ach z i i of Cooay 33 [6] Whi a itga 3) f t xp up t; phdt U poducd fo th itga () difftiatig by p cog ao uifoy: C h y y h y y h y y hy xp a y a y dy dy fo ach h ic ach z ca b itt i th fo z z xp M h z zz [ ) M M: M M i accodac ith Popoitio 3 [6] I i of Equatio (56): 6) f t xp u p t; hdt U f t xp( u p t ; )d t p ad 4) 5) f t xp u p t ; d t hi xp d d h C a y a y y y h C a Copyight Sci

5 S V LUDKOVSKY 67 fo ach h I i of cogc of itga gi abo (-6) th utipaat o-coutati tafo Fu p; i (up)difftiab by p ad oo Fu p; p ad Fu p; i th coidd p -p- tatio I accodac ith [56] a fuctio g p i ocay aaytic by p i a op doai U i th Cayy-Dico agba if ad oy if it i (up)difftiab by p i aoth od -ho oophic Thu Fu p; i -hooophic by p ith a < p < a ad du to Tho 6 [4] Cooay Lt uppoitio of Tho 4 b atifid Th th iag F f u; p; ith u up t; gi by () ha th fooig piodicity popti: ) fo ach ad πz ; ) fo ach o that ad π fo ach ad hi ith p p ad p p fo ach ith o p p ad f t i a fuctio ith by th t t aiab o a odd fuctio by t t ith ; 3) F f u; p; π i F f u; p; Poof I accodac ith Tho 4 th iag F f u; p; xit fo ach pwf : z : a < z< a ad h Th fo th π piodicity of i ad coi fuctio th fit tatt foo Fo i i co co i π i co π co gt co p co p that i p cop ip cop ad i p i p p p i i fo ach t O th oth had ith p p ad p p fo ach ith o p p ad f t t t f t t t i a ith o odd ith fuctio by th t t aiab fo ach t t t h t fo ; t Fo thi ad Foua (4) th cod ad th thid tatt of thi cooay foo 5 a Fo a ubt U i put π U : u: zu z u p p t b p fo ach p b h t: : b\{ p} p z : z b b p h b: i i i i th faiy of tadad g- ato of th Cayy-Dico agba That i goticay π p t U a th poctio o th copx pa C p of th itctio U ith th pa π p t t C p: abp: a b ic p * b ˆ : b \ ca that i 5-7 [6] fo ach cotiuou fuctio f : U it a dfid th opato ˆf by ach aiab z Fo th o-coutati itga tafoatio coid fo xap th ft agoith of cacuatio of itga Haudoff topoogica pac X i aid to b - coctd fo if ach cotiuou ap f : S X fo th -diioa a uit ph ito X ha a cotiuou xtio o fo ach ( ao [7]) -coctd pac i ao aid to b ipy coctd It i uppod futh that a doai U i ha th popty that U i -coctd; π p t U i ipy coctd i C fo ach i p i t p ad u C p fo hich th xit z u t U 6 Tho If a fuctio f t i a oigia ( Dfiitio ) uch that NFu p; i it iag utipaat ocoutati tafo h th fuctio f ad F u a itt i th fo gi by 3(33) f o th Cayy-Dico agba h N Th at ach poit t h f t atifi th Höd coditio th quaity i accopihd : f t π N π N F a p ; xp u a p t ; d p N N N N ) N u N u : F F a p; u t; Copyight Sci

6 68 S V LUDKOVSKY h ith up; t p t up; t p M p; t o N ( ad ) th itga a ta aog th taight i p N fo ach ; a < p a < a ad thi itga i udtood i t t t th of th picipa au dp d p N d p N d pn Poof I Itga () a itgad p dp ctaiy copod to th itatd itga a p d p N dpn h p pn pn p p Uig Dcopoitio 3(3) of a fuctio f it i ufficit to coid th i tafoatio of th a aud fuctio f hich dot fo ipicity by f W put NFu p; : f txp up t; d t If i a hooophic fuctio of th Cayy- Dico aiab th ocay i a ipy coctd doai U i ach ba B z ith th ct at z of adiu > cotaid i th itio It U of th doai U th i accopihd th quaity z a d z z a z h th it- ga dpd oy o a iitia z ad a fia z poit of a ctifiab path i B z a ( ao Tho 4 [4]) Thfo aog th taight i N th tictio of th atidiati ha th fo a N d ic ) z N a d ˆ an N d z N az az z h N fo th (up)difftiab by z U z h z N Fo th cho bach of th i itga pcifid by th ft agoith thi atidiati i uiqu up to a cotat fo ith th gi z -ptatio of th fuctio [4-6] O th oth had fo aaytic fuctio ith a xpaio cofficit i thi po i o-coutati itga pcifid by ft o ight agoith aog taight i coicid ith uua ia itga by th copodig aiab Th fuctio i z co z z ad paticipatig i th utipaat o-coutati tafo a aaytic ith a xpaio cofficit i thi i by po of z Uig Foua 4() duc th coidatio to U t f t itad of f t By yty popti of uch doai ad itga ad utiizig chag of aiab it i ufficit to coid U ith I thi ca fo th dict uti- paat o-coutati tafo (7) ad (4) duc to Thfo coid i thi poof bo th doai U oy Uig Foua 3(33) ad () tio that ay a agba ith gato N N ad N ith i ioophic ith th quatio fid H ic N N ad N N ad NN Th xp Mxp M xp M fuctio fo ach a ub ad a puy iagiay Cayy-Dico ub M Th octoio agba O i atati hi th a fid i th ct of th Cayy-Dico agba W coid th itga g t : π N π N F a p ; xp u a p t ; d p Nb Nb 3) N b Nb b N u fo ach poiti au of th paat < b < With th hp of gato of th Cayy-Dico agba ad th Fubii Tho fo a aud copot of th fuctio th itga ca b itt i th fo: g t [ π N d ] N d f xp u a p t ; xp u a p ; d p Nb Nb 4) N b Nb b N N ic th itga f un a p U fo ay ad < < a a 3 xp ; d i uifoy cogig ati to p i th doai a pa i ( ao Popoitio 8 [4]) If ta ad t fo ach ad S N fo o i La 7 [4] coidig th aiab t th ith a uitab ( -ia) autoophi of th Cayy-Dico agba a xpio fo M p; t ipifi i i th copx ca ith CK : K fo a puy iagi- ay Cayy-Dico ub K K itad of C: i h x x fo ach a ub x But ach quaity i i quiat to Th * * 5) NN qnn NN q q fo ach q If S ; N N ; N ith ad a ub fo ach th 6) N * * S N N S N Copyight Sci

7 S V LUDKOVSKY 69 Th att idtity ca b appid to ith S M p N p N t t ; N N ad o N M p N p N ; N N S p t N p t N ad N p N p N h M p N p N t t ; N N 7) p N i p i p 8) ; t t t ; t ; ; g fo ach t t W ta th iit of b t h b td to th ifiity Eidty fo 9) u p p N p N ; N N p M pn pn( ); N N u N gi by () h M i pcibd by (7) ; ; ; ; fo ach < By ou cotio ; ; fo < hi ; fo > Put ; ) ; u p p N p N N N p p N fo u u N gi by (88) Fo > th paat fo u u N gi by (88) o () ca b ta qua to zo fo up ; gi by Foua () o : pn pn ; N N : Wh t t t t ad p p p p aiab a ad ta th paat N N a p p N p N : pn pn ; N N : N N a p p N p N fo up ; dcibd i (88) Th th itga opato i ( ao Foua (4) abo) appid to th fuctio N π d b N dp N b N b xp ; un a p pn pn N N f t t u a p p N p N t t N N N xp ; Copyight Sci

8 7 S V LUDKOVSKY ith th paat itad of tatd by Tho 9 ad 35 [4] gi th iio foua copodig to th a aiab t fo f t ad to th Cayy-Dico aiab pn pn tictd o th copx pa CN N ic d c d fo ach (a) cotat c ft itgatio ith ith th hp of Foua (6- ) ad 3() gt th fooig: N b N N N ) igb t πn d πn d xp N ; f t t u a p p N p N t t N N xp un a p p N pn ; NN d p Moo fq f q fo ach q ad i () th fuctio f f q tad fo o ad q i accodac ith Dcopoitio 3(33) ad th bgiig of thi poof Mtio that th agba ag N N N o th a fid ith th gato N N ad N i atati Th poduct NN of to gato i ao th copodig gato ( ) N ith th dfiit ub ad th ig utipi O th oth had ( ) h N N N N N N N N W u dco- poitio (7-) ad ta du to Foua () h tad o th ight id of th quaity ic NN ad N N NN fo ach Thu th patd appicatio of thi pocdu by ad to Foua () of thi tho Cooay If th coditio of Tho 6 a atifid th ) f t π Fu a p; xp u a p t; dp d p F NFu a p; u t; Poof Each agba ag N N N i atati Thfo i accodac ith 6 ad Foua N b ) N f xp u a p t ; xp u a p ; d p N N b N N N b N N N f xp un a p t; xp un a p ; dp N b b f xp un a p t; xp un a p ; dp b fo ach ic th a fid i th ct of th Cayy-Dico agba hi th fuctio (88) ad (-4) fo ach o-coutati itga gi by th ft agoith gt i ad co a aaytic ith a xpaio cofficit Thu b b 3) g t π d d f xp u a p t ; xp u a p ; d p d p b b b N N hc taig th iit ith b tdig to th ifiity ipi that th o-coutati itatd (utip) itga i Foua 6() duc to th picipa au of th uua itga by a aiab ad p p 6() 7 Tho oigia Dico agba f t ith fid by it iag f o th Cayy- ith N i copty d F p; up to au at poit N u of dicotiuity h th fuctio ; u p t i gi by (88) o () Poof Du to Cooay 6 th au f t at ach poit t of cotiuity of f t ha th xpio thoughout NFu p; pcibd by Foua 6() Moo au of th oigia at poit of dicotiuity do ot ifuc o th iag NFu p; ic o ach boudd ita i by ach aiab t a ub of poit of dicotiuity i fiit ad by ou uppoitio abo th oigia fuctio f t i Copyight Sci

9 S V LUDKOVSKY 7 aot yh o 8 Tho cotiuou Suppo that a fuctio ; NFu p i aaytic by th aiab p i a doai W : p : a < p< a h N f ith up; t p t o up; t : p M p; t Lt NFu p; b itt i th fo NFu p; NFu p; NFu p; ; N ( ad ) h F p i hooophic by p i th doai u < a p Lt ao N u ; p i th doai < F p b hooophic by p a Moo fo ach a > a ad b< a th xit cotat C a > C b > ad a > ad b > uch that F p; C xp p fo ach p ) N u a a ith p a ) NFu p; Cb xp b p ith p b th itga N N 3) NFu p; dp N N fo ach p cog abo- uty fo ad ad ach a < < a F p; i th iag of th fuctio Th N u f t [ π N ] π N F p ; xp u p t ; d p N N N N 4) N u N u F ( F p; u t; Poof Fo th fuctio N u ; p g a g F p coid th ubtitutio of th aiab < Thu th poof duc to th coidatio of NFu p; itgatio by dp i th itatd itga (4) i tatd a i 6 Ta ad au of aiab p p p p ad t t t t h ; fo ach ( 6 ao) Fo a gi paat : N N p p N p N fo up ; pcibd by Foua () o : N N p fo ; p N p N u p t gi by (88) itad of ad ay o-zo Cayy-Dico ub ha i Fo ay ocay z-aaytic fuctio ai U atifyig coditio of 5 th hootopy tho- fo a o-coutati i itga o i atifid ( [56]) I paticua if U cotai th taight i N ad th path : N t tn th g zd z g zdz N g z i a do- h gˆ z hi z td to th ifiity ic i a fiit ub ( La 3 i [4]) W appy thi to th itgad i Foua (4) ic NFu p; i ocay aaytic by p i accodac ith Tho 4 ad Coditio () a atifid Th th itga opato π N N N o th -th tp ith th hp of Tho ad 36 [4] gi th iio foua copodig to th a paat t fo f t ad to th Cayy-Dico aiab pn pn hich i tictd o th copx pa CN N ( ao Foua 6(4) abo) Thfo a appicatio of thi pocdu by a i 6 ipi Foua (4) of thi tho Thu th xit oigia f ad f fo fuctio NFu p; ad NFu p; ith a choic of i th coo doai a < p < a Th f f f i th oigia fo NFu p; du to th ditibutiity of th utipicatio i th Cayy-Dico agba adig to th additiity of th coidd itga opato i Foua (4) Cooay Lt th coditio of Tho 8 b atifid th ) f t π NFu p; xp u p t; dp d p F NFu p; u t; Poof I accodac ith 6 ad 6 ach ocoutati itga gi by th ft agoith duc to th picipa au of th uua itga by th copodig a aiab: ) N N NFu p u p t p N N NFu p u p t p π ; xp ; d π ; xp ; d Copyight Sci

10 7 S V LUDKOVSKY fo ach Thu Foua 8(4) ith th ocoutati itatd (utip) itga duc to Foua 8() ith th picipa au of th uua itga by a aiab p p 9 Not I Tho 8 Coditio () ca b pacd o i up ˆ p C F p ) ( ) h C ( ): z : z a< z < a i a quc of itctio of ph ith a doai W h < fo ach i Idd thi coditio ad to th accopiht of th aaog of th Joda La fo ach ( ao La 3 ad a 4 [4]) Subqut popti of quatio octoio ad ga utipaat o-coutati aaog of th Lapac tafo a coidd bo W dot by: ) Wf p : a f < p< a f a doai of NFu p; by th p aiab h a a f ad a a f a a i Fo a oigia 3) f t U t f put W p : a f < p that i a Ca ay b h ith th ft hyppa p a o th ight hyppa p a i (o both a) icudd i W f It ay ao happ that a doai duc to th hyppa W p: p a a f Popoitio If iag NFu p; ad NGu p; of fuctio oigia f t ad g t xit i doai W f ad W ith au i h th fuctio u p; t g i gi by (88) o () th fo ach i th ca H ; a a f ad g ith au i ad ach o f ad g ith au i ad ach i th ca of ith 3 ; th fuctio NFu p; NGu p; i th iag of th fuctio f t gt i a doai W f W g Poof Sic th tafo NFu p; ad G p; xit th th itga f t g t xp u pt ; d t f t xp u pt ; dt g t xp u pt ; dt cog i th doai N u W W p : ax a f a g < p < i a f a g f g W ha t hi i th ct of th Cayy-Dico agba Th quatio fid H i aociati Thu ud th ipod coditio th cotat ca b caid out outid itga Tho Lt cot > F p; b a iag of a oigia fuctio f t ith ith u p t o u gi by Foua () o th Cayy-Dico agba ith < Th a ; f t xit t ao iag F p of th fuctio Poof Sic p p p fo ach h ; t ; t fo ach Th chagig of th aiab ipi: d p u p; t u( p / ; ) d f t t f F ; du to th fact that th a fid i th ct Z( ) of th Cayy-Dico agba Tho U U ; t ) F f t t t u; p; F f t t u p t ; ; p; Lt uch that f t t ao fo atifi Coditio (-4) Suppo that ; f t b a fuctio-oigia o th doai ad u p t i gi by () o (88) o th Cayy-Dico agba ith < Th p p S F f t U t u ; p ; U i th phica coodiat o Copyight Sci

11 S V LUDKOVSKY 73 U U ; t ) F f t t t u; p; F f t t u p t ; ; p; i th Catia coodiat i a doai W p : ax a f a f t < p h Poof Ctaiy f t f t t ad ) ) p ps F f t U t u; p; t : t t t : t S fo ach f t t f t t f t fo ach ic t t ; t fo ach Fo h 3) xp up t; p xp u p t; p S xp u p t; ic xp up t; xp p xp M p t; xp p p xpp ic Foua 3(67) [4] ha th quaity i th phica coodiat: p p i p i co i xp pi xp p i p xp p π i p co p π i p π i ps co p i p i a a idpdt aiab fo ach h fo hi ad 3) S cop i p i co p i p i co π i π p p i I th Catia coodiat ta t itad of i (3) If ach : z pt th 3) xp z qt qd xp d z z p qp z z z z! qp xp z p S xp z q z i a difftiab fuctio by z fo h ith q o q ic z That i x S xp i fo ach ad 33) ay poiti ub x > x 34) S xp i xpi xπ ad x S xp i xp i xπ fo ach o-gati a ub x ad h S S th zo po S i th uit opato; I u 35) ( p t ; ) p q Sq T i cop i ip cop i i p cop i i i p p Copyight Sci

12 74 S V LUDKOVSKY i th phica coodiat h ith q o q ad x T : xπ 36) fo ay fuctio ad ay a ub x h Th i accodac ith Foua (3) ha: S xp u p t; 37) q fo ; z qi z! up t; u p t gi by Foua (88) i th Catia coodiat h ith q o q Th itgatio by pat tho (Tho i II6 o p 8 [8]) tat: if a < b ad to fuctio f ad g a ia itgab o th gt ab x x F x f tdt ad Gx B gtdt a h a ad B a to a cotat th b b b F xgxd x FxGx f xgxdx a a Thfo th itgatio by pat gi 4) f t t xp u p t; dt t xp ; f t u p t t f t xp u p t; t d t Uig th chag of aiab t ith th uit Jacobia t t ad appyig th Fubii tho copoti to f i if: a U f t t xp u p t; d dt 5) f t t xp u p t ; d t f t t xp u p t ; d i th f t xp u p t ; dt p p S xp ; d f t u p t t phica coodiat o U 5) f t t xp u p t; dt f t xp u p t ; dt p p S xp ; d f t u p t t Catia coodiat ic xp p t pxp p i th Foua () h ; 6) U fo ach Thi gi F f t u p t ; ; p; f t xp u p t ; dt t dt dt dt dt f t xp u p t ; i th o-coutati tafo by t t t t t a Shift opato of th fo x xpd dx x i a aiab a ao fquty ud i th ca of ifiit difftiab fuctio ith cogig Tayo i xpaio i th copodig doai It i poib to u ao th fooig cotio O ca put co co co co i co i co co co h fo ach < o that T co fo ach > ad T i co fo ach > ad h T T T i th itatd copo up; t itio fo > N Th T gi ith up; t uch cotio th a ut a S o o ca u th yboic otatio T But to aoid iudtadig ha u S ad T i th of Foua (3-37) It i oth to tio that itad of (37) ao th foua ) xp pi pi co Mi ith / : p: p p ad M pi pi fo ; pi pi p ) i M co xp i Mp i ; ; π / u p t p u p t i Copyight Sci

13 S V LUDKOVSKY 75 ad pt ca b ud 3 Tho Lt f t b a fuctio-oigia Suppo that ) u p; t i gi by () o (88) o th Cayy-Dico agba ith < Th a (up) diati of a iag i gi by th fooig foua: F f t u; p; p h F f t u; p; h S F f t u; p; h S F f t u; p; h i th phica coodiat o ) F f t u; p; p h F f t u; p; h S F f t t u; p; h S F f t t u; p; h i th Catia coodiat fo ach h h i h i h h h p Wf Poof Th iquaiti a f < p< a f a quiat to th iquaiti a f t t < p< a f t t ic i t xp bt t fo ach b > iag ) ; ; < < F f t u p i a hooophic fuctio by p fo a f p a f by Tho 4 ao ct U { } t d t < fo ach c > ad Thu it i poib to difftiat ud th ig of th itga: f t xp u p t; dt p h ( f t xp u p t; d t p) h f t xp u p t; p hd t Du to Foua (33) gt: xp u pt ; p h xp u pt ; h S xp u pt ; h S xp u pt ; h 3) i th phica coodiat o 4) U xp ; xp ; xp ; xp ; u pt p h u pt h S u pt th S u pt th i th Catia coodiat Thu fo Foua (3) dduc Foua () 4 Tho If f t i a fuctio-oigia th ) F f t u; p; F f t u; p; p fo ith u p; t p M p; t o i) ii) u p; t p t o ith < i a doai p Wf h ) p p pi pi ith ; fo ach i th fit (i) ad p p i th cod (ii) ca ( ao Foua (8) ()) Poof Fo p i th doai p> a th idtiti a atifid: u( p t; ) U 3) F f t u ; p ; f t d t u p ; p f t d F f t u ; p ; p U U du to Foua (78) ad (4) ic p t ; p ; p ; ad pt p p ad p t ; p ; p ; fo ach h t Syticay gt () fo U itad of U Natuay that th utipaat o-coutati Lapac itga fo a oigia f ca b coidd a th u of i- tga by th ub-doai U : 4) f t xp u p t ; d t f t xp u p t; t d t U { } Th uatio by a poib Foua () gi Copyight Sci

14 76 S V LUDKOVSKY 5 Not I i of th dfiitio of th o-coutati ta fo F ad up; t ad Tho 4 th t i i ha th atua itptatio a th iitia pha of a tadatio 6 Tho If f t i a fuctio-oigia ith au i fo < b th bt t ) fo ach a b > p > a b F f t u; p; F f t u; p b; h u i gi by (88) o () Poof I accodac ith Expio (88) ad () o ha > > u p; t b t t u p b; t If a b p a b th th itga ) bt t bt t U U F f t t u; p; f t xp u p t; dt U f t xp u pb t; d t F f t U t u; pb; cog ppyig Dcopoitio 4(4) dduc Foua () 7 Tho Lt a fuctio f t b a a aud oigia Fp; F f t; u; p; h th fuctio up; t i gi by (88) o () Lt ao Gp; ad qp b ocay aaytic fuctio uch that F g t ; u; p; G p; xp u q p ; ) u p t o ; th ) F gt f d ; u; p; Gp; Fqp; p W ad qp Wf fo u p t t M p t fo ach g h < Poof If p Wg ad qp Wf th i i of th Fubii tho ad th tho coditio a chag of a itgatio od gi th quaiti: g t f d xp u p t; dt g t u p t t f Gp uqp f Gp; f xp u( qp ; d Gp; Fqp; xp ; d d ; xp ; d ic t ad th ct of th agba i 8 Tho If a fuctio f t U i oigia togth ith it diati f t U t o f t U t t t h Fu p; i a iag fuctio of f t U o th Cayy-Dico agba ith N fo u p M p t; gi by () th o ) i p ps ps ps Fu p; p < < ; < < ; () u; p p S p S p S Fu p ; f ) i p ps p ps p ps Fu p; p < < ; < < ; f u; p p S p p S p p S Fu p ; fo ; u p t gi by (88) h f itu ; t f t p td to th ifiity iid th ag g p < π fo o < <π p p i If th tictio Copyight Sci

15 S V LUDKOVSKY 77 tu ; t t ; t f t f t i t t ; t xit fo a < < th i p ps p S p ; S F u p p ) i th < < ; < < ; f t t t ; t << phica coodiat o () p p S p S p S Fu p ; u ) i p ps p ps p ps Fu p; p < < ; < < ; f t t t ; t << p p S p p S p p S Fu p i th Catia coodiat h p iid th a ag u ; t F f t U u p t p () ; Poof I accodac ith Tho th quaity foo: 3) F f t U t u; p; p ps F f t U t u p t; p; fo u up t; p M p t; ; ; ; i th phica coodiat o ; t U 3) F f t t U t u; p; p ps F f t U t u p t; p; F f t u p t ; ; p ; i th Catia coodiat ic 3) f t f t t f t t fo ach f t f t t h p p pi p i p p i i a th gato of th Cayy-Dico agba fo ach th zo po S I i th uit opato Fo hot it f itad of f U Thu th iit xit: ; t F f t u p t ; ; p; 4) d d d d xp up t; i t t t t t f t Mtio that t t t : t fo y ic t fo ach W appy th Foua (34) by iductio to f t f t f t itad of f t Fo Not 8 [4] it foo that i th phica coodiat i p gp <π/ ao i th i p gp <π/ F f t u; p; Catia coodiat U F f t t t ) u; p; U hich gi th fit tatt of thi tho ic u p u t; u ad Copyight Sci

16 78 S V LUDKOVSKY u ( F ) u p ; f fid fo ach p > hi ; F p i d- 5) i u If th iit f t xit h t : t t t : t th ; t t t t t f t u p t F f t u p t p t d d d d xp ; : ; ; ; fo ach Thfo th iit xit: Ctaiy t t t : t t i p gp <π/ U f t xp p M p t; u(; ) f t t f t t t ; t d U < < i p ps ps ; <π p S F u p p g p < < ; < < ; () ( ) u p p S p S p S Fu p ; f fo hich th cod tatt of thi tho foo i th phica coodiat ad aaogouy i th Catia coodiat uig Foua (3) 9 Dfiitio Lt X ad Y b to ia od pac hich a ao ft ad ight odu h Lt Y b copt ati to it o W put X : X X i th ti odd to poduct o of X By L q X Y dot a faiy of a cotiuou ti poy-ia ad additi opato fo X ito Y Th Lq X Y i ao a od ia ad ft ad ight odu copt ati to it o I paticua Lq X Y i dotd ao by Lq X Y W pt X a th dict u X Xi X i h X X a pai- i ioophic a od pac If Lq X Y ad xb xb o bx b x fo ach x X ad b th a opato ca ight o ft -ia pctiy ia pac of ft (o ight) ti poy-ia op- ato i dotd by L X Y (o L X Y pctiy) W coid a pac of tt fuctio D: D Y coitig of a ifiit difftiab fuctio f : Y o ith copact uppot quc of fuctio f D td to zo if a f a zo outid o copact ubt K i th Eucida pac hi o it fo ach th quc ( f ) : N cog to zo uifoy H a ( ) uuay f t dot th -th diati of f hich i a ti poy-ia ytic opato fo to Y that i () ( ) ( f ) t h h f ( ) t h h Y fo ach h h ad y tapoitio : i a t of th y tic goup S t Fo coic o put f () f I paticua ( ) f t f t t t fo a h ith o th -th pac Such cogc i D dfi cod ubt i thi pac D thi copt by th dfiitio a op that gi th topoogy o D Th pac D i ia ad ight ad ft odu By a gaizd fuctio of ca D : D Y i cad a cotiuou -ia -additi fuctio g : D Th t of a uch fuctioa i dotd by D That i g i cotiuou if fo ach quc f D cogig to zo a quc of ub g f : g f cog to zo fo tdig to th ifiity gaizd fuctio g i zo o a op ubt V i if g f fo ach f D qua to zo outid V By a uppot of a gaizd fuctio g i cad th faiy dotd by uppg of a poit t uch that i ach ighbohood of ach poit t upp g th fuctioa g i difft fo zo Th additio of gaizd fuctio g h i gi by th foua: ) g h f : g f h f Th utipicatio g D' o a ifiit difftiab fuctio i gi by th quaity: ) g f g f ith fo : ad ach tt fuctio f D ith a a iag f h i bddd ito Y ; o Copyight Sci

17 S V LUDKOVSKY 79 : ad f : Y gaizd fuctio g pcibd by th quatio: 3) g f : g f i cad a diati g of a gaizd fuctio g h f D Lq Y g D L Y q oth pac : B B Y of tt fuctio co it of a ifiit difftiab fuctio f : Y ( ) uch that th iit i t t f t xit fo ach quc f B i ( ) cad cogig to zo if th quc t f t cog to zo uifoy o \ B fo ach ad ach < < h BZ z : yz : y z dot a ba ith ct at z of adiu i a tic pac Z ith a tic Th faiy of a -ia ad -additi fuctioa o B i dotd by B I paticua ca ta X Y ith Z aogouy pac DUY DUY BUY ad BUY a dfid fo doai U i fo xap U U ( ao ) gaizd fuctio f B' ca a gaizd oigia if th xit a ub a < a uch that fo ach a < < a th gaizd fuctio 4) f txp q t U i i BU Y ach fo a fo y fo t ith t fo ach h q By a iag of uch oigia ca a fuctio 5) F f u; p; : fxp up t; of th aiab p ith th paat dfid i th doai Wf p : a < p< a by th fooig u Fo a gi p Wf choo a < < p < < a th 6) f up t xp xp ; U u p t q t BU Y xp ; : f q t u p t q t ic xp ; h i ach t f xp q txp up t; q t th gaizd fuctio bog to BU Y U by Coditio (4) hi th u i (6) i by a adiib cto Not ad Exap Eidty th tafo ; ; o a choic of U F f u p do ot dpd ic f xp( q t xp u p t; q t f xp q t b t xp u p t; q t b t fo ach b uch that a < b < p< b < a fo ach bcau xp b t t th a ti th a fid i th ct of th Cayy- ) U Dico agba h N Lt b th Diac dta fuctio dfid by th quatio t t : fo ach B Th DF ( ) ( ) { } t up t F t u; p; [ t xp q t xp u p t; q t ) xp ; t U ic it i poib to ta < a < < a < ad h fo ach ) F t u; p; xp up ; I th ga ca: i th paat cua fo ha t : t t I pati- 3) ; ; F t xp ; u p p p S p S p S M p i th phica coodiat o Copyight Sci

18 8 S V LUDKOVSKY 3) F t t t u; p; p p S p p S p p S xp u p; i th Catia coodiat h a ogati itg :!!! dot th bi- oia cofficit!!! ;! fo ach 3 ; t Th tafo F f of ay gaizd fuctio f i th hooophic fuctio by p Wf ad by ic th ight id of Equatio 9(5) i hooophic by p i W f ad by i i of Tho 4 Equatio 9(5) ipi that Tho -3 a accopihd ao fo gaizd fuctio Fo a a th gio of cogc duc to th tica hyppa i o Fo a < a th i o ay coo doai of cogc ad f t ca ot b tafod Tho If f t i a oigia fuctio o F p; i it iag f t o f t t t i a oigia Z ; th ) F f t u; p; p ps ps ; ; p S F f t u p fo up; t : p M p; t gi by ( ) o ) F f t t t u; p; p p S p p S p p S F f t u; p; fo ; u p t gi by (88) o th Cayy- Dico agba ith < Doai h Foua () a tu ay b difft fo a doai of if < th utipaat ocoutati tafo fo f but thy a atifid i th doai a < p < a h a i a f a f t : ; a ax a f a f t : a a h o t fo ach copodigy Poof To ach doai U th doai U y- u ) ( p t ; ) u d ( p t f t f t ; ) dt ticay copod Th ub of difft cto i Thfo fo u p t M p t; du to Tho th quaity u( pt ; ) u( pt ; ) dt f t dt f t d i atifid i th phica coodiat ic th about au of th Jacobia t t i uit Sic fo a < p< a th fit additi i zo hi th cod itga cot ith th hp of Foua () Foua () foo fo : 3) F f t u; p; p F f t u; p; p S F f t u; p; To accopih th diatio u Tho 4 o that ; ; ; ; i F f t u p F f t u p i F f t u; p; F f t u; p; p pi p i u( p t; ) u p; t pp i pi i f t d t Copyight Sci

19 S V LUDKOVSKY 8 h pac If th oigia ith o th -th xit th f t f t i cotiuou fo ith fo ach h f : f Th itchagig of i ad ay chag a doai of cogc but i th idicatd i th tho doai a < p< a h it i o oid Foua (3) i aid ppyig Foua (3) i th phica coodiat by iductio to f t : ith th copodig od ubodiatd to f t o i th Catia coodiat uig Foua () fo th patia diati f t ): ith th copodig od ubodiatd to / f t t t dduc Expio () ad () ith th hp of Statt 6 fo XVII3 [9] about th difftiatio of a ipop itga by a paat ad a Fo th ti Eucida pac Tho fo f t gi oy o o to additi o th ight id of () i accodac ith (3) Eidty Tho 4 ad Popoitio a t accopihd fo ; t F f u; p; ao t ; t Tho i atifid fo F ad ay o that ; t t t fo ach p ad fo ach (th a cotio i i ao bo) t ; t Fo F i Tho 3 i Foua 3() it i atua to put t ad h fo ach o that oy additi ith h h h o th ight id gay ay ai Tho 4 ad 7 ad odify fo t ; t F puttig i 4() ad 7() ad () t ad pctiy fo ach To ta ito accout bouday coditio fo doai difft fo U fo xap fo boudd doai V i coid a boudd ocoutati utipaat tafo F f t u; p; : F f t u; p; ) V V Fo it idty Tho Popoitio ad Cooay 4 a atifid a taig pcific oigia f ith uppot i V t fit ta doai W hich a quadat that i caoica cod ubt affi diffoophic ith a b h a < b a b : x : a x b dot th gt i Thi a that th xit a cto ad a ia itib appig C o o that CW W put t : t t t : t a t : t t t : t b Coid t t t 3 Tho Lt f t b a fuctio-oigia ith a uppot by t aiab i ad zo outid uch that f t t ao atifi Coditio (-4) Suppo that u p; t i gi by () o (88) o ith < Th ; ; ; t ; t ) F f t t t u; p; F f t t u; p; F f t t u; p; i th phica coodiat o p p S F f t t u p F ; t f t t ; u ; p t ; F f t t u ; p ; p p S F f t t u ; p ; i th Catia coodiat i a doai W ; if a o b th th adddu ith t o t copodigy i zo Poof H th doai i boudd ad f i aot yh cotiuou ad atifi Coditio (-4) hc f txp u p t; L fo ach p ic xp up t; i cotiuou ad upp f t aogouy to th itgatio by pat gi b t b b ) f t t xp u p t; d t f t xp u p t; f t xp u p t; t d a t a a t h t t t Th th Fubii' tho ipi: Copyight Sci

20 8 S V LUDKOVSKY 3) xp ; d b b b b b f t t u p t t xp ; d d f t t u p t t t a a a a a i th f t xp u pt ; d t ] [ f t xp u pt ; dt t t b t t a b b p ps a xp ; d a f t u p t t phica coodiat o 3) f t t xp u p t; dt f t xp u pt ; dt f t xp u pt ; dt t t b t t a b b a a p p S f t xp u p t; dt i th Catia coodiat h a uuay t t t t t d t dtdt dt dt Thi gi Foua () h 4) ; t F f t t u p t ; ; p; b b b b f t xp u p t ; dt a a a a i th o-coutati tafo by t dt i th Lbgu ou t o 4 Tho If a fuctio f t t diati f t t o f t t t t h ; fuctio of f t t i oigia togth ith it F p i a iag u o th Cayy-Dico agba ith N fo th fuctio u p; t gi by () o (88) b b > fo ach th ) i p ps ps ps Fu p; i th p < < ; < < ; phica coodiat o ; p p S p S p S F p f () u(; ) u ) i p ps p ps p ps Fu p; p < < ; < < ; i th Catia coodiat h f it t f t p td to th ifiity iid th ag ; p p S p p S p p S F p f ) ; t F f t t u p t ; ; p; i b b b b a t a a a () u(; ) u < π dt dt dt dt f t xp u p t; g p fo o < <π Poof I accodac ith Tho 3 ha Equaiti 3() Thfo if that h a > b Mtio that t t: t t fo y aogouy to appy Foua () by iductio to f t f t f t Copyight Sci

21 S V LUDKOVSKY 83 itad of f t ; t a i o appyig to th patia diati f t t t f t t t f t t itad of f t t copodigy If > fo o th > fo ad u p t ; i p fo uch t () h t t t () () () t t t t () a fo ad () t b i p g( p) <π/ fo Thfo i p {}; f u(; ) ic u p; u; f t h f () u( p t () ; ) () i () t ; t t f t I accodac ith Not 8 [4] F f t t u p t; ; p; i th phica coodiat ad i p g( p) <π/ F f t t t t u p t; ; p; i th Catia coodiat hich gi th tatt of thi tho 5 Tho Suppo that f t t F p; i it iag i a oigia fuctio f t t t t i a oigia Z a < b fo ach W fo Lt boudd W p : a < p fo b fo o ad fiit a fo ach ; W p : p < a fo a fo o ad fiit b fo ach ; W p : a < p < a h a ad b fo o ad ; () () () t t t W put () t t ad q fo () t a fo t () b fo q q q q q q a ax a f a f t t t : a i a f a f t t t : if a < a If a ad b fo ith a gi th If ith a > o b < fo a ad th W ao put h h ig fo ach h ig x fo x < ig ig x fo x > h h h h h : ig ig Lt th cto uat fac () i fo h o that ( ) h ( ) () ( ) fo ach ( ao o dtaid otatio i 8) Lt th hift opato b dfid: ) T F p; : F p; i i π ( ) ao th opato a > ; : SO S F p S S F p; h ( ) [ ) S( ) S ( ) fo ach poiti ub < S I i th uit opato fo ( ao Foua (3-37)) uuay t b th tadad othooa bai i o that Tho Th F f t t t t u p t ; ; p ; F f t t u ; p ; ( ) ; q h ; ; q ; h ig ; q fo fo ach ; ( ) {} q q ; ; F f t t t t u p ( ) ( ) h( ) q ( ) ( ) Copyight Sci

22 84 S V LUDKOVSKY fo ; u p t i th phica coodiat o th Catia coodiat o th Cayy-Dico agba ith < h ) : p ps ps i th : p p S p S p S phica coodiat hi b b ( ; ) ) u p t f t t t t t t dt ) : p ps : p ps i th p a Catia coodiat i opato dpdig o th paat p If () t fo o th th copodig adddu o th ight of () i zo Poof I i of Tho 3 gt th quaity d / d d u( p t; ) u( p t; ) t f t t t t f t t a t t t a i atifid fo fo ach ith < O th oth had fo p W additi o th ight of () cot ith th hp of Foua 3() Each t of th fo h( ) () dt q () () q q u( p; t ) f t t t t () ca b futh tafod ith th hp of () by th coidd aiab t oy i th ca ppyig Foua () by iductio to patia diati f t t f t t f t f t a i ad uig Tho 4 ad a dduc () 6 Tho U b a fuctio-oigia ith au i ith < u i gi by () o (88) Lt f t t ) t t g t : f x d x th ) F f t u; p; U U i th doai >ax F g t t u; p; p a h th opato a gi by Foua 5() Poof I i of Tho 5 th quatio U 3) F f t u; p; F g t u; p; ( ) h( ) ( ) ; ; F g t u p ; ; h ; h ig ; fo ach ; q q i atifid ic g t t t f t U h fo ach Equatio (3) i accopihd i th a doai p >ax a ic g ad g t ao fufi coditio of Dfiitio hi ag<ax a f b fo ach b > h a O th oth had g t i qua to zo o U ad outid U i accodac ith foua () hc a t o th ight id of Equatio (3) ith > diappa ad upp g t U Thu gt Equatio () 7 Tho Suppo that ; t ; t U : : > F p i a iag F f t t u; p; of a oigia fuctio f t fo u gi by () i th haf pac W p p a ith < p p ; π π fo ach i th phica coodiat o fo ach i th Catia coodiat; i ) th itga F p z; dz cog pi h p p pi pi p fo ach U : t t : t t Lt ao ) th fuctio F p; b cotiuou by th aiab p o th op doai W oo fo ach > a th xit cotat C > ad > uch that 3) F p; C'xp p fo ach p S ( ) S : z : z < < fo ach N i h a i fixd i i i ad fo ach Th i 4) pi F p z; d z t ; t U S F f t t u; p; Copyight Sci

23 S V LUDKOVSKY 85 h p p fo ach ; π π ad t ; i th phica coodiat hi ad t i th Catia coodiat copodigy fo ach Poof Ta a path of a itgatio bogig to th haf pac p fo o cotat > a Th U U f t xp u p t; dt C xp p a t t d t < cog h C cot > p Fo t > fo ach coditio of La 3 [4] (that i of th ocoutati aaog o of Joda a) a atifid If t th ic a t t a o-gati Up to a t U of Lbgu au zo ca coid that t > t > If th ao Th cogig itga ca b itt a th fooig iit: i 5) pi F p z; dz i i < pi F p z ; xp z d z S ic th itga fo F S z; dz i abouty cogig ad th iit i xp z uifoy by z o ach copact ubt i h S i a puy iagiay ad Cayy-Dico ub ith S Thfo i th itga i 6) pi F p z; d z i xp ; d d p i f t u p z t t z U th od of th itgatio ca b chagd i accodac ith th Fubii tho appid copoti to a itgad g gi gi ith g fo ach : i 7) pi F p z; dz i d t xp ; d U f t u p z t z pi i up z; t f t dz d t U pi Gay th coditio p p ad π π i th phica coodiat o i th Catia coodiat fo ach i tia fo th cogc of uch itga W ctaiy ha ad bi * 8) coiz pi dz i b p co π bi * 9) i iz pi co dz b p b p b p i π fo ach > ad < p < b < ad ppyig Foua (3-9) ad () o (88) ad (3-37) dduc that: ; d i p i F p z z U t ; t U S f t xp u p t; dt S F f t t u; p; h t t t t t fo ach < t i th phica coodiat o t i th Catia coodiat 8 ppicatio of th Nocoutati Mutipaat Tafo to Patia Difftia Equatio Coid a patia difftia quatio of th fo: ) f t gt h ) f t : a t f t t t a t a cotiuou fuctio h Z : Z i a atua od of a difftia opato Sic ; t t t fo ach th opato ca b itt i coodiat a f t ) : b t f t That i th xit b fo o ith ad b fo > hi a fuctio b t i ot zo idticay o th copodig doai V W coid that (D) U i a caoica cod ubt i th Eucida pac that i U citu h It U dot th itio of U ad c U dot th cou of U Paticuay th ti pac ay ao b ta Copyight Sci

24 86 S V LUDKOVSKY Ud th ia appig t t th doai U tafo oto V W coid a aifod W atifyig th fooig coditio (i-) i) Th aifod W i cotiuou ad pici C h C dot th faiy of ti cotiuouy difftiab fuctio Thi a by th dfiitio that W a th aifod i of ca C C oc That i W i of ca C o op ubt W i W ad W \ W ha a codiio ot tha o i W ii) W W h W W W W fo ach diw diw W W iii) Each W i ith i a oitd C - aifod W i op i W oitatio of W i coitt ith that of W fo ach Fo > th t W i aod to b oid o o-oid i) quc W of C oitab aifod bddd ito xit uch that W ui foy cog to W o ach copact ubt i ati to th tic dit Fo to ubt B ad E i a tic pac X ith a tic put 3) dit B E : ax up up bbdit b E Edit B dit b E: if E b : if b B b B E di W Lt x x h dit B b Gay b a bai i th tagt pac TW x at x W co itt ith th oitatio of W N W uppo that th quc of oitatio fa x x of W at x cog to x x fo ach x W h i x x W hi x x a iay i dpdt cto i ) Lt a quc of ia ou t o W ( XIII [9]) iduc a iit ou t o W that i BW i B W fo ach copact caoica cod ubt B i coquty W \ W W ha coid ufac itga of th cod id i by th oitd ufac W ( (i)) h ach W i oitd ( ao XIII5 [9]) i) Lt a cto ItU xit o that U - i cox i ad t U b coctd Suppo that a bouday U of U atifi Coditio (i-) ad ii) t th oitatio of U ad U b coitt fo ach N ( Popoitio ad Dfiitio 3 [9]) Paticuay th ia ou t λ o U i coitt ith th Lbgu au o U iducd fo fo ach Thi iduc th au o U a i () o th bouday coditio a ipod: f t f t 4) U q q q ( q U ) q Z f t f t fo q h q q q q q q fo ach t U i dotd by t f f ( q) a gi fuctio Gay th coditio ay b xci o o u o of th o thi ia cobiatio ( (5) bo) Fquty th bouday coditio 5) f t f t U f t f t fo a ao ud h dot a a aiab aog a uit xta oa to th bouday U at a poit t U Uig patia difftiatio i oca coodiat o U ad (5) o ca cacuat i picip a oth bouday coditio i (4) aot yh o U Suppo that a doai U ad it bouday U atify Coditio (D i-ii) ad g g U i a oi gia o ith it uppot i U Th ay oigia g o gi th oigia g : U g o h U \ U Thfo g g i th oigia o h g ad g a to oigia ith thi uppot cotaid i U ad U copodigy Ta o doai U atifyig Coditio (D i-ii) ad (D-D3): D) U U ad U U ; D3) if a taight i cotaiig a poit ( (i)) itct U at to poit y ad y th oy o poit ith y o y bog to U h U U ad U a cox; if itct U oy at o poit th it itct U at th a poit That i D4) ay taight i though th poit ith do ot itct U o itct th bouday U oy at o poit Ta o g ith upp g U th upp gu U Thfo ay pob () o U ca b coidd a th tictio of th pob () dfid o U atifyig (D-D4 i-ii) y outio f of () o U ith th bouday coditio o U gi th outio a th tictio f o U U ith th bouday coditio o U Hcfoad uppo that th doai U atifi Coditio (DD4 i-ii hich a ath id ad atua I paticua fo thi a that ith a o b fo ach oth xap i: U i a ba i ith th ct at zo U U \ U ; o U Copyight Sci

25 S V LUDKOVSKY 87 U U t : t ith a ad ub < < But ubt U() i U ca ao b pcifid if th bouday coditio dad it Th copx fid ha th atua aizatio by a atic o that i i Th qua- tio fid a it i -o ca b aizd ith th hp of copx atic ith th g- i ato I J K i i L o quiaty by 4 4 a atic i Coidig atic ith ti i th Cayy-Dico agba o gt th copxifid o quatioifid Cayy-Dico agba o C ith t H z ai bi o z ai bj ck L h abc uch that ach a cout ith th gato i I J K ad L Wh f ad g ha au i H ad 4 ad cofficit of difftia opato bog to th th utipaat ocoutati tafo opat ith th aociati ca o that F af af f 6) fo ach a H Th ft iaity popty F af af f fo ay a H J KL i ao accopihd fo ith opato ith cofficit i o Ci I i o H J K L I J K L ad f ith au i ith ; o ic a f ith au i C i o H J KL ad cofficit a i but ith 4 Thu a uch aiat of opato cofficit a ad au of fuctio f ca b tatd by th ocoutati tafo Hcfoad uppo that th aiat ta pac W uppo that g t i a oigia fuctio that i atifyig Coditio (-4) Coid at fit th ca of cotat cofficit a o a quadat doai Lt b oitd o that a ad b fo ach ; ith a o b fo ach > h i a ad itg ub If coditio of Tho 5 a atifid th F f t u; p; a p p p F f t t u; p; ( ) ; q h ; ; q; h ig( ); q fo fo ach ; ( ) {} ( ) h( ) q ( ) ( ) q q ; ; p p p F f t t t t u p ( ) F g t t u; p; fo up; t i th phica o coodiat h th opato Catia p a gi by Foua 5() o 5() H uat fac () i fo h o that () ( ) fo ach h ( ) ( ) i accodac ith 5 ad th otatio of thi ctio Thfo Equatio (6) ho that th bouday coditio a cay: q ( ) q q f t t t fo () a q fo q h fo q ( ) q q f t t t () () But h ig t di coquty ca b cacuatd if o () ( ) () ( ) () f t t t fo q h h a ub copod to > ( ) ic q fo ad q > oy fo > ad > That i t() t( ) a coodiat i aog uit cto othogoa to () Ta a quc U of ub-doai U U U fo ach N o that ach ( ) U i th fiit uio of quadat N W choo th o that ach to difft quadat ay itct oy by thi bod ach U atifi th a coditio a U ad 7) i dit U U Thfo Equatio (6) ca b itt fo o ga doai U ao Fo U itad of gt a fac U() itad of () ad oca coodiat () ( ) othogoa to U() itad of t () t ( ) ( Coditio (i-iii) abo) Thu th ufficit bouday coditio a: Copyight Sci

26 88 S V LUDKOVSKY ( ) ( ) f ( t () ( ) U ( ) t 5) ( ) fo q h a q fo q h h ig q fo > ; () () ( ) t a o fuctio o U() t U() I th haf-pac t oy 5) f t t t h a cay fo q < ad q a abo Dpdig o cofficit of th opato ad th doai U o bouday coditio ay b doppd h th copodig t aih i Foua (6) Fo xap if t t U U th f U i ot cay oy th bouday coditio f i ufficit U If U th o ay bouday coditio appa Mtio that u( p a; ) F f a ; u; p; f a 53) () hich happ i (6) h a t ad h Coditio i (5) a gi o dioit fo difft (i) ubaifod U() i U ad patia diati a aog othogoa to th coodiat i o thy a cocty pod I phica coodiat du to Cooay 4 Equatio (6) ith difft au of th paat gi a yt of ia quatio ati to uo fuctio S( ) F f t u; p; fo hich F f t u; p; ca b xpd though a faiy h () q () () q ( ) ; ; ; ( ) q S F g t u p S F f t t t t u; p; : Z () ad poyoia of p h Z dot th ig of h( ) itg ub h th copodig t F i zo h () t fo o I th Cat- ( ) ( ) 8) F f t u; p; P p S F g t u; p; ( ) P p S F f t t t t u ; p ; h( ) q ( ) ( ) q q ( q)( )( ) ( ) U( ) ( q)( ) ( ) ( ) ia coodiat th a ot o piodicity popti gay o th faiy ay b ifiit Thi a that F f t u; p; ca b xpd i th fo: h P p ad ( ) P( q)( )( ) p a quotit of poyoia of a aiab p p p Th u i (8) i fiit i th phica coodiat ad ay b ifiit i th Catia coodiat To th obtaid Equatio (8) appy th tho about th iio of th ocoutati utipaat tafo Thu thi gi a xpio of f though g a a paticua outio of th pob gi by (3) ad it i pcibd by Foua 6() ad 8() Fo ; ; ; ic P p ad P F f u p Coditio 8() a atifid ( ) ( q)( )( ) p a quotit of poyoia ith a copx o quatio cofficit ad h( ) a aiab ao G ad F t o th ight of (6) atify th Thu ha dotatd th tho 8 Tho Suppo that F f; u; p; gi by th ight id of (8) atifi Coditio 8(3) Th Pob (3) ha a outio i th ca of oigia fuctio h g ad ( ) a oigia o i th ca of gaizd fuctio h g ad ( ) a gaizd fuctio Mtio that a ga outio of () i th u of it paticua outio ad a ga outio of th hoogou pob f If ( ) ( ) ( ) g g g f f f f g ad f o U atifi (5) ith ( ) th f g ad f o U atifi Coditio (5) ith ( ) 8 Exap W ta th patia difftia opato of th cod od h h h h h h a : a h h h a h th quadatic fo i odgat ad i ot aay gati bcau othi ca coid Suppo that ah ah h h fo ach h 3 Th duc thi fo a by a itib ia opato C to th u of qua Thu 9) h h h h h h t t C ith a h a t t h a ad h fo ach h If cofficit of a cotat uig a utipi of th typ xp h hh it i poib to duc thi quatio to th ca o that if ah th h ( 3 Chapt 4 i []) Th ca ipify th opato ith th hp of a ia ta- Copyight Sci

27 S V LUDKOVSKY 89 foatio of coodiat ad coid that oy ay b o-zo if a Fo ith cotat cofficit a it i -o fo agba o ca choo a cotat itib a atix c copodig h h to C o that a h fo h ad ah fo ) h h F f t u ; p ; a p F f t t u ; p ; h h > h < Fo ad th opato i iptic fo ith a ad th opato i paaboic fo < < ad th opato i hypboic Th Equatio (6) ipifi: h () () () () F f t t th u; p; p F f( t t u; p; h {};( ) () () h h h ; ; t t F f t t u; p; F f t t u; p; p F f t t u; p; F f t t u; p; F g t u; p; i th phica o Catia coodiat h h ith o th h -th pac S I i th uit opato th opato h p a gi by Foua 5() o 5() pctiy W dot by S x th dta fuctio of a co tiuou pici difftiab aifod S i atifyig coditio (i-i) o that x S xd x ydy S x o fo a cotiuou itgab fuctio h dis < dy dot a ou t o th diioa ufac S ( Coditio () abo) Thu ca coid a o-coutati utipaat tafo o U fo a oigia f o U gi by th foua: ; t F f t t u; p; ) U U t ; U F f t t u; p; Thfo t i F i () copod to th bouday Thy ca b ipifid: F f t t u; p; F f t t u; p; ) ; t ; t ; F t t f t t u ; p ; h t i a pici cotat fuctio o qua to o th copodig fac of othogoa to gi by coditio: ith t a o t b ; t i zo othi If a o b th th copodig t diappa If bd ito ith a i i th thi iduc th copodig bddig of o U ito Thi pit to a futh ipificatio: a p F f t t u; p; F f t t t u; p; h () () () () ) h h h h {};( ) () () h h h F a t f t t u p t ; ; p; F P t f t t u; p; h t dot a a coodiat aog a xta uit oa M t to U at t o that M t i a puy iagiay Cayy-Dico ub at i a pici cotat fuctio qua to a h fo th copodig t i th fac hh ith h >; Pt p: Pt : h p fo t hh h i π i ad ic co π co fo ach Ctaiy th opato-aud fuctio Pt ha a pici cotiuou xtio P t o That i 3) U U F t f t t u p t ; ; p; U : t f t t xp u p t; dt Copyight Sci

28 9 S V LUDKOVSKY fo a itgab opato-aud fuctio t o that t f t i a oigia o U h thi i- tga xit Fo xap h i a ia cobiatio of hift opato S ( ) ith cofficit ( ) t p uch that ach ( ) t p a a fuctio by t U fo ach p W ad f t a oigia o f ad g a gaizd fuctio Fo to quadat ad itctig by a coo fac F f t u ; p ; a p F f t t u ; p ; h h 4) h U U U U U U U xta oa to it fo th quadat ha oppoit dictio Thu th copodig itga i F ad F tictd o uad cac i F Uig Coditio (i-ii) ad th quc U ad quadat outid abo gt fo a bouday pob o U itad of th fooig quatio: F t ) P t p f t t u; p; F a t f t t u; p; F p f t t u; p; F f t t u; p; F g t u; p; h Pt p: Pt: ah hp th h fo ach t U ( ao Sto foua i XIII 34 [9] ad Foua (443) bo) Paticuay fo th quadat doai ha a t a h fo t hh ith h > t fo t ith > ad zo othi Th bouday coditio a: f t t f t t 4) U U U U Th fuctio a t ad t ca b cacuatd fo ah : h ad aot yh o U ith th hp of chag of aiab fo t t to y y y h y y a oca coodiat i U i a ighbohood of a poit t U y ic U i of ca C Coid th difftia fo h a dt dt d t ady dy h h h d th dt h h ad it xta poduct ith th 4) a t a t U h h h U ad 43) t t U U U F U Thi i ufficit fo th cacuatio of 83 Iio Pocdu i th Sphica Coodiat Wh bouday coditio 8(3) a pcifid thi Equatio 8(6) ca b od ati to F f t U t u p t; ; p; paticuay fo Equatio 8(44) ao Th opato S ad T of ha th piodicity popti: 4 ; ; S F p S F p 4 ; ; T F p T F p ; ; ad S F p S F p ad ; ; T F p T F p fo ach poiti itg ub ad W put 6) F p ; : S 4 S 4 F p ; fo ay 6) F p; : S 4 Fp; Th fo Foua 8(6) gt th fooig quatio: ; 63) a p pt p pt pt p pt p T F p pb b> a ( ) ; q h ; ; q ; h ig( ); q fo fo ach ; ( ) p pt p pt p T p pt p T fo ach h F p; F f t t u; p; ad pb b> G p; F g t t u; p; Th quatio a od fo ach a it Copyight Sci

29 S V LUDKOVSKY 9 i idicatd bo Taig th u o gt th ut F p; F p; F p; 64) Sic up t S S S ; 4 up; t up; t S Th aaogou pocdu i fo Equatio (4) ith th doai U itad of Fo Equatio (63) o (4) gt th ia quatio: x 5) () () () h i th o fuctio ad dpd o th paat () a o cofficit dpdig o p x () a idtiat ad ay dpd o fo h o that x () x () ; h 3 fo h > h x () 4 x h () fo ach h > i accodac ith Cooay 4 ctig o both id of (63) o (4) ith th hift opato T ( ) ( Foua 5(SO)) h h 3 fo ach h > gt fo (5) a yt ( ) of ia quatio ith th o fuctio ( ) : T( ) itad of : 5) () () T( ) x() ( ) fo ach () Each uch hift of ft cofficit () itact ad x() ( ) x( ) ith od h h h od 4 fo ach h > h fo othi Thi yt ca b ducd h a iia additi goup ith o-zo g g h g : Z hz G: ( ) : od od 4 ; gatd by a cofficit i Equatio (5) i a pop ubgoup of 4 h dot th fiit additi goup fo < h Z Gay th obtaid yt i o-dgat fo aot a p p p o i W h dot th Lbgu au o th a pac W coid th o-dgat opato ith a copx C i o quatio H J KL cofficit Ctaiy i th a ad copx ca at ach poit p h it dtiat p i o-zo a outio ca b foud by th Ca u Gay th yt ca b od by th fooig agoith W ca goup aiab by Fo a gi h ad ubtactig a oth t fo both id of (5) aft a actio of T ( ) ith ad ad h fo ach h > gt th yt of th fo 6) x x b x x b hich gay ha a uiqu outio fo aot a p : x b b ; 7) x b b fo a gi t h a pcifid fo ach h h b b Wh ith h h h <h th th yt i of th typ: 8) ax bx cx3 dx4 b dx ax bx3 cx4 b cx dx ax3 bx4 b3 bx cx dx3 ax4 b4 h abcd o C i o H J KL hi b b b3 b4 I th att ca of H J KL it ca b od by th Gau xcuio agoith I th fit to ca of o C i th outio i: 9) x h ad c3 b4 bb b33 b44 b4 bb3 b43 3 b3 b4 b3b4 4 b b3 b34 b4 3 a b ccd ac abd 3 abbc d bd acd 3 3 ab c ad a c bcd 3 adb cdbd abc 4 Thu o ach tp ith to o fou idtiat a cacuatd ad ubtitutd ito th iitia ia agbaic yt that gi ia agbaic yt ith a ub of idtiat o to o fou pctiy May b paii outio o ach tp i ip bcau th doiato of th typ houd b aot yh by p poiti ( (6) (4) abo) Thi agoith act aaogouy to th Gau agoith Fiay th at to o fou idtiat ai ad thy a foud ith th hp of Foua ith (7) o (9) pctiy Wh fo a ad h i (6) o (4) a h od (ai oy x fo h o ai x ad x 3 fo h >) o fo o h > a h od 4 (ai oy x ) a yt of ia quatio a i (33) ipifi Thu a outio of th typ pcibd by (8) gay aot yh by p W xit h W i a doai W p : a < p < a p > of cogc of th ocoutati utipaat Copyight Sci