Table Of ContentYALE UNIVERSITY
MRSH.E PSEAL YS ILLIMAMNE MORIALLE CTURES
LECTURES
ON
CAUCHY'S PROBLEM
IN
LINEARP ARTIALD IFEFRENTIALE QUATIONS
SILLIMAMNE MORIALL ECTURES
PUBLISHEBDY YALEU NIVEliSIPTRYE SS
ELECTHICITY AND MATTEH. By .JosEP.IJIo nNT noMSON,D.scL.L,. D.P,I I.D.,
F.R.SF.e,l loowfT rinitCyo lleagned C avendiPsrho fessoofEr x perimenPthayls ics,
CambridgeU niversi(tFyo.u rtPhr inting.)
THE INTEGHNI'IVEA CTION OF NERVOUS SYSTEM. By CHARLES
THE
S.S HERRINGTOND,. sc.M,. D.n,o N.L L.DT.O R.F,. R.sll.o,l tP rofessoofPr h ysiology,
UniversiotfLy i verpoo(lS.i xtphr inting.)
HADIOACTIVE TRANSFOHl\IATIONS.B y ERNESTH uTHERFORDD,. sc., LL.D.,
F.R.sA.f,a cdonaPlrdo fessoofrP hysicMsc,G illU niversi(tSye.c onpdr inting.)
EXPEHIMEN'l'AL AND 1'HEORETICAL APPLICATIONS OF THEHl\10-
DYNAMICS TO CHEMISTRY. By DR \VALTERN ERNST,P rofessaonrd
Directoofrt heI nstitouftP eh ysicaClh emistirnyt heU niversiotfBy e rlin.
PROBLEMS OF GENETICS. By WILLIAMB ATESONM,. A.F,. R.sD.i,r ectoofrt he
JohnI nnesI lorUcultuIrnaslt itutAifoenr,t oPna rk,S urrey, Engl(aSnedc.o nd
printing.)
STELLAR MOTIONS. WithS peciHaelf erentcoel\l otionDse terminebdy Means
oft heS pectrograpBhy .\ V1LLIAWMA LLACEC Al\IPBELsLc,. D.L,L .D.D,i rectoofr
theL ickO bservatoUrnyi,v ersioftC ya liforni(aS.e conpdr inting.)
THEORIES OF SOLUTIONS. By SvANTEA RRITENIUPSn,. n., scM..DD..D,,i rector
oft heP hysico-ChemDiecpaalr tmenot.t fh eN obelI nstituStteo,c kholSmw,e den.
(Thirpdr inting.)
IHRITABILITY.A PhysiologiAcnaall ysiosf t heG eneraElff ecto fS timuliin
LivinSgu bstancesB.y l\IAX VERWORN,M .D.P,H .D.P,r ofessaotrB onn Physio
logicIanls titu(tSee.c onpdr inting.)
PROBLEMS OF Al\IEHICANG EOLOGY. By WILLIAMN ORTHR ICE,F RANKD .
ADAMS, ARTHUPR. COLEMAN,C HARLES WALCOTT, WALDEMALRI NDGREN,
D.
FREDERICKL ESLIER ANSOME,A ND \VILLlADM. MATTHEW. (Seconpdr intgi.n)
THE PROBLEM OF VOLCANISM. By JOSEPHP AXSONI DDINGSP,H .B.S,C .D.
(Seconpdr inting.)
ORGANISM AND ENVIRONMENT AS ILLUSTRATED BY THE PHYSIO
LOGY OF BREATHING. By JOHN SCOTTH ALDANE,M .D.,LL..D,F.R.s.,Fello
ofN ew CollegOex,f orUdn iversi(tSye.c onpdr inting.) w
A CENTURY OF SCIENCE IN AMERICA. \VithS peciaHle ferencteo the
AmericanJ ournal of Science By EDWARD SALISBUR·YD ANA,
1818-1918.
CHARLES SCHUCIIHEERRTB,E RTE . GREGORY,. JOSEPBHA RRELL,G EORGEO TIS
SMITH,R ICHARDS WANN LULL, LOUIS PIRSSON\,V JLLIAEM. F ORD,R . B.
v.
SosMAN,H ORACEL .W ELLS,H RY W. FooTE, Lman PAGE,W ESLEYR .C oE,
AND GEORGEL .G ooDAI,E. AR
THE .EVOLUTION OF MODERN MEDICINE. By Sm \VILLIAOl\SIL ER,B ART.,
M.D.F,. R.S(.T hirpdr inting.)
RESPIRATION.B J. HALDANE,M.A.,1\1F..DR..,S H.O,N.L L.DB.I RM.A ND EDIN.,
y s.
Felloowf N ew CollegOex,f ordIl;o norarPyr ofessor, BirmiUnngihvaemr sity.
LECTURES ON CAUCHY'S PROBLEl\I IN LINEAR PARTIAL DIFFEREN
TIAL EQUATIONS. By JACQUESH ADAMARD,L L.D.A,1 embeorf theF rench
Academyo fS cienceFso;r eigllno norar:Mye mber otfh eA merican Acadoefm y
Artsa ndS ciences.
AFTER LIFE IN ROMAN PAGANISM. By FRAN CuMONT.
THE ANATOMY AND PHYSIOLOGY OF CAPILLZA RIES.B y AuGusTK ROGH,
·
PH.D.L,L .n.P,r ofessoofrZ oo-physiolCoogpye,n hageUnn iversity.
LECTURES
ON
CAU CHY'S PROBLEM
IN
LINEARP ARTIALD IFFERENTIAELQ UATIONS
BY
JACQUESH ADAMARD,L L.D.
Memboeftr h Fer enAccha deomfSy c iences
ForeHiognno rMaermyb oeftr h Aem eriAccaand emy
ofA rsta ndS ciences
NEW HAV EN :Y ALE UNIVERSITY PRESS
LONDON HUMPHREYM ILFORDO,X FORDU NIVERSITPYR ESS
:
MDCCCCXXIII
PRINTED ING REAT BRITAIN
THE SILLIMAFNO UNDATION
N they ear a legacoyfe ighttyh ousadnodl lawrass l efttot he? resident
1883
I andF elloowfsY aleCo lleignet hec itoyf N ew Havent,o b eh eldi nt ru,s t
asa gifftr omh erc hildriennm ,e moryo ft heibre lovaendd h onoremdo ther,
l\frsH.e psaE lyS illiman.
On thifso undatYiaolne C ollewgaes r equestaenddd irectteoed s tablains h
annuaclo ursoef l ecturdeess ignteodi llustrtahtepe r esenacned p rovidence,
thew isdoamn dg oodneosfsG od,a sm anifesitnet dh en aturaanld m orawlo rld.
Theswee ret ob ed esignaatset dh el\ frsH.e psaE lyS illiml\Iaenm oriaLle ctures.
Itw ast heb elioefft het estattohra ta nyo rderplrye sentaotfit ohne f actosf
naturoerh istocroyn tributtote hde e ndo ft hifso undatmioorne e ffectivtehlayn
anya ttempttoe mphasitzheee lemenotfds o ctrionroe f c ree;da ndh et herefore
providtehda lte ctuorned so gmatoircp olemitchaelo losghyo ulbdee xcludfreodm
thes copoeft hifosu ndatiaonnd,t hatth es ubjeschtosu lbdes elecrtaetdh ferro m
thed omainso fn atursacli enacnedh istogriyv,i snpge ciparlo minentcoae s tron
omy,c hemistgreyo,l oagnyd a natomy.
Itw asf urthdeirr ecttehda te acha nnuaclo ursseh oulbde m adet heb asis
ofa volumteo fo rm parto fa sericeosn sttuitinag m emoriatloM rs.S illiman.
Them emorifauln cda mei nttoh ep ossessoifto hneC orporatoifYo anl Uen iversity
int hey ea1r 90a1n dt hep resevnot lumceo nstittuhteefi sf teenotfth h es eries
:
ofm emorilaelc tures.
PREFACE
HpEre sveonltui masr e eu smoef rmeys ewaorrocknth h hey per
Tbolciacis nel inpeaarrdt iieaffrle netqiuaaltI ih oanvhsea. d the
happionfsep sesao k1si onmge opfia ttroa tn sA meircaaund iaetn ce
ColumUbniisavi e(tr1y 9a1na1dl) hs aotd h heo noofut rr eastoimneg
poianttt hsUe n iverosfRio mte(i 1e6s9a) 1n Zdu r(ic1h9* 1.I7a m)
mucihn detboYt aelfoder h avingm etg hioevp epno rttodu envietlyo p
the wohifotw l,ie tt hhre e ciemnptr ovwehmieIcnh hta svb ee eanb le
tom ake.
'rohrei ogtfih fnelo lowiinnvgea sttiiiotsgnbo se fo unidnR iemann,
Kirhcohaffn d stViolllrt aefum'rnosadr mee nMteamlo oinr srs ipchael
ancdy nldirwiacval\elIs ey.n deahvaboseu etrnpo urtshuee owtfoh rek
Italgieaonm eatnsedotr oi, m praonvedex tiettnhd ia mtta bye come
applitcoaa l(blnl oer hmyaple)r ebqoulaiticin osntosefo,a n dtl oyo ne
oft heOmn.t hoet hhearnt dh,pe r esweonrmtka bye c onsiadse red
ac ontinoufma yLt eir;osonun 1sP· r olpaa gdaeEtsti duoeenslt e Esq ua
tiodnelHs 'y drodyuneaa,nm ediv,qe ansr, e plsaecrviaepnlga goefts h e
lacshta pTthelera .t itnedrew,ea adsfi , r asttt eimnwp hti,Ic o hn ly
succeiensd heodw tihneg diffiocftu hplert oibetlshe semo t liuoofn
whiIca hmn oawb tloep resent.
Furtehxeetnrs icoonuasll bdse go i vteosn u ch researches, including
equatoifho ingsoh redrse yrsst,oe fem qsu atainoednv ses,mno e a p
plicattoni oonin-nslee aqru at(itsohtneus od fwy h ihcahbs e eunn der
takienrn e cteinmtte hsa,nt kots h teh oeroyfi t negerqaula st:)i on
whiscuhb jheocwte,sIv , e hrdaevbleei raltteea flsyi dteh,pe r aaisrmy
onceo nstaiw thuobtlyeie st .s eslIhf ablehl a pipsfyo mgee ometers
succienee xedtn ditnhfgeo llmoewtihntogotd hse se new cases.
AtfeVro elrtrfanu'dsa mel\nIteamloo fti hrAesc tJaf atahtceiam,
vol. anhdi fsru thceorni tbrutwieos nhso,hu alvdte om ention,
XVIII,
asd elvoepainncdgo mplVeotlirtnaegp'r osi onfvt i etwh,we o rokfs
TedoCnoeu,la ondndA' d hemaTrhtle.a ttveorl'uLsme eseq tuiaons
audxe ripvaeretsi eclalrteaesscrt iiqrueees(l lSecsi Ceoneltclitai on,
a
* almseon taib orni neoftr ee aadtt hIen etrnatCioonngarloe fMs ast he
I
maticiSatnrsa sab(toS uerpgt e1m9b2e0r) .
Picarreds'esa rchewses- hwahqluiloci thnte h epilra cea-laseroses ential
t
ins evepraaroltf ts h per esweonrtkS .u cihas l stohc ea se foRro uLxe.
Vlll PREFACE
ParGiasu,t hilelrai-rnVsci)l auc daerseb fiubgllr piahoirceavli aenwd,
anotohnehera b se egni vbeyVn o elrthria meslifnh iLse ctduerleisv ered
atS tohcokl(mp ublaitsH heerdm a'snP,na riWse)d .in do tth iintk
necseasrtyog ivaet hdio rneev,e tnoa dtd hmee ntoiflo ant weorr ks,
ancdo ntoeunrts ewlivteehvs e nqtuuoatla itnfoi ootnnso tes, apologising
ina dvatnoct eha eu trhswo howme m ayh avfeo rgotten•.
Reasmounssat l bseog ivfeontr h ceh nagoef t wtoe rms hwahdi ch
beepnr evsilioynu trodauncdae ddo pitneS dc ienOcneei. s" fu nda
menatsl olutrieopnlb"ay c" eedl netmaesroytl iuont"h;e other consists
inr eplatchiweno gr" dc onorcmraelab,ty"te dh e fi(ndd'eArd hemar)
himfsb,ey l" trsavnersTahle. "fih rabsset e dno nieno rdteoar v oid
confusion "wufintdha mteshnoetl anulst i"in otroduPcoeidn cbayr e
anhdi ssu cce(sasssoo rlsu toifho onmso genieneotguresaq lu at;i ons)
thsee cofonrrd e asoof"n esc oomnyo ft houtg,ah"st hneo tiinoq nu es
tioanl rdeyoa cciurnts hC ea lcuolfVu asr iawtihoeinrtise sd, P- noted
byt hweo r"dt ranssa"vl e.r
wisthoe xpreshse amrytt iheasntk tsw yoto ou nAgm eircan
I
geomeMtreW rasl,sa hn l\d.f Mru rr,aw yhomh avbee esnop leased
I
tos eaet P ardiusr ingA ctahdeey meiacr Thyev ery
1920-1921.
kilnydu ndteorotkor evise thoeft hEgenr gelaiptsaehror tf m y
manuscrfiaeprmt a.n fayu lotfls a ngumaagyhe a ve edsectaepced
I
tiobnu,tt h astuh ce rraorres nontu mmreooruiesds u teot heuisre ful
anfrdi enhdellyp .
*Ouro wnM emoiornts h seu bjheacvtbe e einn rsteeidnt he Sci'ent.
.Annales
Ee..N ormS.up .( 1904-1a9n0dt5 h)e (v.o l 1908.)' Ve
.Act.Aaf atltematxixcx1a,
wanttop oionuttt h atth lea tctoenr tasienvsee rrarloi rnns u mercioceaffilc ients,
vi.iz nfo rmu(l3a0 p'.3) 4,9 w,h erade e nomin2am tuosbrte c ance(lfalcetdo2 r
a
havisnigm iltaobr ela yd deidnt hper eceldiinnaegn) di, na lfolr mulrwe lattoi ng
m eve(nc orproensditnoog u rB ooIkV )w,h icmhu sbte c orreacsti end the
presveonltu me.
H.
J.
192.1
July
ama lgsro eyai tnldebttoPe rd.oA f.L .U nderohfl\i :1liln,n esota,
I
forh iksi nadd viicnec so rrectiongfl afanugludtausgr ei tnhge
reviosfpi roonsoa ,fn edx prteohs ism bmeyst th anks.
1923.
May
CONTENTS
PREFACE PAGE
vu
BOOKI. G ENERAPLR POERTIOEFSC AUCHYP'RSO BLEM
I. CAUCHY'S FUNDAMENTAL THEOREM. CHARACTER-
3
ISTICS
II.D ISCUSSION OF CAUCHY'SR ESULT 23
BOOKI IT.H EF UNDAMENTFAOLR MULAAN D
THEE LE�IENTSAORULYT ION
I. CLASSICC ASESA ND RESULTS 47
58
II.THE FUNDAME NTAL FORMULA
III.THE ELEMENTARY SOLUTION 70
1.GE NERARLE MARK.S . . . 70
.
2.SOLUTIOWNIST HA N ALGEBROISDI NGULARITY 73
3.THEC ASEO FT HEC HARACTERISCToINCO ID. 83
ADDITIONNAOLT EO N THEE QUATIONOSFG EODESICS 111
BOOKI IIT.H EE QUAITONWSI THA N ODDN UMBER
OFI NDEPENDVEANRTI ABLES
I. INTRODUCTION OF A NEvV KIND OF IMPROPER
INTEGRAL 117
1.DI SCUSSIOOFNP RECEDINRGE SULTS . . . 117
2.THE FINITPEA RTO FA N INFINISTIEM PLIEN TEGRAL 133
3.THE CASEO FM ULTIPLIEN TEGRALS 114
4.S E IMPORTAENXTA MPLES. 105
II.THE IOJ\NITEGRATION FOR AN ODD NUMBER OF INDE-
PENDENT VARIABLES . 159
III.SYNTHESISO F THE SOLUTION OBTAINED 181
IV.APPLICATIONTSO FAMILIARE QUATIONS 207
BOKO IV.r rHEEQ UATOINSW ITH AENV ENN UMBER
OFI NDPEENDENVTA RIABLAENSD T HEM ETHOD
OFD ESCENT
I. INTEGRATIONO F THE EQUATIONI N 2m1V ARIABLES 215
1.GENERALFO RMULJE 215
2.FAMILIAERX AMPLES . . . . . 23.6 .
3.APPLICATITOONA DISCUSSIOOFNC AUCHY'PSR OBLEM. 247
II.OTHER APPLICATIONOSF THE PRINCIPLOEF DESCENT. 262
1.DESCENFTR OMm E ENT Om ODD . .. . .2 62
2.PROPERTIEOSF THVE COEFFICIENITNS T HE ELEMENTARY
SOLUTION. . . .. . 266
3.T REATMENOTF N ON-ANALYTEIQCU ATIONS 277
INDEX 313
Seveforramlu blerei,on fgg e naelar ncdo nstut sahena,v bee en
denobtye ds pseycmibvaoill:z s .,
Stanfdors Introduicne d
Book Seciton
(e1) Equatioofnv ibartinsgt rin.g.s. . .. ... I 4
(C1) CorresponCdaiuncgh cyo nditio..n. s . .. I 4
(e3,)( C3) Thes amefo r equatioofns ound ... . .. I 4
(e2), (C2) Thes amefo r cylindrwiacvaels .. . . .. I 4.a
(E) Generfoarlm oft hel inepaarr tidailff erential
equatiooftn h e2 ndo rde.r. . .. . ... I 12
(A) Partidailff erenetqiuaalt ifoornc haracteristIic s 13
(F1) Fundamental foforrm uRliae mannm'est hod II 36
(g) Greenfo'rsm ula ... . .. . . . . .. II 40
(F) Fundamentfoarlm ulian g eneral.. . ... II 40
(@)A djoienqtu ation ... .. . ... ... II 41
(£) Generlailn eeaqru atiionnt wov ariabl.e..s II 42
(L1)(L,2) Differenteiqaula tiofonrsg eodes.i•.c s ... II 55
