PNAS-1935-Stratton-316-21

PHYSICS: J.A.STRATTON SPHEROIDAL FUNCTIONS OFTHESECOND KIND ByJ.A.STRATTON DEPARTMENT OFPHYSICS, MASSACHUSETTS INSTITUTE OFTECHNOLOGY ReadbeforetheAcademy, Monday, April22,1935 InarecentnoteinthesePROCEEDINGS,I henceforth tobereferred toas I,certainsolutions oftheequation (1-z2)w'-2(a +1)zw'+(b-c2z2)w=° (1) weredefined. Corresponding toadiscrete setofcharacteristic valuesof theparameter b1thereexistsolutions ofthetype Re,(c,z)=(CZ)-a-1/2EalJx+a+1/2(CZ)t (I-10)n convergent throughout theentirez-plane andbehaving as(cz)-a-i sin cz-+arasz-*co.Intheneighborhood ofz=0aformalsolution maybeobtained intermsofanexpansion inhypergeometric functions, 54,(c,z)-'d,Tx(), (I-8) whichforthecharacteristic valuesb1isalsobounded atthesingularities z=k1.Withtheaidoftheintegralrepresentation (I-18)itwasshown that(I-8)and(I-10)areproportional, Sea1(c,z)=XlklRe1,z(c,z), (I-26) andthattheexpansion coefficients arerelatedbythesimpleexpression (I-22). Anindependent particular solutionof(1)behaving atinfinityas(cz)-l coscz-+ar)maybedefinedformally by R21(c,z)=(cz)-a'/sEj'axNx+a+,12(cz). (I-ll)n Thisexpansion, however, diverges ingeneralatthebranchpointsz= =k1andconsequently isusableonlyforlargevaluesoftheindependent variable. Itistheobjectofthepresent notetodefineasecondsolution validintheneighborhood ofz=0,andtodetermine itsanalytic connection withthefunctions (I-10),(I-ll). Byanalogy withthedefinition (I4)ofthefunctions T,let T"(S) =(1-Z2)-a/Qa( (2)316 PROC.N.A.S. (2) VL215PHYSICS: J.A.STRATTON wheretheQn+.areAssociated Legendre functions ofthesecondkindas definedbyHobson.2 IfR(n+a+1)>0, eahiri(n +2-a+1)PTae;"r(n+a+1)(1-t2)X+a(z-t)-X2a1 dt.(3) SinceboththeTx'andtheT"thusdefinedsatisfythesamerecursion formulas, asecondformalsolution of(1)maybedefined, Se2,&(c,z)=E',Txa(z), (4)U whichmaybeexpected toconverge intheneighborhood ofz=0,since E'dxisbounded whenbassumes oneofthecharacteristic valuesb1. Itisknownthat(1)issatisfiedbytheLaplacetransform w(z)=fe(l-t2)au(t)dt, (I-18)C whereu(t)itselfsatisfies(1)andthecontourCistobedetermined bythe bilinearconcomitant eis(1-t2)a+l[icu -du 0. (I-19) AsinIletu(t)=Sea,1(t)=Xgk1Re1j(ct). Ift-*oXandIargctI<7r, Se,,a(t)-Xikl(ct)-'-1 sin(ct- +a) Letitbeassumed thatI[c(z+1)]>0,I[c(z-1)]>0.Thisisinaccord withtheconditions ofmostphysicalproblems wherein zisrealandI(c)>0. Thecontour maynowstartandterminate att=+co.Inparticular a solution mayberepresented by *r~~(-I+J+,C' t)e.w(z)=J e'0'(1t2)Se1,(t)dt. (5) r(-1+,1+) LetIn,=Jf(~l+.i+)eis'(1-t2)Ta(t)dt Thenonreplacing Tabythedifferential expression Tx(t)=2 -1)r(n+1)na+1)(1-f2)-1x(1-t2)X+a,(6)()2x+,r(n+1)r(n+a+1) dtx( ),(6 andintegrating partially onefinds r(n+2a+1) seU+G)Vi(cz)U f'(-4x=2ior(n+1)r(n+a+1) Ji'(t2 -1)x+adt. (7)VOL.21,1935 317 PHYSICS: J.-A.STRATTON IthasbeenshownbyWatson' that r(-n -aJa)(-i+i+)J-ts-a_l/2(CZ)=2"a+.4 sl(t2_1)x+adt2 -2x+a+1irv"2ir e"~t (8) andhence IK=5r(X+2±1(-_)'+'2A/fir5sinair(cz)-a-'/2J X.a-/2(Cz).r(n+1) (9) Multiply bothsidesof(9)byd4,replace4ontherightsidebya.according to(I-22),andsumoveroddorevenvaluesofnasIisoddoreven.There results f(-1+l w(z)=Je"'I(lt2)aSel4(t)dt= 2k,(-1)l+l sinar(cz) /2 --a-/(cz) (10) Theintegralthusevaluated maybereadilyexpressed intermsofthe functionsRe',,andRe'j,.Invirtueofthedefinition oftheNeumann functionNx+,a+i,2 whennisaninteger, Nx+a+l=/ -tanawrJx++a +,-((1)'J-n.a-1/2, (11) cosair itfollowsthat (-l1)'+(cz) -a/ -t'agJ- x-a(cz)=cosar[tanarRela,I+Rea,j.(12) Inthedomain 1<zcoonehastherefore J e"(1-t2jSa,(t)dt=k1sin2air[tanairRea5,+Re,a].(13) Thecontourintegral mustnow-beevaluated forvaluesofzlyingin domain .-1<z<1.Toaccomplish this,letSe,j(t)bereplaced by X1kgR4,g(ct) intheintegrand of(13)andconsiderintegrals ofthetype Kx e -)a(Ctf4T/I J+a+q/m(d)dt. (14)- TheBesselfunction maybereplacedby-''itsrepresentation asa-defihite integral, (C+a+ +a$t) (15)=2x"a/2-Ir(n +a+1)-1ied1 s+d, (5PROC.N.A.S. 318 PHYSICS: J.A.STRATTON andonreversing theorderofintegration thereresultsfor(14), 2x+aV21Frr(n +a+1)fds(-s S eict(s-5)(t2 l)a(ct)ndt. (16) Letthecontour ofintegration betakenaroundthepoints(-1,+1)such thatt>1overtheentirepath.Then(t2-1)"maybeexpanded ina convergent seriesofinversepowersoft,anditmaybeshown4that l+yl+)eict s(s-S)(t2 --)a(Ct)Xdt = 2,r+ mi2r(-a)Lr(m -a)C2M(z)2m-2a-x+. (17) C2+ m=Om!r(-a)r(2m -n-2a) K=12e'2(1)1+'sin2a 2n+aVwr(n +a+1) c9a+i 1(-1),r(m-a)r(n+ 2a+1-2m)mft(1-S2)n+am-o m2t(-2it)(18) (Z-S)2m-2a-X-ldS Nowby(3)itappears thatII (1_-S)+a(Z_)2m2a ds=2+a+r(n+a+1) -avTa-" FP(n+2a 2m+1) (1nM andonsubstituting in(18)andreducing, oneobtains sn2a- "(-1)mr(m a)ma-2m Knm=2*- --ea+1e-2me (+20) 1rcr~~~~~m=o m!r(-a) .(20) (21) (22)Finally, invirtueof(14)and(13),thereresults -a"(-+,I+) E'a,KI =e +eic (1a2)GS41(t)dt, n olklJ or tanawrRe',a+Re,l = e-a2ie -61)+112 .Ema=, O 2(-1)mr(m a)C2YTa'2m m!1'-a) .+5 Theright-hand sideof(22)isnexttobeexpressed asamultiple ofthe function Sr,edefined in(4).Inthesimple casea=0corresponding to axiallysymmetric spheroidal functions, thesummation overmbreaksIl1VOL.21,1935 319 32YICYSICS: J.A.STRATTON offwiththefirstterm.Replacing a.byd.,according to(I-22)yieldsat once Re=-e2Set,l. (23)C Inordertodetermine theproportionality factor,letuswrite 'axe (-2)"x(m -)2"Tj+ =;q(c),'dxTx,(24)n m-om!r(-a) n andevaluate attheordinary pointz=0.Byanobviousmodification of Hobson's expressions forthefunctionsQx+.onthecrosscutbetween 16wehave rn+2a +1) Ta(o) =2a-1sn+2a 2 ' (25) r(n22} andacorresponding expression forTx,J+(o).Introducing thesevalues into(24)oneobtains anexpression forthejoiningfactor,ulintermsof theknownexpansion coefficients. r(n+2a+1) PI(C)E'dlxe2_ , = n~~~~~rn+2a2+1 r(n+2)n+ a-m+ m.om!r(-a)2'm n+2m+2)rU2) Theanalytic connection between thefunctions ofthesecondkindis therefore tanarRea,l+Rei,a=2(-1g4 X,p,S4g. (27) Incaseaisaninteger,asinthespheroidal wavefunctions,Sea,,represents theanalytic continuation ofReajintotheorigin. Inparticular te-2;(a=0), (28) and b(bc+1)(a=1). (29)320 PROC.N.A.S. PHYSICS: R.C.TOLMAN Ifaishalfanoddinteger, asinthecaseoftheMathieu functions (a=-1/2),tanawrin(27)becomes infinite. Thefunction (CZ)-a.-1/2 alJ-.-a-11,(cz),however, isthennolongerindependent ofRe4,'and,as mightbeexpected, itiseasytoshowthatSe2,jisnowproportional to Sea,'.Anindependent expression forSe!,1applicable inthiscasemaybe obtained byalimitingprocess, thedetailsofwhichwillbepublished later. IJ.A.Stratton, "Spheroidal Functions," Proc.Nat.Acad.Sci.,21,51-56(1935). 2E.W.Hobson, TheoryofSpherical andEllipsoidal Harmonics, p.193. 3G.N.Watson, TheoryofBesselFunctions, p.164. 4Whittaker andWatson, ModernAnalysis, 4thEd.,p.245. 6Hobson, loc.cit.,p.229. THERMAL EQUILIBRIUM INAGENERAL GRAVITATIONAL FIELD ByRIcHARD C.TOLMAN NORMAN BRIDGE LABORATORY OFPHYSICS, CALIFORNIA INSTJTUTE OFTECHNOLOGY Communicated May9,1935 1.Introduction.-In theabsenceofanyappreciable gravitational field theconditions satisfiedbythetemperature ofamediumhavingnothermal flowfromoneportion toanother, arewellknowntobethoseofuniformity throughout thesystem. Inthepresence ofagravitational field,however, theconditions imposed atthermalequilibrium onthepropertemperature- asmeasured atdifferent pointsofthemediumbylocalobservers-are considerably morecomplicated, owingtotheassociation ofinertiaand weightwithallformsofenergyandtheconsequent tendency forheatto flowfromregionsofhighertothoseoflowergravitational potential. Nevertheless, forthespecialcaseofstaticgravitational fields,thesemore complicated conditions havealreadybeenobtained, ",2anditisthepurpose ofthepresentarticletoinvestigate theconditions forthermalequilibrium inthegeneralcaseofanykindofgravitational field. Theinvestigation showsthatareasonably straightforward treatment canbefoundwhichleadstoanapparently satisfactory, covariant expression ofthegeneralconditions forthermalequilibrium. Thepossible valueof sucharesultcanbetwo-fold. Inthefirstplaceitsatisfies ourdesirefor coherent theory,sinceourpreviousexpression oftheconditions forthermal equilibrium, whichappliedonlytostaticsystems andwasmadeinthenon- covariant language ofakindofcoordinate systemthenappropriate, can nowberegarded asaspecial caseofthegenerally covariant expression applying toanykindofsystem. Inthesecondplace,theexpression ob-VOL.21,1935 321