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SPECIAL FUNCTJO;\,S OF MATHE:\IATICAL PHYSICS AND CII:EMISTRY M:\'I'III~;MI\TICJ\L UNIVElls!'ry 'I'I;:XTS AT.l~X,\~Dlm DANIEl. C. AITKEN, D.Se" F.Il.S. Eo nU'I'JrEIlFOHD, D.Se" On. :'IATll. D"TI!UMISANTII ANn i\IATlIICI!S STATllIT/CAI. :'IATlllmATICS W,Wt;8 EU:CTIIICITY 1'/t().JI!C'1·/V'-: GI':O)lHT/lY 1:o."Tl'.G/lAT'ON 1'.\/11'110/. DII"'P./I/!STI.\TIOS ISFlslTl.: SIWII!S, Prof. A. C. Aitken, n,Se" 1'.11.5, I'rof. A. C. Aitken, D.Se.• F.rI.S, Prof. C, A, Coulson, D.Se., F.n.S, Prof. C. A, COllison, D.Se., F.II.S. 'I'. K Faulkner, Ph.D, Il. P. Gillcspie, Ph.D. n. P. Gillespie, Ph,D. ProF. ,1. :'1. IlyHloll, IJ.Se. 1:o."TEGlIolTIO:S 01' OllDINAII\' DlI'I'l!.H1!:o.'·IM. ~UAT/OSS ISTllOllUCTIOS E. L. Ince, D.Se. '1"0 "IlI! 'I'llI!OILY 0/' I'IS/TI! Gnoups \\'. Letlcr"mllll, Ph.D .• D.Se, ASAL\'T/C'\L CI!OMlITRY 01' TURI!I! DUlr.SSIOSS ProF. W. I-I. M'eren, Ph.D., F.Il.S. FUSCTIOSS 01' A emll'l.!!X VAR/AIII.I-: D. E. CUSS/CAl. :\II!ClIASICS • V"CTOll MCTIiOIll! • Vor.u~,,: AS/) h'TIWIIAI. E. G. Phillips, I\I.A., M.Se. nutherrortl, D.Se., Dr. :'lalh. D. Eo Illllherrortl, D.Se., Dr. MUlh. .pror. W, W. Ilogosillski, Ph.D, S/'/!CIM, FUSC'flOSS Ot· :.IATllE)IATICAI. I'U\'SICS .\SI) CllI!)IlSTlLY Prof. I. N. Sneddon, :'1.:\., n,Se, ll"rry Spain, B.A., .\I.Sc., I'h.D. l'roF. II. W. Turnuull, F.ltS. Tlmony 01' EQUATIOSS hi f~tqHlrlllion TJIlwny 01" OUmNAUY DIl'I'ImENTIAI. EQU.... I'IO.'1S J. C. Butkill, Se.D., F.RS. GEluI,\s·Escl.ISII :'IATIll!~IATICAI. VOCA .. UI .... IlY S. Macintyre, M.A., I'h.D. TOl'OI.OO\· • K ,\1. PUllcrson, Ph.D. SPECIAL FUNCTIONS OF MATHEMATICAL PHYSICS AND CHEMISTRY fly IAN N. SNEDDON .'1.:\., D.Se. 1'f10FF.$son 01' MATUf:!ltATICS IN TilE USl\'RHSITY COU.EG!: 0.' !'\ORTII STArrOROSlllRI: OLIVER AND nOYD EDINllUUCIi AND LONDON NEW YOIlK: INTERSCIENCE PUlll.lSIIEIlS, INC. 1950 FIRST EOiTlOS tu:;o ",UNTIP III 1l0LLA10.I When /I is 11 ncgntivc fraction F(II) is defined by means of equation (ii); for example By mellllS of the result (ix) we can derive all ill\.crcsting expression for I!:ulcr's constnnt, y, which is defined by the equation y = lim (1 n_oo + ~ + ... + -.:. - log 'II) = 0.5772 (5.3) /I From (ix) we have ~{lo~r(=+ l)}= '" liz lim (100'11- - ' - _ - ' - - ... _ -'-) .::+1 z+2 :::+11 n ...., . , . . so that letting z --'10- 0 we obtain the result (.iA) lLnd from (5.1) we find y= - f: e- l logfdl. (5.5) 12 THE SPECIAL FUNCTIONS OF PHYSICS AND CHEMISTRY §S Integrating by parts we see that f ,oc f log tllt = log:: + for. -dl , t so that - Y = lim .~ (f• - t ""/:-I dt + log z . ) (5.6) Closely related to the glllumn fUllction are the cxpOllcntial-integrnl ei(x) defined by the equation ei(x) = f O'-" - d.. '" (x> 0), (5,7) II· and thc lognrillllllic-integrlli li(x) dcfined by li(X)=f'l dU o og • (5.8) I~ Si(.r) Ci(x) -1'- -1 - 2 "--=~,,--,--~:-::c'-:-----:-~~----' Fig. t Variation of Ci(~1 nnd Si(z) with ~, which are themselvcs connected by the relation ci(z) = - li(c"'). (5.9) Other integrnls of importance arc the sine and cosine integ-nils Ci(x), Si(r), whieh arc uefincd by