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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