DOUBLE GAMMA FUNCTION. 
381 
S'* s“ 
sin ITS 
g-^T-,r|T| _ jP 
where M is finite, however large |r| may be. 
Hence the absolute value of 
sill ITS 
S^s,k{u 
O)^, (W 
tends to zero as | t j tends to infinity; and this theorem is true if a is replaced hy 
a + fi- m^oj.T, where and vi.^ are positive integers. 
Hence the absolute value of 
~ Y (ft “b W?|(y| -|- 1 . 9 , k) 
Sill TTS ' ~ 
(where y is the function introduced for brevity in § 57) tends to zero as |rj tends 
to infinity. 
But 
re X) 
i.{s, a \ oji, oj.) = — - - X 1 'h f>'), 
Hi, = 0 Hij = 0 
and therefore 
lyS 
Sill TTS 
C: (•?, <( I Wo) = 
sin TTS 
00 x> 
■ - - p"!,X (« + »bwi + I-b Z^)- 
iHi = 0 nii = 0 
Now the double series on the right-hand side is absolutely convergent for all 
finite values of | t j, and the absolute value of each term tends to zero as j r j tends 
to infinity. 
Therefore 
■sP.rNo (s, c [ wd 
sill TTS 
remains finite as |sj tends to infinity, '£\{s) being finite and not greater than 2. 
When ’ii(''') is greater than 2, we have 
(. 9 , a 1 Wp CO,) 
CC CO -Sq/? 
V V ^ 
sin TTS 
»i,= 0 ))!,2 = o(ff T ^»iCOj -i- >11,(0,)' siu TTS 
and therefore the expansion on the left-hand side is finite however large | s | may be, 
provided \z\ < 1. 
§ 86. Consider now the Integral 
1 r ^,5 7r^o(.s-,a|M,,oJo) 
' 2 irL j * s sin tts 
The subject of integration is a uniform function of s, wherein 2 '^ is to have its 
principal value with respect to the axis of — (w; + ojo), z is to lie within the region 
bounded by axes to —&![ and and a is to be positive with respect to the oi’s. 
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