( 26 ) 
we must conclude 
fae > ef mee An LA) PC 
2nie url 20% Bg uz—! (u — (9)? 
DE EEE ‘oy 
= (2-1) 2>— = z— G Oo) 
sa 2: (mo dmo) 
Em =O, den 
Eme Sy te fale 
The above double series still differs from that which served to 
define Z(z) because it does not include the terms of the simple series 
=’ (mo) 
nk Oe 
Hence replacing this series by the equivalent expression 
(1 Hemi ) w+ C (2) there results 
1 
=> 2z—-1)[Z (2; @, 0) — (1 + e—7”) w* C(z)]. 
We will now seek to express the integral J in a different way, 
The function w (u) can be expanded in a trigonometric series. By 
a known formula we have 
and owing to the fact that along the path of integration the ratio 
u/o remains real, we are at liberty to substitute in the integral J 
the cosine-series for the function w(u). So we are led to a series 
of integrals, each of which of the form 
1 1 k 
IS f cos ae aus 
@ 
Always supposing the real part of «> 2, we find by the usual 
methods of integration 
