( 27 ) 
— ait Wz 1 7t od 
cos —  ——__—_——_- ee : 
rrd (e—1) 
ie ge 
I=2e 7” cos (EZ je = hel 
@ Ee) (z en pn 1—q?k 
Combining this equation with the one we got before, we finally have 
Zes o,o) = (1 + e—7%) oo F(z) + 
2 2 2k 
++ — zl -*) e—*t cog — S 2 jet . 
T(z) Re 1=gik 
Now, in deducing this equation we postulated that the real part 
of z was greater than 2, but as the right hand side defines a one- 
valued function of z, holomorph all over the z-plane, we may regard 
this equation as the proper definition of 7 (<), thus establishing the 
existence of an integral transcendental function that is only partially 
represented by the doubly infinite series. The resemblance between 
the functions Z(z) and (1 + e—7) wm € (<) is manifest. The former 
is evolved from the doubly infinite system mo + m'@', the latter 
arose entirely in the same way from the simply infinite system mo. 
Therefore we may conceive Z(z) as an extension of the function 
(Lei): £(z), the relation between these functions finding its 
analogue in the relation between the elliptic functions and the simply 
periodic ones. 
Seeking for the values of Z(e) corresponding to integer values of 
its argument, we find in the first place that 7(z), therein behaving 
as (Ll +e—7)@—*C(z), vanishes whenever < is either a positive 
odd or a negative integer > 1. For z=0, we get 
Z(0)= 20(0) = —1, 
for <= 1 there results 
Z(1) = Lim (l—e-*#) ot 6 (1 + 6) = — 
a 
As for positive and even integer values of 2 we have 
EN 4 EEn an a ea n gn In—l 
Z (Qn) = 20? 6 Gn) + ram =) DZ Em! 
