13 
S, even in case S lies wholly or partly in the domain of only 
conditional convergence; of course it must be supposed that all points 
of S lie at a distance greater than some fired arbitrarily small 
quantity } from the straight line parallel to the axis of imaginary 
quantities which forms the limit between the domains of convergence 
and divergence of the series. Finally we shall prove that this series 
represents an. analytic function in S, and for this it will be sutfi- 
cient to show that the series of differential coefficients, with regard 
to rz, of the terms of the first series converges in the same domain 
as this first series. 
The series of derivatives may be represented by 
oo 
oe en n! ay 
See et ty ell MED) 
0 
where, in the notation of NieLseN '), w(e) denotes the logarithmic 
derivative of the Gammafunction: 
dlog T(z) F(z) 
es dx i‘ I (z) 
Summation by parts gives, if the notation (44) is used again 
i M!dmW(e+m+1)__ sad ml am I (c+-m-+1) w (em 1) 
De 7 “= 
wp (2) 
(48) 
m 
I (@ +m + 1) c+m-+ 1) P(e +m +1) 
ri 
st V(e+l+1)w(@+l+1) yw SnmI(e+m+ lo . ; 
== P(e +1+1) phils raat, c)yp(a + m+ 2)—1], 
where the equation of finite differences 
1 
satisfied by y(#) is taken into account. Since, for «= the prin- 
cipal part of We) is equal to loge, the first term of the second 
member has, for &(«—c)>4d, zero as a limit for /—=o, and the 
series in that member converges absolutely. Hence the series in the 
first member converges also and we have ; 
Has m! an (a + m +1) ay ND Symi (ce + m +1) 
pie, U (« 4m + 1) nae ae: P(e + m + 2) 
n n 
ed) w (w+ m42)—-1]. 
From this equality it may be easily deduced that, corresponding 
to any e, there is a number N such that for n > N and for all z 
in the domain S 
1) Handbuch der Gammafunktion, p. 15, Leipzig, Teubner. 
