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2. The integral (1) ezists for a certain value «=c of 2. 
We put 
(= ff van eN (| 
0 
Then, by 2, g(é) is continuous in the closed interval (0,1), and 
zero for t=O. Further, by Ll, g(t) is differentiable at all points of 
that interval, except, possibly, at t= 0, and we have 
OOAD mee re ss ly Ng he 
Hence, if d >0, we may write 
i 
1 1 
fr (t) edt = f g (tte dt =[9 () tee], — (a@—c) fg) tret dt 
é ) ) 
If, now, z is a complex number with real part & (2) greater than 
c, the number d in this equation may be made to approach to zero, 
and thus we find 
1 
1 
fro t? dt = g (1) — (ed) fi ged. —. (5) 
0 
0 
From this it follows: If the integral (1) exists for a certain value 
«=c of x, it exists in the whole half-plane defined by R(x) > R(c)'). 
Further it follows from (5) that the integral in the left-hand 
member represents a continuous function of « in any domain S 
lying wholly in the finite part of the half-plane R(c) + d, where 
(f > 0). In the same manner as above it is found that the integral 
1 
[ioe tog ta NEE EN) 
0 
exists for R(x) > R(c) and represents the derivative of a (a) at any 
point of this half-plane, so that @(z) is also an analytic function. 
These consequences, too, are mentioned by PINCHERLE. 
The proof Lercu gives of his theorem equally starts from the 
equation (5). In the following reasoning, however, we shall use an 
1) This theorem is fundamental in the theory of generating functions. After 
PINCHERLE different authors have proved it, though often under less general 
suppositions. The reasoning in the text is due to Lerou. This reasoning is founded 
upon the continuity of f(t), which, presumably, is forgotten by LERcH, when, at 
the end stating his theorem, he says that f(t) may be as well discontinuous. 
(Of course we do not mean to say that generalization is impossible). 
