168 
integral (15) vanishes for the sequence of values (11), hence ¥ (s), 
and therefore also f(s), identically vanishes in the interval (0,1). 
The theorem of J.ercn has thus been proved completely. 
4. The first part of the theorem, that «(w) becomes identically 
zero, if this is the case for an arithmetical progression of z-values, 
may be proved in a direct manner, without first proving the second 
part; and it is an immediate consequence of the proposition : 
A function a(x) defined by an integral of the form (1) can, under 
the suppositions 1 and 2 mentioned at the beginning of § 2, be 
expanded in a binomial series 
aj Sal"). re Nagler dae 
0 
where B is a number lying in the domain of convergence of the integral. 
Suppose, for a moment, this proposition to be true. If, then, « (x) 
becomes zero for the sequence of values (11), we take B= £ in the 
equation (16). Substituting for # the values §&, 5 +1,§+ 2,... in 
succession, we find that all coefficients c, of the binomial expansion 
vanish and therefore that a(x) vanishes identically. 
The first part of Lwercn’s theorem is very easily proved in this 
manner and it would therefore be desirable that we might derive 
from it the second part in a short manner. But as yet we are not 
in a position to do this. The above demonstration is, after all, rather 
short, but besides, on grounds that, with a view to conciseness, we 
prefer not to state, we do not think it likely that the zdentica/ vanishing 
of a(x) is more effective for the purpose than the vanishing for an: 
arithmetical progression of values of the argument. 
Nevertheless the first part of Lerca’s theorem has an interest in 
itself, because remarkable consequences may be inferred from it. 
Among these Lerch mentions the truth that simple functions such as 
Ta): (k > 9) 
cannot be the determining functions of generating functions, in other 
words that they cannot be represented by integrals of the form (1), 
neither can products of these functions with others which remain 
within finite limits in the finite part of a certain halfplane R(x) > c. 
The proposition concerning the expansion of the integral (1) in a 
binomial series may be proved in different manners. In the first 
place integrals of that form belong to the general category of 
functions of which I showed, in an earlier communication in these 
sin ka, cos ka, 
