23 
(11) is real and not greater than 1, a fired integration-path W for 
the remaining integral may be chosen, as soon as the number n 
attains a certain magnitude, and for this we may take the imaginary 
axis in this case. The proof that im Rk, =O for n= o is then very 
simple, so that the above mentioned particular cases in which a 
function can be expanded in a binomial series, may be derived in 
a short manner from Cavcuy’s integral. 
As further regards the question, how far the inequality (7) is 
necessary for the expansion of a function in a binomial series, the 
way in which the-sufficient condition has been obtained gives us the 
conviction that the aggregate of functions determined by the latter 
condition is as large as possible. In order to come to certainty 
concerning this it is necessary to investigate how a function represented 
by a binomial series behaves in the domain of convergence of that 
series. This investigation may be effected by means of the statement, 
contained in the theorem of PiNcnerre, that a binomial series 
necessarily represents a coef/icientfunction, at least in the domain of 
absolute convergence of that series, for to this only the proposition 
of PINCHERLE applies. 
For simplicity we assume for the binomial series the original form 
(1), which is the one considered by Pincnerre. If the characteristic 
A' of the coefficients c, is less than —1, then, as already mentioned, 
the binomial series can be represented by the integral (6) in the 
half-plane R(x) > 0. It can now be proved that this integral satisfies, 
in the domain mentioned, the condition (7), with y = 0, the exponent 
l being subject to the inequality 
Bed Wv! ar be Bll et) 
where d is an arbitrarily small positive number. This condition can 
further be specified by evtendiny a certain property of the operation 
I by means of which, according to the view of PincHErLE, the gene- 
rating function gp(t) passes into the coefficient-function w(z); we 
mean the property expressed by the equation 
1) (2) 
I{g™@] = Ea 
EU" p (O1, 
This equation is given by PrNcnerue (le. p. 30) for the case r is 
a positive integer. If 7 is replaced by a and g(t) by g(t): (t-—1), 
the formula passes into 
r(: * op 
1D oe. |- GE OEE 
_ (t—1)* (re +e) (t—1)® 
The last equation appears indeed to be true for arbitrary positive 
