21 
the function w («) can be expanded as a binomial series 
7 a BN) 
w (2) = Si a ( ) ak le ed Dd CLO) 
at n 
0 
if L+8—k>y, and otherwise the expansion is possible in the domain 
R (2)> y. 
The special value —w of the argument « of a is such that the 
expression at, w—y being given, has the greatest modulus com- 
pared to those for other values of a on the circumference of the 
circle (1,1). Lf the inequality (7) holds for fixed a-value on the circle 
(1,1), then expansion of w (w) in the series (10) is possible in the domain 
| BNB ln na en (9% 
of 1+ 8—15y, and otherwise in the domain R(#)> y. 
The sufficient condition for the expansion of a function in a 
binomial-series contained in the above theorem seems, indeed, very 
simple. If a function w (7) can be represented by the equality 
D= Et Oye ee) A ee EI) 
where c is a fixed number within the circle (1,1) and u (z) a function 
remaining within finite limits in R(z) > y, then it satisfies the ine- 
quality (7) for a value of / differing arbitrarily little from — oo, 
and therefore it can be expanded in a binomial-series in the domain 
R(«)>y. For c=1 formula (11) gives an expression which shows 
that all functions regular in the finite part of the half-plane R (x) > y 
and vanishing at infinity may be expanded as a binomial-series in 
that domain; further all functions becoming infinitely large at 
infinity of an order lower than a certain finite power of z; so all 
irrational and logarithmic expressions. 
The way in which we have arrived at our theorem is substanti- 
ally the same as that followed in the ordinary theory of functions 
of a complex variable, in order to obtain the expansion of a function 
in a power-series; it is founded upon the fundamental theorem of 
Caucny. According to this we have . 
ih w (z) dz 
ot) 
(2) 271 zE 
W 
(12) 
where the integral is taken round a closed curve W, within and 
upon which w(z) is regular, and which contains the point z= 
in its interior. If we wish to deduce from this integral an expansion 
according to positive integral powers of w—a, a being a number 
1) This series is taken instead of (1) for the sake of generality. 
