1227 
that the value of an analytic expression, coinciding at first with 
that of the function ~, (z), can be made to change continuously into 
the value of the function g,(z) and this possibility more or less 
seems to be incompatible with the theory, according to which the 
functions g, (z) and gy, (z) are wholly uneonnected. 
In the present paper I propose to treat two simple examples in 
which the transformation of ¢,(z) into a series of polynomials is 
not necessary, and that, as I believe, yet give an insight into the 
tendency of Borer’s remarks. 
Let the given analytic expression be the series of LAMBERT 
n=o, | gn 
F (2) = = — 
n=1 Nn 1— 2" 
’ 
where the exponent s may supposed to be real. 
Clearly, whatever be the value of s, we can expand F(z) into 
an integral series, and as for |z| <1 we have 
Lf. 2 1 1 
wie a ee 
F(z) defines an analytic function p‚(z) inside the circle C of radius 
unity. However, if s >>1, we may write 
n= 1 il gn n=o | “gn 
Pg) = == j- —, —}|——O(s)— YS —.—— 
n=l ns ns all 
erst ind 
and from F(z) we derive also an integral series in —, that is a 
z 
second analytic function p‚(z) existing in the region outside C. 
The functions p‚(e) and @,{z) represented in distinct regions by 
the same analytic expression satisfy the relation 
1 
nl te(ZJ=—s (CD 
but the main question is, whether either of them is, or is not an 
analytic continuation of the other. The decision can be based on 
a transformation of F(z). Corresponding to the rational numbers 
a” the interval (0,1) we can arrange the so-called rational 
on? 
points a, =e ? on the circle C as a sequence (a,) and denoting 
by g the denominator of the rational fraction that corresponds 
to a, it will be seen that we have 
— 1 
F SE 1 EE EE 
(z) 2$(s+ brie PES Dd 
hor 
