Mathematics. — “On analytic functions defined by certain LamBert 
series.” By J. C. Kiuyver. 
(Communicated at the meeting of March 26, 1921). 
The definition of the analytic function was based by Wetrrsrrass 
on his theory of power-series. From a given analytic expression we 
deduce an element of the analytic function, that is a power-series 
converging within a determinate circle, and by the continuation of 
this element an analytic function is defined existing within the 
region that is covered by the set of the circles of convergence. One 
and the same analytic expression in distinct regions may define 
several functions. So, for instance, TANNERY’s series 
n 
n = 0 22 
= 
ni) ag 
k=% 
for: |zj <1 will represent the analytic function p,(2)—= = == 
Gl 
z 
TE whereas for |z| >>1 the expression defines the analytic func- 
+z 
k= 0 1 
tion p‚ (2) = — JF e*k*= — 
hil ars 
defined in a separate region, can be continued over the whole plane, 
but manifestly they remain everywhere essentially distinct. 
In fact, from the general theory it follows that the concept of 
an analytic function is not co-extensive with the concept of function- 
ality as expressed by an analytic expression and it is precisely this 
fundamental idea that, as Bors. repeatedly pointed out, sometimes 
leads to conclusions which are not always in every respect satisfactory *). 
Bore supposes that a given analytic expression F (z) defines a 
function @, (z) inside a certain closed curve C and moreover a second 
function g,(z) in the region outside C, the singularities of these 
functions being everywhere-dense on the curve, so that C for both 
functions constitutes a socalled natural limit. He then shows that 
the series of polynomials representing ~, (2) under certain conditions 
remains convergent, absolutely and uniformly, when the variable z 
along certain radii crosses the boundary C. Otherwise said, it occurs 
Both functions, each of them 
1) Legons sur les Fonctions monogénes uniformes d'une variable complexe. 
Chap. Ill. 
