418 
given by Monten with the purpose of refuting the assertion of 
Pomprsu that it should be necessary — in order to render the limiting 
function f of a within G convergent series of analytical functions 
analytical in G — to suppose that it is permitted to integrate the 
series termwise along every curve of integration in G''). In fact 
MonrEL’s series converges to zero throughout the whole plane whilst 
the termwise integration along the real axis from O to 1 yields 
not O but 1, so a uniform convergence in every limited region is 
here altogether out of the question. We state, besides, that it is 
neither permitted to differentiate Monren’s series termwise for z= 0, 
as this would yield the result + oo instead of zero. The example of 
Monte, is embraced by none of the foregoing theorems. The want 
is now felt of a necessary and sufficient condition that a sequence 
of analytical functions, convergent in the interior of G, should 
have an analytical limiting function. Necessary it is that the con- 
vergence of the sequence be quasi-uniform in every closed region 
within G, because of the continuity of f and /. If, conversely, this 
quasi-uniform convergence is given, then it follows that f is conti- 
nuous within G, but not that f is analytical, as an example of 
Monte”) shows. In the following a necessary and sufficient con- 
dition will be deduced among other things. 
ib 
We consider a sequence of real functions f,(e), 7,(a),..., each 
of which is continuous for 0<#<a; the sequence converges for 
O<ac<a. Let f(#) be the limiting function. 
1. The function B (ry) =f fy) is definite in the rectangle 
O<#<a, OS y<a and is there continuous. The sequence of functions 
DP (x,y), B,(v,y),... converges at every point of the rectangle where 
not simultaneously # > 0 and y = 0, or vw =O and y > 0. The limiting 
function is Dey) = f(x)—f(y) at every point where w > 0 and y > 0 
and vanishes at <= 0, y =O, since there every ®, =O. Let it be 
given that f(w) converges to a finite limit as w — 0. We consider a 
set of points P, (z,, ,), P, @a» Ya). Where 0 <a, 5 a,0<y, Sa, 
line, =0, liny,=0; two arbitrary positive numbers may be 
k= k=o € 
given, e and N. Let MN, denote an arbitrary integer > JN. 
We have (0,0) = 0. In consequence of the continuity of Py 
0 
a number Jd, can be assigned such that at every point /, where 
1) P. MonTEL. Bul. des Se. math., série 2, tome 30, part 1 (vol. 41, 1906, p. 191). 
3) Pe MONTEL, li c, p. 190. 
Thase p. 98. 
