425 S 
of |, (2) ai (2) 
a> Z, 
/ (2 converges also. If C’ denotes a circle situated within z <a, 
For #=y9 =z fe (w, y) coincides with f(z), hence 
with centre z,, then /* (2,2) converges quasi-uniformly within C. 
This function is continuous in C and for z= z, coincides with f (2). 
In consequence of the quasi-uniform convergence the limiting function 
F@) —fG) 
is also continuous within C. This limiting function is 
22, 
for z+ z,. 
Hence 
ee = ey 
2 zy od 
whence it follows that f'(z,) does exist and f' (z,) = lim f, (z,). Since 
n= @ 
| 
z, may be chosen arbitrarily within |z/ <a the theorem is hereby 
established. 
9. Lf the sequence of functions f(z), f, (2),...., each of which is 
analytical for \z| <a, in this region converges everywhere to an 
analytical function f(z) and if besides for any z the following 
relation holds 
f' (2) = lim f ((2), 
LOF 
am) 
at the points of every closed set internal to |x| <a, \yl <a. 
In consequence of J, being convergent J’ (x,y) converges to 
(a)—f ‘i sh 
LW for Fy and to f'(z) for «—y—~z, since there f, wy) = 
J (2). The function /* («, 4), which is every where continuous therefore 
then the function f* («, y) = converges quasi-uniformly 
converges to a function which, since f is analytical, is also every- 
where continuous. This involves the quasi-uniform convergence of 
a pn S , . ta aan N a . . | 7 
J, (% y) =f, ©) at the points of every closed set within «| <a, 
vl <a. 
10. The theorems of § 8 and 9 may be resumed concisely as follows: 
A necessary and sufficient condition that it should be permitted to 
differentiate termwise a convergent series of analytical functions is 
, » / 4 IJ y » * id 
the quasi-uniform convergence of the series = f* (a, y). 
11. Lf within 2) <a a convergent series of analytical functions 
28* 
