170 
A third proof has the advantage of showing that expansion of (1) 
according to factorials of «#—c is possible when the integral only 
exists for «== c, even when the straight line A(#)—= A(c) were the 
limit between the domains of convergence and divergence in the 
w-plane, and when the integral did not exist at all points of that 
line. The proof consists in repeating the process which led to the 
theorem of Leren an infinite number of times. We write 
t 
g(t) =] u T (u) du, (= vo fo u) du, | 
; . . (18) 
7.0) = 2 {i , (u) du, .- + gn(t) = fan (u) ne 
0 0 
Then formula (10) may be generalized in the following manner: 
a (x) = 9 (1) > a (1) (w —c) EE Is (1) re mr Us (1) (ort) ae . 
1 
= (— Int oid (1) Cz) doe (—1y" lens d'n (t) gend... | 
Deer 
The remainder has zero as a limit for R(«) > R(c), for if G i 
the maximum modulus of the limited function g (é) in the ie) 
(0,1), we have in succession 
lg, OIS Ge lg, |< GY, ... ar (OG 
hence 
| 
fo n teen dt afs 1 (é) teen dt 
<n foe ae) 1 dt 
0 
je (rd) >0 
———, voor R(«r—e 4 
R(«—e) 
gi) 4 —Ri = —] 
Now ( ) is for n =o equivalent to n Eee ‚ and thus the 
n 
modulus of the remainder in formula (19) is for all m-values’ less 
than 
H (a—e) 
! 3 20 
R (we) nR (x«—c) ey 
