(Sane) 
1\s-1 
] lo En oo 
( | ys! 2, (—1)"-! 
7 6) ees 
1 4. | ‘ 
will hold. 
However the theorem under discussion does not serve to show 
that the above eqnation remains correct for 0 << s<1. Here as well 
as in other cases this theorem needs amplifying and as such the 
following theorem can sometimes serve. 
When /'(2) is developed in a series of continuous functions we 
shall be able to deduce 
b 
b 
ale (cde = Sf (w) da, 
0 
a 
a 
out of 
F(x) = Z u, () 
0 
as soon as is given: 
ist. 2 u,(v) is convergent for a<#< hb. 
0 ia 
b 
iv 2) 
)nd, SS Un (w) da converges. 
0 
a 
3'¢. The function w,(7) does not change its sign in the interval a< a < 6. 
i | Unt (z)| . oa : : 
gidst pete): | is monotonic with respect to w and that for all values 
| Un \t) | 
of the index ” in the same sense. 
In order to prove this theorem we must show in the first place, 
that the series to be integrated converges uniformly for a<a< b. 
In the main this follows out of the fourth datum, which states that 
for all values of m the inequality 
| 
Unt (7) 
n(x) | Un(y) | 
will exist when #<y, or that for ail values of » that inequality 
will hold when «> y. 
I first suppose that tae inequality holds for «< y. On the ground 
of the third datum we find 
< | Uni (y) | 
| 
| 
| 
| 
| 
| 
| 
Uni (#) 
An = —— and Cn) = ———— 
to be positive numbers and the inequality expresses that nti S Gn. 
