( 852 ) 
When the upper limit 4 of the range of integration belongs no 
more to the interval of the uniform convergence, we can prove 
still pretty simply in some suppositions, that the integrating term-by- 
term gives a correct result. 
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We are sure of this, when the series > u, (+) converges uniformly 
0 
for a<a<h, when the series of integrals converges absolutely and 
when moreover each term w, (+) has a constant sign in the whole 
domain of integration. 
For, now we find in the first place for a <1 < bh 
f t 
. = . 
il (a) dg a u (2) Om: 
0 
a a 
Of this last series the absolute value of each term is smaller than 
the corresponding term of the convergent series 
b 
a) 3 | 
= | Uy (x) dx |, 
a | 
from which ensues that with respect to ¢ the series 
t 
D 
. 
= ( Uy (a) de 
Ow 
a 
converges uniformly in the domain a <t <5. 
The principal property of the uniformly convergent series furnishes 
then immediately 
b t b 
DL 5 AL Er 
fre dz == Tam 2 ele ae =S td 
i) 0 0 re 
a a a 
Very often this theorem proves sufficient. Thus we find that for 
O<a«< 1 the equation holds: 
and the development at the right-hand side is in this interval 
uniformly convergent. 
The series of the integrals 
as (—1)-! 
F(s) > —_— 
1 
n° 
converges absolutely, if only s >> | and under this condition there- 
fore the equation 
