THE B AND T FUNCTIONS 
273 
has shown how the series 1) may be made to converge more 
rapidly. We have for any x in 21 
log (\ + x~) = x — 2 ( — 1)" 
n 
This on adding and subtracting from 1) gives 
log <?(1 + *) = - log (1 + x) + (1 - 0)x+ I(- - 1)| 
Changing here x into — x gives 
log 0(1 -x)=- log (1 - x) - (1 - 0> + t(H n - 1)* 
Subtracting this from the foregoing gives 
.x n 
n 
log 0(1 + x') — log 0(1 — x') 
1 + X 
"I I ™ /y.2m+l 
= - l0 « tzt x + 2(1 - 0)x -?2»TPT (H ^ - X) 
From 225, 4 
7XX 
log 0(1 + x') + log 0(1 —x)= log — 
Sin 7TX 
This with the preceding relation gives 
log 0(1 + 2:) 
1 + 2: . i 7T2: 
v.2m+l 
(2 
= (1 - <7>-i!ogi±S + log1, -l) 2 - m + x 
valid in 21. 
This series converges rapidly for 0<2;<|, and enables us to 
compute Gr(u) in the interval 1<m<|-. The other values of 0 
may be readily obtained as already observed. 
228. 1. We show now with Pringsheim* that Gr(u) = T(u). for 
u > 0. 
We have for 0 < u < 1, 
V(u n)= e~ x x u+n ~ 1 dx 
=/"+/* 
* Math. Annalen, vol. 31, p. 455.