INTEGRAL FUNCTIONS DEFINED BY TAYLOR’S SERIES. 
207 
round the contour AOBjI^D of the figure. This contour consists of the real axis 
from A (+ oo) to O, the origin, the line OBj, a circular arc B^ of radius e round B as 
centre, and the line B 2 D, which passes to infinity parallel to OA. 
If we take throughout that value of 
— log (1 — — 
x 
3-1 
y 
X 
0-1 
which is one¬ 
valued on the plane dissected by a cross-cut drawn away from the origin from B to 
infinity, and which is therefore represented in this region by the series (summable 
, we may use Cauchy’s theorem. 
and divergent when | y | > | x 
We thus see that 
G '<^=SrW, 
_l_f 
xV (/3) J, 
y\ p \ r , i~y 
X 
71 = 0 
-logC-f 
0-1 
log f — — 
X 
1 
0-1 
e x y dy 
0-1 
e r, clr) + Iz. 
The integral I 3 is the integral round the arc BxB^,. It vanishes in the limit when 
e = 0, since 2ft (d)> 0. The two line integrals are taken along OA and B 2 D respectively 
parallel to the positive half of the real axis. In the second integral we have made 
the substitution y = y—x. 
The first integral by the general theorem of § 5 admits the asymptotic expansion 
Z (-) n c n T((3 + n) ' 
x 13 ,i=o r (ft) x n 
§ 21. We proceed to consider the second integral. 
We have to seek to find on BD the value of log [— log ( — y/x)^\ which on OB 
admitted the expansion 
= log^ + (...) ..., 
log 
log 1 
X 
X 
and which is represented by the continuation of this summable divergent series. 
Let x = re' 0 , so that, its principal value being taken, log ( — x) = log r+ l (0 — tt) 
6 being the angle of the figure. Let y = pe L{<i,+0 ~ ir \ so that is the angle given in 
the figure. We assume that r is large, and consider the shaded area bounded by 
p = 1 and p = r. 
When —yfx is real and positive, 1 77 1 being less than \x\, log [— log (— y/x)] is real. 
We will show that, for values of y on B 2 D within the shaded area, we must take 
log 
-log 
= log {log r+L (6— 7 r)} 4 - log 
where, within the shaded area, the final logarithm is such that 
exp 
log p+i — 7r) 
log r+ l (9—tt) 
( 2 ), 
(/3- 1 ) log {1 - l -°f p+ - 6 
1 log r+ l (0— 77 ) 
2 M 2 
