INTEGRAL FUNCTIONS DEFINED BY TAYLOR’S SERIES. 
263 
axis of the contour which is also described both ways. We suppose that the cross¬ 
cut which renders (— z)^ 1 uniform has been deformed with the contour. 
The value of the integral in the two directions along the final line P is, putting 
z — R+£? 
In = -Ufip)(R+ £)*-' exp{-R0-£fl + e-*-%} « H, 
1 (p) Jo 
the principal value of (R + £)' 3_1 , which has a cross-cut along the negative half of the 
real axis, being taken. The integral is a line integral along P. 
On this line 1\(£) > 0. Hence the maximum value of . the real part of e _R ~fir is 
e~ n | x | K, where K is a finite quantity independent of P and | x |. 
Hence 
| la | < { e_R MK - ^dl£(0)} |P 3_1 | 
exp ( — 0£) | \d£\. 
The last integral is convergent and tends to a definite finite value Q as Pt increases. 
When |a;| is large let us take \x\ = e R . Then 
I 2 |< 
1 
r(£) 
exp K {log | x 
i»w-i Q 
J \x\^ (e) ' 
When H (6) >0, |I 2 | will obviously tend to zero as R tends to infinity, and this is 
true of | e~ z I 2 j for all values of 9 which are not real and negative, if H ( x ) > 0. 
§ 16. We have now to consider the value of the original integral along the contour 
which consists of the small circuit round the origin and the real axis described both 
ways from 0+ to R. We denote this integral by R. Make the transformation 
1— y = e~ z . Then corresponding to the original modified contour we have a new 
contour Q as in the figure. This consists of a small circuit round the origin and the 
real axis described both ways from 0+ to R/, where R' = 1— e~ R . The former line 
P from R to oo becomes an infinite spiral from R' round the point 1 up to this point, 
the whole spiral being contained within a circle of radius e -R . 
We now have 
Ii 
— e 
tr(i-ff) 
277 
[log (i yYf l { l ~y) e 1 e xy dy, 
the integral being taken round the origin from R' back to Pt'. The many-valued 
functions which intervene in the subject of integration are such that, when \y\ < 1, 
[log (1 — 2 /)] p_1 (1 “ 2 /)° _1 = (~yY~ l 2 c n (-y) n 
n =0 
( 1 ), 
