Integral functions defined by Taylor's series. 
291 
wher 
’e 
G p (x‘,0)= t 
x 
„i=o m! (m + dy 
Now we have obtained the asymptotic expansion (§ 22) 
x n =o x i (q — p — n) n=0 n\ r(p— n) [log (— a?)] 
where the c’s are defined by the expansion, valid when y < 1, 
' ] og(i-.y) ' 
- -y . 
( 3-1 
(l- 2 /r= 2 c n (-y)L 
w=0 
If | arg x | < 7T, we see, as before, that it is sufficient to take p such that 
2 a < 2 p+l<«, or «<2p+l<fa, in order that the integral may vanish for an 
infinite contour for which H (s) > —k. 
When 1 <C«*=C2, we may take p = 0. And when 0 < a < 1, we may take p = 0, 
provided | arg x | < § our. 
Under these conditions the integral is asymptotically equal to 
- X 
771 = 1 
_1_ 
r (1 — am) (9—my x m 
r (aO + ay ) sin {tt (a — 7 ) (y+6)} 
sin n(6 + y) 
The contour of the integral embraces the positive half of the real axis and contains 
no singularities of the subject of integration except the origin. The terms neglected 
in the equality are of order less than any algebraical power of l/|a?|, when \x\ is 
large. 
To obtain an asymptotic expansion for the integral, we may, under the integral 
sign, employ the summable divergent expansion 
V (aO + ay) sin {tt (a ~q) ( y+0) \ 
sin tt (6+y) 
OD 
t e n {-ij) n . 
The integral is then represented by x 6 % — 7 -^- 
1 y H = 0 r(/3-n)[iogx]^ +1 
All these asymptotic expansions are negligible compared with the dominant terms 
of Gp { x 1 ' a e 2m,lla ; a.6}. 
Hence when 0<a<2, and |arg x \ <onr/'2 , we have asymptotically 
oc 
where e c n is given by the expansion 
“ «c„r(l-/3) 
«=o x n/a T (l— ft—n) ’ 
[ 1 °g( 1 -^);T 1 (i~:?/)“ 0 ~ 1 = 2 e c n (- y y, 
valid when \ y\ < 1. 
2 p 2 
