INTEGRAL FUNCTIONS DEFINED BY TAYLOR’S SERIES. 
279 
00 
The Function f(x) = 2 
> 1=0 
1 
x n e n+e 
r (n+ 1) 
§ 37. To illustrate the general theory which has just been developed , we will disc 
the function 
ass 
f(x) = 2 
x n e n+e 
n~ o r ('ll T 1) 
in which 0 is not zero or a negative integer. 
With our previous uotation we have /3 = 0, b m — 1/m!, and l may be taken as 
small as we please. 
Hence, when H (x) > 0, we have 
s=0 X 
J<r(aQ 
X a 
5 
where | (a. 1 ) j tends to zero as \x\ tends to infinity, and where 
S = \jt % mC s-nT (1 ft m ) __ V ( ) S m m c s-m r (s) 
p=om=o m!r(l— /3—s) m = o m!r(m) 
Now the coefficients m c n are given by the expansion, valid when \y\ <1, 
(log (1 —2/)/(—2/)} m_1 (1 —= t m c n (-y) n . 
71 = 0 
Hence S s is the coefficient of ( —;j)' 1 in the expansion of 
r (s) 
" s i-Y r 
_m=o m ! (m— l)! 
(i -yT 1 
in ascending powers of y when \y\ < 1. 
The reader will notice the connection between the series in the square brackets 
and Jo {log (1 — 2/)}, where J 0 (x) is Bessel’s function of zero order. 
When :H(cc)>0, we have now obtained the asymptotic expansion 
f(x) = e x i^ 
§ 38. Consider next the case when H (x) < 0. 
Then, il |arg(— x) \ <7t/ 2, we see that f{x) is equal to 
1 
27 TL 
| (— x) s T ( — s) e s+9 ds, 
provided we neglect terms whose modulus when multiplied by | x | l , where l has any 
