434 
MR E. W. BARNES ON INTEGRAL FUNCTIONS. 
Now so long as the integral 
2 sin 7 t 9 
| G (xz) e z (— x) 8 1 dec 
is within the regions surrounding z — 0, for which the original series can be summed, 
the differential coefficient of the function which it represents is the function repre¬ 
sented by the integral 
i 
2 sin 7 t 9 
(— x) e 1 dx, 
for we do not transgress the rules which govern differentiation under the sign of 
integration.* 
Therefore, within the region for which an asymptotic equality is valid, such 
equality may be differentiated. 
Similarly such equality may be integrated. And the process of differentiation or 
integration may be repeated any number of times. 
§ 32. We have hitherto limited ourselves to the consideration of asymptotic series 
of the type 
o 0 -f- oqz + . . . -f- ci n z n 
_ n'/ 
in which L£ \/a n — co and IA MTk — o. 
7? = 00 71=00 ^ 
The first condition is necessary that the series may have zero radius of con- 
vergency, that is to say, that it may be asymptotic. 
The second condition was requisite in order to ensure the applicability of the 
exponential process. 
V 
It is convenient to call an asymptotic series for which L t —T — o an asymptotic 
n = cn % 
series of the first order; one for which this limit is greater than zero, but L t 
n = cc 
\/ a n _ n 
o ^ 
a series of the second order, and so on. 
We have given in the preceding paragraphs the theory of summation of series of 
the first order. But suppose that we wish to sum one of the most simple asymptotic 
z (~y 
n=l 
noullian function. 
By Cauchy’s theorem, re-discovered by Hadamard, we know that 
——where S n (a) is Hermite’s Ber¬ 
ne" ’ v ’ 
series, that for lo 
F (z + a) 
T(z)^ ’ 
namely 
* Jordan. ‘ Cours cf Analyse,’ 2nd edition, vol. 2, pp. 151-157. 
