438 
ME. E. W. BAENES ON INTEGEAL FUNCTIONS. 
which we take the series to represent is the function of which the series is the 
expansion within the circle of convergency. The series will therefore denote, with 
our present conceptions, the function 
p—an 
- r (i -«) 
The series with which we commenced may therefore he regarded as giving rise to 
the analytic function 
_ -L r (1 _ s) f (_ xzy-'e-'zdx = -~T (l-s) tV~ l dt 
27r v 7 J 1 - e~ x ‘ v 7 2?r v 7 J 1 — e~ l v 7 
on making the substitution t = xz. 
This function admits when \z\ is small, the arithmetically asymptotic expansion 
from which we started. 
When z~ 1 is a large real positive integer, the series and integral become fundamental 
in the asymptotic definition of the extended Riemann £ function. 
But there can be obtained by other processes an indefinite number of analytic 
functions, each of which has an essential singularity at z — 0, near which point it 
admits the given series as an arithmetically asymptotic expansion. We proceed to 
indicate one alternative process by which such an analytic function can be obtained at 
a single step. 
§ 36. For this purpose we use certain results of the theory of the connection between 
linear difference and differential equations. 
Consider the function 
f (ot, P\i • • • , pm > '' ) 1 
■«+...+ 
+ 1... a + r — 1 
r!p 1 ...p tu ...p 1 + r— 1 ... p m + r 
i (-x) r +, 
It is evidently a transcendental integral function which is a solution of the 
differential equation 
($ + a ) H- & ~k Pi — 1) • • • (d + Pm — 1) 
y = o, 
d 
wherein the operator $ = x —. 
If y be any solution of this equation, form the function 
— f y (— £c) ;_1 dx, 
2 sin ttz J J v 7 
where the contour of the integral and the prescription for (— x) z ~ l are exactly those 
employed in the definition of the integral for T (z) previously employed (§ 24). 
On integrating by parts, we have 
