ME. E. W. BARNES ON INTEGRAL FUNCTIONS. 
441 
We can thus sum any natural series of convergency zero whose n tV coefficient is 
of the same order as a finite power of n ! # 
§ 38. But we can go further than this : we can construct inverse functions which 
will enable us to sum any series of convergency zero. 
For we have seen that 
i 
7r ~" '! l nw( X ) / ,\<l+0-l 
(sin 7 rey™ K ’ 
dx = [r (n + 6)J m . 
Suppose, now, that we construct the function f(x) — % — 
m— 1 
This function will be an integral function of x, for we have 
2w F sw ( - x)' 
(sill ird)*' 1 ' 
which is absolutely convergent for all values of | x [. 
But, if we operate by our integral on this function, we have ^ jjf(te) (— x) n+e ~ l dx 
= S [r (n + 6)~\ hn ; and the function N [r (n -f- #)]' m is infinite if ( n + 6) be 
T7l= 1 
1)1 — 1 
positive. 
^ co Jj / 7T~ m F 0 i ( — 
If, now, we take f (x) = S . . 5 " 8 , : c -, Vvdrere b m is so chosen as to make the 
J v 7 (sm TrOy m 
771= 1 
CO t ... 
series N b m [r (n + converge for all finite values of n, it is obvious that f(x) 
771= 1 
will itself converge for all values of x, and so be an integral function. 
We may now take for the associated function 
X ( 2 ) — c 0 + c L z + . . . + c a z u + , where c n 1 = 2 h m [r (n + 9)J m ; 
771= I 
and by suitable choice of the coefficients b we may make c a vanish to an order as 
great as we please. 
We can then sum the series « 0 + a x z + ...-[- a n z n + . . . , where a H is infinite 
with n to an order as high as we please. In other words, we have invented the 
analytical machinery necessary to sum any (natural) asymptotic series. 
§ 39. As an example, suppose that we wish to sum a series a 0 + a x z + . . . 
+ a n z n + . . . , where a n is infinite like where 0 < a < 1. 
* This theorem corrects a mistake in my paper, ‘ Theory of the Gamma Function,’ p. 112. 
VOL. CXCIX. —A, 3 L 
