440 
MR, E. AY. BARNES ON INTEGRAL FUNCTIONS. 
Thus rrr ( 2 ) = 1 , and we have finally for all values of a. p l} . . . p m and 2 the 
identity 
l f T7i / W \_1 7 _ m / \ h (a — z) T( Pl ) ...F(p m ) ^ 
2 sin 7T2 j F “ pk,m ” P ^ ( ■''' '''' “ ^ ^ r («) • r ( Pl - 2 ) ... r( Pm - *y 
This identity is the direct generalisation of the identity 
2 sin ir: 
( e~ s (— x) : 1 dx = T ( 2 ), 
and we may therefore expect to be able to use it to extend our former process of 
“ summing” asymptotic series. 
§ 37. We may, in fact, show at once that we can sum any series of convergency 
zero f ( 2 ) = « 0 + a x z + . . . + a n z n + • . . , in which L t a n — (n I) 4 , where k is any 
n = x> 
finite quantity. 
For this purpose we put a = p 1 = . . . = p m = 1 ; 
| •' i • | -| ^ [7T ,H F m ( X ) 
F™ (.T) = 1 + xrx- + ... + 7 — + . . . , and we have — - - n — 
mK ' 1 (1 !) W1 1 1 (rl) m 15 27tJ [sm Try]™ 
(—x) : 1 dx = \Y ( 2 )]” 
Then, with our former notation, we take the auxiliary function 
X ( z ) = 2 c nZ n , where l/c n = [T (n + 0)]“ = - 
l [7T m F m ( — X) 
n =0 
2tt J [sin 7r^J” 
( — )" m X Y 1 (~ £c)“ dx. 
\ 
And now f(z) is defined by the integral 
in which 
W f q (_ xz \ (-*)(- 1 c ( r 
2t r J 1 ’ [simr0r ’ 
G(w) = 2 (~)" m a u c n u\ 
n —0 
We take m > k, and then G (u) will be an integral function. 
For L t a n c H — n ,l{ - k ~^e~ n ^ ~ m) + - •• ; and therefore L t \Za n c u — 0. 
n= 00 n= 
* In connection with the proof of this formula, the reader may with advantage refer to:— 
Mellin, ‘Acta Mathematical 8, pp. 37-80; 9, pp. 137-166 ; 15, pp. 317-384. 
,, ‘Acta Societatis Fennicse,’ t. 20, pp. 1-115. 
Poincare, “American Journal,” vol. 7, pp. 203-258. 
Pincherle, ‘ Accad. del Lincei,’ ser. iv., t. 4, pp. 694-700. 
Pochammer, ‘ Mathematische Annalen,’ Bd. 38, pp. 586-597 ; Bd. 41, pp. 197-218. 
„ ‘ Crelle,’ Bd. 71, pp. 316-352. 
Orr, ‘Cambridge Phil. Trans.,’ vol. 17, pp. 182-199. 
