IT 



114 Prof. G. N. Watson on the 



If we use Frullani's formula 



I (e-* — e- nt )r l dt = log n, 



and write nt = u, the formula for 7rS n becomes 

 a ^ i ^ f P °° 1~" e~ u d-uT[ 



*b w p ^-irfu . ( -)r + i Br+1 r^ 



(2m) U 2 ™" 1 ^ g«+l "* 1 (2r+-2)!n 2r+I J «" + l 



It will be observed that n no longer appears under the 

 integral signs, so that this result may be expected to give 

 the required asymptotic expansion when the integrals have 

 been evaluated. 



To evaluate the integrals, let u > — 1 ; then 



du 



o 



Jo «*+i Jo LB«"'«"+iJ«= 



_(-r LB^ r J„ ^+lJ„=o' 



the change of order of differentiation and integration being 

 easily justified sincea>— 1 (Bromwich, 'Infinite Series/ 

 pp. 436, 437). 



Now, a being greater than —1, we have 



/too r»oo *■ 



i e -o- u (e u + iy l du=\ e~ au 1 e~ u -e- 2u + . . . + ( — )P' 1 e 



+ ( — ) p — — i \du 

 K ' e u +l J 



— pu 



1 +... + ( -> p - 1 



a + 1 a + 2 a +p 



V Jo «"+l ' 



Now 



0<J o ^f M <j o ,-c--^« 



and this tends to zero as p tends to infinity. 



