﻿On the Expansion of a Function. 65 



=0 y=0 0=0 \ P J 



S=0 v=0 \ P / 



My 

 We therefore obtain 



0=o y =o\ p / L a =o • \ a /J 



2 V (-l)^ +v _ a (^ + a -?) 



a=0 \ a • ■/ 



University of Pennsylvania, 

 Philadelphia, Pa., U.S.A. 



"VIII. Note on the Expansion of a Function . 

 By I. J. Schwatt *. 

 npO expand 



1 Iog"(l + «) (1) 



in powers o£ x. 

 We have 



log (1 + ■'•)= TT7= S (~D-W.r= 2 (-D-^r T-. • (2| 



JO J-TtA Jo a =0 ao=0 a ~r - 1 - 



Now 



log»(i + «)=-2f X log(l + «)-^-=2r I (-1)"°-^ t (-l)i*i<fe 



Jo 1 + flJ Jo a =o a + l ai=0 



C* x oo oo + ai //'ao+ai + 1 



= 2 ss (-ir +1 — — r ^. 



Jo « =0 ai =0 «0+-L 



Letting a + a i = £» tnen 



Z* 1 oo oo _ r + l 



lo g2 (l + .,) = 2 2 2 (-l?^—d* 



JO oo=0 0=a o a 0"1" - 1 - 



oo J3 „ r 0+ 2 



= 2 2 £ Z— 1^— — — - 



0=o a o=o l ; (« +l)(/5 + 2)- 



Writing a for /3 and a x for a 0j we obtain 



oo ^,ao + - «o 



log 2 (l + ^ = 2 2 (-lh orT o S . . (3) 



a =0 a 0"i" - a X =0 «1 + J- 



* Communicated by the Author. 



Phil. Mao. 8. 6. Vol. 29. No. 169, Jan. 1935, F 



