Expansion of a Function. 491 



Letting tc + a — fi, we have 



,2k— 1 00 CO CO 



e=] <£/c — 1 a=1 K =i s=k+1 *# — - 1 - 



23-3 



/3 — 1 1 



= 2 (-l)V- 3 2 H— ■, 



8=2 k=1«* - L 



Therefore 

 (tan-Vy=2 ! 2 (-l)- 1 ^ 2 g-^- (2) 



Again 



Jo K=l 2tf ^=1 2/^ — l a =l 



J^X X 1 K 1 CO 



2 ^ 2 srn 2 (-i)'**-*fe. 



i'* SO j8-l 1 K 1 



= 3 ! 2 (-l)V^- 2 2 ±- 2 — ^— , <fe. 



J 0=2 ic=l^«i=l^i— 1 



Hence 



co t 2k+1 k 1 Ki 1 



(tan-V) 3 -3!2 (-l)-i^-— 2^2 ^-r. .... (3) 

 We now assume that 



co 2k u + P -2 p-l/ K a—l 1 \ 



(fan-^)P=pl 2 (-l)^- 1 ^-^ 5 n 2 9 ■ A (4) 



7 K0 =i 2/e Q +p— 2 a=1 \ fffl _ 1 2/e a -f- p ~ a— 2/ 



then 



T» * r 2« + ?5-2 p— 1 / K a-l) 1 \ 



J k„=1 2/c + ;9-2 a=1 \ Ka=1 2/r a 4- j p-a-2/ 



x 2(-l)v-Vv--^ ; 



y=l 



