Sum of a Series oj Cosecants. 115 



Thus we have 



\ e u + 1 ~«+l~«t2 + a + 3 



= *{*Gk+i)-+(Wi)h 



where ^ is the logarithmic derivate of the Gamma-function. 

 It follows at once that 



Now, when m>-l, 



r(i)=(-r% i^ + 2 ii + • • •} 



= ( — ) m ~ 1 m lcr m+ i, say ; 

 and 



t (»., ( i) =( _)»,- lm ;|_i_ + _t_ + . . . j. 



= ( - )—«-+>« ! |„„ +1 _ -^ -^ - . . .} 



=( _ ) „,- 1(2 ». + i_ 1)m!(T)vi+i . 



Consequently 



Finally, we have to evaluate 



, r(§) . r.d) 

 = ,og r(I)" log r(i) 



= log 7T - log 2 ; 



the change of order of integration being justified, since 

 «>— 1 on the range of integration. 



