438 Mr. J. W. L. Glaisher on the Verification 



an obvious application of Fourier's theorem to the definite in- 

 tegral 



e~ a2 * 2 cos bx dx = — — e 4" 2 . 

 2a 



I 



All these methods, however, involve high analytical processes ; 

 and as (2) is a very singular identity, it is perhaps worth while 

 to notice how it and (1) admit of being obtained by elementary 

 mathematics, in fact almost by algebra. 

 From the well-known formula 



1111 1 



cota=-H 1 1-- — ^+ ^— +... 



a a — ir a-\-7r a — 2ir a+2ir 



we deduce by ordinary algebra 



a 2 + # 2 + a 2 + (x-irf + « 2 + (x + tt) 2 + ' " 



cot (ai + x) -f cot (ai—x) 

 "~ 2a~i J 



and as 



COt U = l< l-\ : : r 



L e ut — e- w J 



= -i{l+2e 2ui + 2e 4ui + 2e 6ui . ..}; 

 therefore 



cot (ai + x) + cot{ai-x) = 1 c _. 2a+2 ^ c - 2fl - 2ft - , _ } 



whence 



1 ■ ] , 1 , 



a 2 + x* ^ a*+ (x-<nf * a 2 + {x + tt)*^' * ' 



= -{l+2e- 2a cos2a? + 2e- 4a cos4#-f...} : (3) 

 a L ' 



herein write a 2 =nc 2 , and there results 

 1 1 1 



/xy (x— ttV , /a? + 7rV 



+ ... 



= _£__ n +2e- 2c ^cos2# + 2e- 4c *' w cos 4x+ . . . }. (4) 

 v ^ 



Now differentiate both sides of this identity n — 1 times, and 

 then make n infinite. We have 



