of an Elliptic Transcendent Identity. 441 



from 



r°° COS bx 7 7T - 



Jo ** + ** 2 « 



and it is evident that the same formulae enable us to connect 

 together integrals that are very dissimilar in form. Thus in 



r 



c 



e~ cx cos bxdx = 



c* + b* 



take c q =n and we have 



I 



cos bx dx ■■ 



| Vn n-\-b 2 



(d\ n ~ l 

 — -j- ) , and there results from (6) and (7), 



I \/- •«""•« 4 c° s bxdx = a/— 



■» ,,-62 



V n V n 



viz. 



r»co #2 



j e~ ^ cos bxdx= x/tt .e~ b *, 

 Jo 

 which is (8). 



Or we may use (6) and (7) to deduce the values of compli- 

 cated integrals from others of a very elementary character : thus 



r°° dx i /r 00 dx f°° <fe -\ 



j_ 7T /_1_ _1_\ 



7T 1 7T 1 



2y>£) ' \/u+\f(3 ~ 2^ ' • (» + 2«)' 



since ci + /3= n } u(3 = a 2 . Operate with ( — — J , and we have 

 J o ^' e ~ ^VT'* ~2<rV2 ,, "V 1 2/(rc + 2«)»-* 



-2a 



■ w 1.3... (2n-3) . 

 ~2a* 2 7l " 1 .?i w -^ * 

 ^r r(2n + l) 2fl 



2«2 2 ».wr(w+i) 



7T >/(4?rw). (2n) 2w .e-2» 



2a 2 2 ".w. \Z{27rn).n n e- 



7T 



a*/2n 



n -n+k t€ -2a 



