298 Mr. J. W. L. Glaisher on a Class of Definite Integrals. 



The former result can be verified by integrating both sides of 

 the equation 



e v x^ax = —= -. — 



2 sjc 





with respect to c. 



The integral I g-(«* a +ft*+«0 d x j s s i m ply expressible in terms 

 Jo 

 of the error-function ; for the former 



A2/»co'/A\ 2 -. b2-4ac h 



= e- c+ ^\ e~ a \ x+ ^) dx = J-e-^-^v^~. . (10) 



I J si a 2y# 



Integrate both sides of the well-known equation ■ 



,CO J IT - r - 



e~ a2x * cos 2rccdx= -77— e « 2 . . . . (11) 



i 



2a 



with regard to r between the limits r and 0, and we have 



that is, 



J. 



e-^ 2 sin2r<r — = ^-V7rErf- (12) 



a? 2 "' « 



By differentiating (11) with regard to r and dividing by r, we 

 obtain 



,GC „2i2 sm 2ra? 7 vV _^i 



e -a x- # ^^p _ _^ e a 2 # 



r 



From this we deduce, by integrating w r ith regard to r, 



JV* 2 Si(2 W )<fe=^-|JirfJ . (13) 



De Morgan (Diff. and Int. Calc. p. 675) obtained a formula 

 which, when slightly altered in form and generalized, may be 

 written 



00 e-™ 2 cos 2bx 



I 



« 2 + a? 2 



dx 



= ^e a2c {e- 2a *Erf(^^ (14) 



and differentiating with regard to b, we have 



r" > e- cx *sm2bx 



a 2 + x 2 



xdx 



= ^e fl2c {e- 2aS Erf(«v/c- ^)-e a "*Erf(flVc + ^Y|.(15) 



