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

 that 



I. 



00 sin (2a + l)x^. , 2 2 , 

 Ei { — q 2 x*)dx 



sm<# 



== -2 a/ttO- + Erfci + ^Erfc- . . . + -Erfc-\ '(45) 

 \J£q q 2 q a qj 



and 



Pf* Ei(- g y)% 



J ' 1 — 2rcos# , -f7* 2 



the latter series extending to infinity. 



A method similar to one frequently given for the evaluation 



v»oo 



of I e~ x2 dx enables us to express the product of two error-func- 

 tions as a single integral for 



Erfc0Erfc6=l I e~ x2 -y 2 dxdy, 

 Jo Jo 



which on transforming to polar coordinates becomes 



! = ^ ( tan " 1 "{l-e- a2s ^e)ddA' f^l-e'-* 8 «»***) dO^ 



tan" 



_!I_I^f^_i^fi^ . . . (4,7) 



after a couple of obvious transformations. 

 Taking a=b, we find 



whence 



■ r^^^^^^Erf^gfErfa)^;.-. ■ ;< (48) 



The equation (48) is not new, being only a simple transforma- 

 tion of one given by Raabe in 1847*. Raabe shows that 



J°° x^^= el "^{^^ |a ~v / ^)}, 



* "Ueber Producte und Potenzen bestiminter einfacher Integrals 

 Ausdriicke, durch mehrfache dargestellt," Crelle's Journal, vol. xlviii. 

 p. 1 37.;_See also De Haan's Nomelks Tables, No. 2. T. 29, and No. 5. T. 80. 



