Mr. J. W. L. Glaisher on a Class of Definite Integrals. 301 



Lies of the integrals; for example/ 

 in 



f °° 1 * 2 h 



I e - a **-2bx dx= _±_ e - Erf o 



</ a \/ n 



-esu 

 I e~ a * 2 (cos 2bx — i sin 2bx)dx = -4- e~« Erf — 



Jo A J 



gest the real values of the integrals; for example, write bi (« = \/— 1 

 as usual) for b in 



—ax 2 —'2bx f j v , T — 



and there results 



1 -i 2 ^, „ Ai 



o 

 and 



y # A 

 whence 



e -aa;2 co§ 2^^ ^j» __ e a 



2^0 



and 



j; 



1 e-«* 3 gin 2fo? ^ = —t- e a \ e x2 dx. 

 Jo *"fl Jo 



The former result is well known ; and the latter is easily verified 

 by differentiating with respect to b and forming a differential 

 equation, from which the value of the integral cau be determined. 

 As another example, writing bi for b in (14), and noticing 

 that „ 



I e- e * 2 sin2&r^=0, 

 we find J -co 



dx 

 J -cc d i -\-x' i - 



' .oo « 2 + ^ 



^« A {e**Erf(aV'c + ^ + c- 2o * l "Erf^fl^/c--~M. (2 4) 



Erf(flVc+4r-)-Erf(«v'c) = -l v V* 8 «fo 



= --t-\ e-v~ 2ax dx=: e ~^[ e^- 2abt db, 

 ence 

 Erf(«v , c+-^) + Erf(«i/c-^ = 2Erf«^ 



a—2a 2 c r*b b* 



— 2 — 7— i e~c sin 2abdb, 

 Vc Jo 



Erf («•,+ -* )-Erf («•,- £)=»gjf 'eJcosS^, 



