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

 Putting #=0 in (28), we find 



which on integration between limits gives 



~ + (Erf b-m a) + | (Erf 2b -Erf 2 a) + . . . = ^log£ 



^'W-S)-K-?> E (-¥)-K-">-}| 



and many similar formulae could, no doubt, be obtained. 



It is a matter of some importance in regard to. the use of the 

 function to be enabled to replace it, when the argument is ima- 

 ginary, by a real integral; this may, of course, be done in many 

 ways, but probably the most convenient forms will be found to 

 be those deduced from the integral 



I 



e r°*-***dx = _t_ e a Erf -4". 



ya y a 



Writing in this integral a (cos a. + i sin a) and b (cos /5 + i sin /3) 

 for a and b respectively, there results 



e -a*2co S a-2tocos/}| cos ^ aa A ^ u _j_ 2 0X S [ n £) 



— 2 sin (fffl? 8 sin a + 2Zw sin /3} <&? 



=^- cosw - , h{r in ^-«)-l a } 



+ i sin {J sin (2/3-«) - » «}] Erf ±{cos (#- *-.) 



+y.Bnf/8- !«)}.. (38) 



Numerous special forms are deducible for Erf (A + Bt) by 

 giving particular or zero values to a, b 3 a, and /3 ; « cos ce, how- 

 ever, must not be made negative. 



Two of the most simple forms are given at the end of the 

 previous paper; from them we see that, to form a Table of the 

 error-function for arguments of the form a + bi } it would be ne- 

 cessary to tabulate 



r»b r*b 



e -a* i e x* s i n ^ax dx and e~" 2 1 e x2 cos 2ax doc, 

 Jo Jo 



or 



/*oo pea 



e - 1 e~ x2 sin 2bx dx and e* 2 1 e~ x * cos 2bx dx. 

 J« - J«2 



