164 Dr. H. S. Carslaw : A Problem 



circular arcs vanish in the limit, these complex integrals 

 may be replaced by integrals along the real axis of a. 

 The first two become 



if 00 i r 



s- 1 e ~ ka2t e~ ia ( x - x,) da+—- \ e~ ka2t e™^+z') da, 

 27rj-n 2ttJ_ 00 



whether x is greater or less than x\ 

 These may be replaced by 



1 r _ (x-x'y -i _(x+x'y->- 



— : [V ^ +e 

 "let |_ 



We have therefore only to show that the third integral of (7) 

 vanishes in the limit when t diminishes indefinitely. 

 This is best done by considering the complex integral 



!• 



s>ia(x+x') 



a + ili 



over the path (P) . 



This integral coriverges to its value for £ = 0, asf converges 

 to zero. When we take the value for f = 0, namely 



I 



gia(z + x') 



— .. da, 

 a + lli 



and consider the integral taken over the closed circuit of 

 fig. 3, since the integrand vanishes at infinity in the upper 



Fisr. 8. 



— 00 , — , +00 







The a-plane. 



part of the a-plane, and there are no poles inside the circuit, 

 we see that 



r p ia(x+x') 



da 



J a-j-ih 

 over the path (P) is zero. 



