376 Prof. Carslaw on the Use of Contour 



in the case when r>r ! ; while when r<r', r and r' must be 

 interchanged. 



The method of this paper consists in obtaining a contour 

 integral for this expression, U (£R), and then adding to it 

 terms which introduce no singularity, and cause the boundary 

 conditions to he satisfied, 



Fig-. 1. 



7#£ /Tf A I Af/S '<W TH£ 7T'.- a 1 LA/V£ 



Path (C) w th£ -hi-pl/we 



Consider the integral 



J 



cosm(7r— 0+0) n x ' " . 



•m mir 



taken in the m-plane over the path (0) of fig. 1. 

 When m is large we have the approximate values 



>z V 1 



J "M-(f) film) 



= (£)" l 



\2/ V'2'irm ,e m 



log 



using the asymptotic value for II (m)*, so that we see that 

 J m (z) vanishes at infinity, when the real part of m is 

 positive. 



In the same way. using the relation, 



n(m)n(-m) = 



mir 



* Cf. Whittaker's 'Modem Analysis/ § 110. 



