606 Prof. H. S. Carslaw on Bromivicli s Method of 



But the integral over the dotted path vanishes in the limit. 

 Therefore v = 0, when t = 0. 



It follows that the value of v given by (5) satisfies all the 

 conditions of our problem. 



Fte/3: 



+oo 



The solution obtained in (5) as a contour integral is trans- 

 formed into an infinite series by Cauchy's Theorem, using 

 the path (Q) of fig. 2. 



For we have 



Ji 



sin ax e 



2iirj sin aa a 



tt ^ n 



doc, over the path (Q), 



. n7T 

 sin — #g 



a 



3. The same rod. The end x = kept at zero ; radiation into 

 a medium at v Q at the end x = a. The initial temperature 

 zero. 



Here we have to solve : — 



§7 =/ %" 0< - T<a - 



v =0, when .t = 0. 



*1 

 'dx 



+ h(v — v ) = 0, when :v = a. 



v=0, wlim < = 0. 



0) 



(;2.) 



(3) 

 (4j 



