BOUNDARY VALUE OF INFINITE SERIES 193 



27rig=27rt7(x)- J f{z) 



x-aY+' dz i ^.Jz-aY'^'- dz 



z—x 



First, let x converge to a point on the outer boundary c\ in the manner 

 prescribed as in Taylor's series. When n = co the integral about d is 

 zero smce [s — a| isless than \x — a\. The integral about Cx is equal to 

 that integral taken around Si less the integral around (a). Around Si 

 it is zero, when n = co, since |a- — a| < U — a|, while around (a) it is 

 equal to 



^,±\^\i'P r ;/'i(z) / «-a \''+^ dz 

 n-\! J(a) z-x \z—a) z-a 



= 2« 1 + • . , . 



\ n+1/ a—x 



Hence when n = m and x on Ci the value of the series is 



f{x) — e^ (11) 



X — a. 



Second, let x converge in like manner to a point on the inner boundary 

 of convergence o>. Then the integral about Ci is zero, when n = ^, 



