BOUNDARY VALUE OF INFINITE SERIES 195 



f{z)^'{z) 



, ip{z) —<p{z) \<pix) — (p{a)/ z- 



VVhen we require x to converge to the outer boundary Ci by a regular 

 sequence thus prescribed, 



X 



c,' V fe) - 'Pix) \<p{z)-<p (a) f 



iz)-^(a)Y+' dz 



<p(x)-^(a) ^ /^ I k 

 <p{a) — <p (a) \ )l + l 



(p being the number in (4), then Avhen n is infinite it is easily seen, in the 

 same way as in §3, that the value of the series at the point 



x=<p ' I (/5 (a) + [ (js (a) - (P (a) ] e '9 

 on the outer boundaiy Ci is 



fix)-e^ ^f"\. ^'(a). ■ (13) 



(p{X) — <p[a) 



When x is on the inner boundary Co the sign of f is changed. When 

 f (x) has a number of simple or multiple poles on the boundary the corre- 

 sponding more general forms result as in the simpler series. 



Thus in all the forms of the power series the initial theorem appears to 

 be established. We proceed now to consider what is perhaps the most 

 interesting case, that of Fourier's series about the boundary conditions 

 of which much discussion has taken place. 



5. Fourier's Series. The n-sum in Fourier's series is given by tt »S2„ + i 

 equal to 



f -' . , „ , sin(2n + l)s , , f - .. „ ^ sin (2??. + 1)2 , ,. ,-, 



fix + 2z) — ^-^ dz + f(x- 2z) ^^ dz. (14) 



Jo sm z Jo smz 



We propose to investigate the limit of this sum when x converges to 

 + T in the manner defined by the regular sequence. 



^' (15) 



2n + l 



wherein c is an arbitrary positive constant for the present restricted to being 

 not greater than t. 



