192 UNIVERSITY OF VIRGINIA. PUBLICATIONS 



1 + n+l 



J z—x\z — al 2 — a 



2«(i+^r^^--^- (8) 



\ n+1/ oc—x 



.V 



Hence at the point x = a-\- (a — a)e''9 on the circle c the value of the 

 Taylor's series is 



, /(^)_^e^+»^ (9) 



X — a 



Since k and 6 are arbitrary this can be made to take any assigned value 

 N. If f{z) has 11 simple poles ai, . . . , am on c, then the value of the 

 series is 



■'^ x~ ar 



For each multiple pole i3 of order m + 1 on c there will be a term under 

 the S equal to 



1 / d V" ^r (/3) 



where in the neighborhood of /3, i/'r(2) = (3 — /3)'""'"V (2) is regular. If in 

 (9) x moves along the circle c to a the value (9) will be co unless k = Q, 

 B = Q, and then has the value ip'{a). 



3. Laurent's Series. Let a and be two isolated simple poles of f{z) 

 and a the center of circles ci and C2 passing through a and /S respectively. 

 The point a being a common center, let Si be a circle outside a and s^ a 

 circle inside (3 such that / (z) is regular on the boundaries si and s^ and 

 through out the ring between them. With a as a center draw a circle C2 

 just outside ^ and another Cijust inside a. Draw a small circle (a) 

 around a and connect it by a double line path to a point on si, also a small 

 circle (/3) around /S and connect it by such a path to a point on S2. Let x 

 be a point in the ring between Cj and C2. Taking the integral 



J- 



/(2)^ (10) 



z—x 



around Ci and C2 in the usual way we have the ?i-sum of Laurent's series 



