194 UNIVERSITY OF VIRGINIA PUBLICATIONS 



since | .r — a | is less than \z — a\; while the integral about d is equal to 

 this integral around S2 less the integral around /3. The integral around s^ 

 is zero, when n is infinite, since l2 — a| < |a; — aj, and the integral around 

 ((3) can be written equal to 



n+lj J(i3)Z-x\P-a/ 3-/3 



\ 71+1/ .P-X 



Hence the value of the series when n = <^ and a; is a point on the i ner 

 boundary of convergence is 



/(^)_/^. (12) 



x — p 



If there be a number of simple or multiple poles on the boundary terms 

 similar to those in Taylor's series appear. 



4. Teixeira's Series. Passing over the particular cases, Burmann's, 

 Laplace's, Lagrange's, etc., we proceed at once to point out the slight 

 changes in Laurent's series that are necessary in order to establish the 

 results for the most general form of power series. Let 



S = 2) ^r['P(x)-<p{a)Y 



be the w-term sum in Teixeira's series. The concentric circles in Fig. 2. 

 now become the parametric contour lines 



\^{x)-<p{a)\=\, 



the different values of the parameter X corresponding to the radii in the 

 Laurent circles. With these changes we use the same symbols for the 

 respective contours as for the circles respectively. The ?i-sum is derived 

 from the integral 



" . , z-x dz 



J' 



<p{z) —<p(x) z—x 



taken around Ci and Co, and 2iri S^ is equal to fix) less the sum of the 

 two integrals 



