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UNIVERSITY OF COLORADO STUDIES 



The development of /(z) =— — around the point a is therefore 



_E IX + !z£ + (£=5Y + (£z?Y+. . I . (6) 



i— z i— a|_ i— fl \i— a/ \i— a/ J 



This series is convergent when \z — a|<|i— a\ ; i.e., when z is 

 within a circle having a as a center and the distance from a to i as a radius, 

 Fig. 2. 



X 



Fig. 2 



2. Analytic continuation. 



From Fig. 2 it is plainly seen that the series (6) converges at points z 

 outside of the original circle of convergence ; in other words, the domain 



of convergence has been extended ; the junction —^ has been continued. 



Designating the expansion of (6) about the point a by }(z/a) we may 

 next take a point b within the circle of convergence of (6) and expand 

 }(z/a) about the point b; designating this last expansion by j(z/a/b) we 

 have 



(z-by 



}(z/a/b) =}(b/a) + (z-b)j'(b/a) + 



Xb/a)+...+^J^j"(b/a) + .. 



where 



(7) 



KW 



fl)= ^r I+ *- + (*-)vi ' ; 



1— a\_ 1— a \i—a/ J 1—0 



