ABSTRACT
Taylor series on subsets of its neighborhood of convergence, when a function f has a convergent Laurent series expansion on an annulus ARr (z0), it will
converge absolutely on ARr (z0) and uniformly on A R1 r1 (z0) where A
any proper subannulus of ARr (z0). To establish this, we must consider the convergence of a given Laurent series a bit more carefully. Since the analytic part of the Laurent series is a power series, it will converge absolutely on its neighborhood of convergenceNR(z0). Recall that the convergence is uniform on NR1 (z0) for any 0 < R1 < R. To analyze the singular part, we define
r ≡ inf
{ |z − z0| :
(z − z0)j converges
} .
