May 2012
Intermediate to advanced
954 pages
36h 6m
English
Content preview from Engineering Mathematics, Volume II, Second Edition
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where f(n)(z0) represents nth derivative of f (z) at z0.
The coefficients 
are called Taylor's coefficients. The infinite series is convergent if | z – z0 | < δ, where δ is the distance from z0 to the nearest point of C. If δ1 < δ, then the series converges uniformly in the region | z – z0 | ≤ δ1.
Proof: Choose 
so that 0 < δ1 < δ2 < δ (Figure 3.12). Then, by the given hypothesis, f (z) is analytic within and on a circle G defined by | z – z0 | = δ2.
Figure 3.12
Let z0 + h be any point of the region defined by | z – z0 | ≤ δ1. Since z0
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