Engineering Mathematics, Volume II, Second Edition

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