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MathidiaMathematics Academy
Complex analysis
Lesson 6 of 75
Introduction

Analytic power series

Objective. Relate holomorphic functions and local power series.

  1. 1. Learn
  2. 2. Worked example
  3. 3. Practice
  4. 4. Feedback
  5. 5. Continue

Learn: the key idea

An analytic function on a domain D is one that equals its own Taylor series in a neighbourhood of every point, and this property is equivalent to complex-differentiability (holomorphy) throughout D. If f is holomorphic on a disc of radius R about z₀, then f(z)=Σaₙ(z−z₀)ⁿ converges for |z−z₀|<R, with coefficients aₙ=f⁽ⁿ⁾(z₀)/n! given by Cauchy's integral formula. This local power series representation means holomorphic functions are infinitely differentiable and determined entirely by their values near one point via analytic continuation. For example, f(z)=1/(1−z) is holomorphic on |z|<1 with power series Σzⁿ, matching its Taylor expansion exactly.

Worked example

Find the radius of convergence of the power series expansion of f(z)=1/(z²+4) about z₀=0.

Diagram of a circle
  1. 1f has singularities where z²+4=0, i.e. z=±2i
  2. 2Distance from z₀=0 to nearest singularity ±2i is 2
  3. 3Radius of convergence equals distance to nearest singularity
  4. 4Radius of convergence = 2

Answer: 2

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Common mistake. Students think real-differentiable functions always have convergent power series like holomorphic ones, but e^{-1/x²} (extended by 0 at x=0) is infinitely real-differentiable yet its Taylor series at 0 is identically zero, not equal to the function; this failure cannot happen for genuinely holomorphic functions.

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