Complex Power Series

Definition

A complex power series is an infinite series of the form $\sum_{n=0}^\infty a_n(z - z_0)^n$. In complex analysis, a function is analytic in a region if and only if it can be represented by a power series locally.

• How to read: “The sum from n equals zero to infinity of a n times the difference z minus z zero raised to the power of n.”
• Meaning: A Taylor-like expansion centered at $z_0$; converges inside a disk where the terms decay fast enough.

Why It Matters

They prove that local information at a single point can determine a function’s behavior across the entire complex plane.

Core Concepts

• Radius of Convergence ($R$): Every complex power series has a disk of convergence $|z - z_0| < R$. Inside this disk, the series converges absolutely and uniformly.
  • How to read: “The absolute value of the difference z minus z zero is less than R.”
  • Meaning: The series converges for all $z$ within distance $R$ of the center; diverges outside.
• Analyticity: If $f(z)$ is analytic at $z_0$, it possesses a unique Taylor series expansion: $f(z) = \sum_{n=0}^\infty \frac{f^{(n)}(z_0)}{n!}(z - z_0)^n$
  • How to read: “The function f of z equals the sum from n equals zero to infinity of the n-th derivative of f at z zero divided by n factorial, multiplied by the difference z minus z zero raised to the power of n.”
  • Meaning: All derivatives at one point determine the function throughout the convergence disk.
• Rigidity: Unlike real functions, complex differentiability once implies differentiability infinitely many times. The existence of the first derivative $f'(z)$ guarantees the existence of a power series representation.
• Singularities: The radius of convergence $R$ is exactly the distance from $z_0$ to the nearest point where $f(z)$ fails to be analytic.

Connected Concepts

• Power-Series Solutions
• Power Series
• Complex Numbers: Operations