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Power Series Cheat Sheet

Radius of convergence

Let a power series be defined as:

\[f(z) = \sum_{n = 0}^{\infty} {c_n} (z - a)^n\]

For complex coefficients $c_n$, contant $a$ and variable $z$. The radius of convergence is a non-negative real $r$ or $\infty$ such that:

• If $\abs{z - a} > r$, the power series diverges
• If $\abs{z - a} < r$, the power series converges

The name radius of convergence alludes to the fact that $\abs{z - a} = r$ is a circle of radius $r$ in the complex plane, and that inside that circle the power series converges.

Laurent Series

A function can be decomposed into two, based on its Laurent series. The series containing the terms for $n < 0$ are called the singular (or principal) part of $f(z)$ and those for $n \ge 0$ are the analytic part.

For analytic functions (which are holomorphic) the singular part is 0. Conversely, if its singular part is non-zero, we know it contains a singularity.

Reference: Zeros and Poles.