$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$

# 9: Residue Theorem

$$\newcommand{\vecs}{\overset { \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vecd}{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$

• 9.1: Poles and Zeros
• 9.2: Holomorphic and Meromorphic Functions
A function that is analytic on a region A is called holomorphic on A . A function that is analytic on A except for a set of poles of finite order is called meromorphic on A .
• 9.3: Behavior of functions near zeros and poles
A zero of order n , a function behaves like (z−z0)n and near a pole of order n , a function behaves like 1/(z−z0)n . The following make this a little more precise.
• 9.4: Residues
In this section we’ll explore calculating residues. We’ve seen enough already to know that this will be useful. We will see that even more clearly when we look at the residue theorem in the next section.
• 9.5: Cauchy Residue Theorem
The Cauchy's Residue theorem is one of the major theorems in complex analysis and will allow us to make systematic our previous somewhat ad hoc approach to computing integrals on contours that surround singularities.
• 9.6: Residue at ∞
The residue at ∞ is a clever device that can sometimes allow us to replace the computation of many residues with the computation of a single residue.

Thumbnail: Illustration of the setting. (Public Domain; Ben pcc via Wikipedia)

9: Residue Theorem is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Jeremy Orloff via source content that was edited to conform to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.