# Residue

Residue[expr,{z,z0}]

finds the residue of expr at the point z=z0.

# Details and Options

• The residue is defined as the coefficient of (z-z0)^-1 in the Laurent expansion of expr.
• The Wolfram Language can usually find residues at a point only when it can evaluate power series at that point.

# Examples

open allclose all

## Applications(1)

Illustrate Cauchy's theorem for the integral of a complex function:

## Properties & Relations(1)

Use FunctionPoles to find the poles of a function:

Use Residue to find the residues at the poles:

ResidueSum gives the sum of the residues at all poles:

## Possible Issues(1)

Residues are not defined at branch points:

Wolfram Research (1991), Residue, Wolfram Language function, https://reference.wolfram.com/language/ref/Residue.html.

#### Text

Wolfram Research (1991), Residue, Wolfram Language function, https://reference.wolfram.com/language/ref/Residue.html.

#### CMS

Wolfram Language. 1991. "Residue." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/Residue.html.

#### APA

Wolfram Language. (1991). Residue. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Residue.html

#### BibTeX

@misc{reference.wolfram_2024_residue, author="Wolfram Research", title="{Residue}", year="1991", howpublished="\url{https://reference.wolfram.com/language/ref/Residue.html}", note=[Accessed: 17-September-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_residue, organization={Wolfram Research}, title={Residue}, year={1991}, url={https://reference.wolfram.com/language/ref/Residue.html}, note=[Accessed: 17-September-2024 ]}