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Residue[expr,{z,z0}]

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

Details and Options
Details and Options Details and Options
Examples  
Basic Examples  
Scope  
Applications  
Properties & Relations  
Possible Issues  
See Also
Tech Notes
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History
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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

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Basic Examples  (2)

Scope  (3)

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_2025_residue, author="Wolfram Research", title="{Residue}", year="1991", howpublished="\url{https://reference.wolfram.com/language/ref/Residue.html}", note=[Accessed: 04-August-2025]}

BibLaTeX

@online{reference.wolfram_2025_residue, organization={Wolfram Research}, title={Residue}, year={1991}, url={https://reference.wolfram.com/language/ref/Residue.html}, note=[Accessed: 04-August-2025]}