Residue
Residue[expr,{z,z0}]
finds the residue of expr at the point z=z0.
Examples
open all close allBasic Examples (2)
Residue[1 / z, {z, 0}]The residue of a product of Gamma functions:
Residue[Gamma[z]Gamma[z - 1]Gamma[z - 2], {z, 0}]An unknown function is assumed to be analytic at all points:
Residue[f[z] / z ^ 5, {z, 0}]Scope (3)
The function 
has no first-order pole at the origin, so the residue is zero:
Residue[1 / z ^ 2, {z, 0}]The residue of a meromorphic function with a pole at the origin:
Residue[1 / Sin[z] ^ 5, {z, 0}]It agrees with the coefficient of 
in the Laurent series expansion:
Series[1 / Sin[z] ^ 5, {z, 0, 0}]Find the residue the point 
of a transcendental function:
Residue[Zeta[z] / (z - 1) ^ 10, {z, 1}]Applications (1)
Properties & Relations (1)
Use FunctionPoles to find the poles of a function:
FunctionPoles[x ^ 3 / (x ^ 4 - 2), x]Use Residue to find the residues at the poles:
Residue[x ^ 3 / (x ^ 4 - 2), {x, #[[1]]}]& /@ %ResidueSum gives the sum of the residues at all poles:
ResidueSum[x ^ 3 / (x ^ 4 - 2), x]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