Residue Calculator
Calculate residues of complex functions at poles using numerical integration.
Complex Function f(z) = u + iv
Example: 1/z = x/(x^2+y^2) - i*y/(x^2+y^2)
Pole Location
Numerical Settings
Residue Formula
Res(f, z0) = (1/2 pi i) oint_C f(z) dz
Residue at z = 0 + 0i
1.0000 + 0.0000i
|Res| = 1.0000
ReRe(Residue)
1.000000
ImIm(Residue)
0.000000
|R|Magnitude
1.000000
argArgument
0.00 deg
Residue Theorem Application
Contour integral around pole:
oint f dz = 2 pi i * Res
= -0.0000 + 6.2832i
Common Residues
1/z at z=0: Res = 1
1/z^2 at z=0: Res = 0
e^z/z at z=0: Res = 1
About Residues
Definition
The residue of a function at an isolated singularity is the coefficient of 1/(z-z0) in the Laurent series. It determines the value of contour integrals around the singularity.
Applications
- Evaluating difficult real integrals
- Counting zeros and poles
- Inverse Laplace transforms
- Quantum field theory