Functions of Complex Variables

The Wolfram Language transparently works with complex variables throughout, not only numerically, but also symbolicallyoften relying on original results to handle intricate branch cut and other issues.

Complex Function Visualization »

ComplexPlot use color to plot the phase and magnitude of a function

ComplexPlot3D use color and height to plot the phase and magnitude of a function

ReImPlot  ▪  AbsArgPlot  ▪  ...

Integration

Integrate symbolic integrals taking account of complex variables

PrincipalValue option for specifying whether to take principal values

NIntegrate numerical integration around contours in the complex plane

Residue residue at a pole

Symbolic Manipulation

ComplexExpand symbolically expand into real and imaginary parts

Reduce reduce polynomial systems over complex variables

Basic Functions »

Abs  ▪  Arg  ▪  Conjugate  ▪  Re  ▪  Im  ▪  ReIm  ▪  RandomComplex  ▪  ExpToTrig  ▪  ...