The Wolfram Language transparently works with complex variables throughout, not only numerically, but also symbolically—often 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

ResidueSum — sum the residues of a function over a region

FunctionPoles — find the poles of a meromorphic function

ContourIntegrate — compute a contour integral in the complex plane

NContourIntegrate — compute a numerical contour integral in the complex plane

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  ▪  ...