This is documentation for Mathematica 8, which was
based on an earlier version of the Wolfram Language.
View current documentation (Version 11.2)
Functions of Complex Variables
Mathematica 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.
Visualization
ContourPlot, DensityPlot plot over the complex plane, potentially coloring by phase
ParametricPlot plot conformal mappings
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
    
Abs  ▪ Arg  ▪ Conjugate  ▪ Re  ▪ Im  ▪ RandomComplex  ▪ ExpToTrig  ▪ ...
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