Functions of Complex Variables

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.

(OrangeLink[Complex Function Visualization », paclet:guide/ComplexVisualization])

(RefLink[ComplexPlot, paclet:ref/ComplexPlot]) — use color to plot the phase and magnitude of a function

(RefLink[ComplexPlot3D, paclet:ref/ComplexPlot3D]) — use color and height to plot the phase and magnitude of a function

Integration

(RefLink[Integrate, paclet:ref/Integrate]) — symbolic integrals taking account of complex variables

(RefLink[PrincipalValue, paclet:ref/PrincipalValue]) — option for specifying whether to take principal values

(RefLink[NIntegrate, paclet:ref/NIntegrate]) — numerical integration around contours in the complex plane

(RefLink[Residue, paclet:ref/Residue]) — residue at a pole

Symbolic Manipulation

(RefLink[ComplexExpand, paclet:ref/ComplexExpand]) — symbolically expand into real and imaginary parts

(RefLink[Reduce, paclet:ref/Reduce]) — reduce polynomial systems over complex variables

(OrangeLink[Basic Functions », paclet:guide/ComplexNumbers])