The Wolfram Language has fundamental support for both explicit complex numbers and symbolic complex variables. All applicable mathematical functions support arbitrary-precision evaluation for complex values of all parameters, and symbolic operations automatically treat complex variables with full generality.

x+I y — the complex number

I () — (entered as ii "imaginary ", or jj)

Complex — convert a pair of reals to a complex number

Re — real part

Im — imaginary part

ReIm — the list

Abs — absolute value

Arg — argument (phase angle in radians)

AbsArg — the list {,arg(z)}

Sign — normalized direction ()

Conjugate — complex conjugate (also entered with superscript co)

ConjugateTranspose — Hermitian conjugate of a matrix (also entered with ct)

ComplexExpand — expand symbolic expressions into real and imaginary parts

PowerExpand — expand symbolic expressions ignoring branch cuts

ExpToTrig, TrigToExp — convert between complex exponentials and trig functions

GaussianIntegers — option for polynomial and number theory functions

Reduce — reduce equations and inequalities over complex numbers

RandomComplex — random complex number

Related Tech Notes

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• Complex Numbers
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• Expressions Involving Complex Variables

Related Guides

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• Visualizing Complex Data and Functions
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• Functions of Complex Variables
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• Angles and Polar Coordinates
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• Quaternions Package
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• Algebraic Numbers
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• Complex Dynamics

Related Links

• Demonstrations related to Complex Numbers (Wolfram Demonstrations Project)