Complex Numbers

Mathematica 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.

ReferenceReference

x+Iy the complex number

I () — (entered as EsciiEsc "imaginary ", or EscjjEsc)

Complex convert a pair of reals to a complex number

Re real part

Im imaginary part

Abs absolute value

Arg argument (phase angle in radians)

Sign normalized direction ()

Conjugate complex conjugate (also entered with superscript EsccoEsc)

ConjugateTranspose Hermitian conjugate of a matrix (also entered with EscctEsc)

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

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