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.
x+Iy — the complex number
x+
y
I (
) —
(entered as
Esc ii Esc "imaginary i", or
Esc jj Esc)
Complex — convert a pair of reals to a complex number
Arg — argument (phase angle in radians)
Sign — normalized direction (
z/
z
)
Conjugate — complex conjugate
z* (also entered with superscript
Esc co Esc)
ConjugateTranspose — Hermitian conjugate of a matrix (also entered with
Esc ct Esc)
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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