The absolute value and argument of a complex number:

Real numbers are a special kind of complex number:

AbsArg[list] gives a list of ordered pairs:

AbsArg accepts all number types:

AbsArg works with symbolic representations of numbers:

Purely symbolic expressions can be partially simplified:

AbsArg supports nested lists and ragged arrays:

AbsArg works with SparseArray and structured array objects:

AbsArg increases the depth of an array by 1 and adds a new inner dimension of length 2:

AbsArg[array] gives an array of {abs,arg} pairs:

This can be turned into a pair {Abs[array],Arg[array]} using Transpose:

ComplexExpand assumes variables to be real:

In general, variables are assumed to be complex, which may prevent simplification:

Use Simplify and FullSimplify to simplify the results of ReIm:

AbsArg converts complex numbers to their polar representation:

ToPolarCoordinates converts pairs of real numbers to their polar representation:

AbsArg can be viewed as the composition of ReIm and ToPolarCoordinates:

AbsArgPlot plots the magnitude of a function colored by the phase:

ComplexPlot plots the phase of a function using color and shades by the magnitude:

ComplexPlot3D plots the magnitude of a function as height and colors using the phase:

Substituting a list l for z in the output of AbsArg[z] is different from directly evaluating AbsArg[l]:

For any array, the two results are related by a transposition of the inner and outer levels: