Real numbers:

Complex numbers:

Plot over a subset of the reals:

Plot over a subset of the complexes:

Plot the real and imaginary parts of Sign over the complex plane:

Define Rademacher functions:

Plot (vertically shifted) Rademacher functions:

Check orthogonality over the unit interval:

Sign with simple arguments automatically evaluates to simpler form:

Sign is idempotent:

Use FullSimplify to simplify expressions involving Sign:

Simplify under additional assumptions:

Assume real‐valued variables:

Use Sign as a target function for ComplexExpand:

Use Sign in definite integration:

Integrate along a line in the complex plane, symbolically and numerically:

For complex values, the indefinite integral is path dependent:

The indefinite integral for real values:

Use in integral transforms:

Obtain Sign from integrals and limits:

Convert to Piecewise:

De‐nest:

Sign is a function of a complex variable and is therefore not differentiable:

As a complex function, it is not possible to write Sign[z] without involving Conjugate[z]:

In particular, the limit that defines the derivative is direction dependent and therefore does not exist:

Use RealSign, which assumes its argument is real, to obtain a differentiable version of Sign:

For purely real or imaginary approximate arguments, Sign returns exact answers:

For general complex arguments, Sign tracks the precision of the input:

Sign can stay unevaluated for numeric arguments:

Machine‐precision numerical evaluation of Sign can give wrong results:

Arbitrary‐precision evaluation gives the correct result:

A larger setting for $MaxExtraPrecision can be needed:

Sign applied to a matrix does not give the matrix sign function:

Form repeated convolution integrals starting with a symmetric product of three sign functions:

Approximate Sign through a generalized Fourier series:

Calculate rational approximations of Sign: