BUILT-IN MATHEMATICA SYMBOL

Sign

Sign[x]
gives

,

, or

depending on whether x is negative, zero, or positive.

MORE INFORMATION

Mathematical function, suitable for both symbolic and numerical manipulation.

For non-zero complex numbers z, Sign[z] is defined as z/Abs[z].

Sign tries various transformations in trying to determine the sign of symbolic expressions.

For exact numeric quantities, Sign internally uses numerical approximations to establish its result. This process can be affected by the setting of the global variable $MaxExtraPrecision.

Sign automatically threads over lists.

EXAMPLES

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Basic Examples (3)

Sign threads element-wise over lists:

In[1]:=

Out[1]=

In[2]:=

Out[2]=

In[3]:=

Out[3]=

Sign threads element-wise over lists:

In[1]:=

Out[1]=

In[1]:=

Out[1]=

Scope (3)

Sign works with symbolic representations of numbers:

Sign gives "directions" of complex numbers:

The absolute value is always 1:

TraditionalForm formatting:

Generalizations & Extensions (4)

Sign returns exact answers for exact numerical arguments:

Infinite arguments give symbolic results:

Sign threads element-wise over sparse arrays:

Series expansions for real arguments:

Applications (2)

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:

Properties & Relations (10)

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:

Possible Issues (4)

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:

Neat Examples (3)

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: