Sign

Sign[x]

gives -1, 0, or 1 depending on whether x is negative, zero, or positive.

Details

Mathematical function, suitable for both symbolic and numerical manipulation.
For nonzero 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 can be used with Interval and CenteredInterval objects. »
Sign automatically threads over lists.

Examples

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Basic Examples (4)

Real numbers:

Complex numbers:

Plot over a subset of the reals:

Plot over a subset of the complexes:

Scope (32)

Numerical Evaluation (6)

Evaluate numerically:

Complex number inputs:

Evaluate to high precision:

For real inputs, the result is exact:

For complex inputs, the precision of the output tracks the precision of the input:

Evaluate efficiently at high precision:

Sign threads elementwise over lists and matrices:

Sign can be used with Interval and CenteredInterval objects:

Specific Values (5)

Values of Sign at fixed points:

Value at zero:

Values at infinity:

Evaluate symbolically:

Find a value of for which the :

Visualize the result:

Visualization (4)

Plot on the real axis:

Plot the real and imaginary parts of the function:

Visualize Sign in three dimensions:

Plot the real part of the function:

Plot the imaginary part of the function:

Function Properties (12)

Sign is defined for all real and complex inputs:

Function range of Sign for real inputs:

The range over the complex plane is the unit circle plus the origin:

Sign is an odd function:

Sign has mirror symmetry :

Sign is not a differentiable function:

The difference quotient does not have a limit in the complex plane:

There is only a limit in certain directions, for example, the real direction:

Use RealSign to obtain this real-differentiable result:

Sign is not an analytic function:

It has both singularities and discontinuities:

Over the complex plane, it is singular everywhere but still discontinuous only at the origin:

Sign is nonincreasing:

Sign is not injective:

Sign is not surjective:

Sign is neither non-negative nor non-positive:

Sign is neither convex nor concave:

TraditionalForm formatting:

Function Identities and Simplifications (5)

Expand assuming real variables x and y:

Simplify Sign using appropriate assumptions:

Express a complex number as a product of Sign and Abs:

is equal to :

for all non-zero :

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 (5)

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:

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:

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Sign

Details

Examples

Basic Examples (4)