Sign

Notations

Traditional name

Traditional notation

Mathematica StandardForm notation

Primary definition

Specific values

Specialized values

Values at fixed points

Values at infinities

General characteristics

Domain and analyticity

sgn(z) is a nonanalytical function. The real and the imaginary parts of sign(z) are real-analytic functions of the variable z.

Symmetries and periodicities

Parity

sgn(z) is an odd function.

Mirror symmetry

Periodicity

No periodicity

Homogeneity

Scale symmetry

Sets of discontinuity

The function  sgn(z) has discontinuity at point z==0.

Series representations

Residue representations

Other series representations

Limit representations

Integral representations

Contour integral representations

Transformations

Transformations and argument simplifications

Argument involving basic arithmetic operations

Addition formulas

Multiple arguments

Ratio of arguments

Power of arguments

Exponent of arguments

Complex characteristics

Real part

Imaginary part

Absolute value

Argument

Conjugate value

Signum value

Differentiation

Low-order differentiation

In a distributional sense for x∈ℝ .

Integration

Indefinite integration

Involving only one direct function

In a distributional sense for x∈ℝ .

Definite integration

For the direct function itself

Integral transforms

Fourier exp transforms

Fourier cos transforms

Fourier sin transforms

Laplace transforms

Representations through more general functions

Through Meijer G

Classical cases involving the direct function

Classical cases involving Cosh

Classical cases involving Sinh

Representations through equivalent functions

With related functions

With Re

With Im

With Abs

With Arg

With Conjugate

With UnitStep

Inequalities

Zeros

Theorems

Rademacher functions

The functions (Subscript[r, n])(x)=sgn(sin((2^n)π x))  form an orthogonal sequence over (0,1).

History

The function sgn is encountered often in mathematics and the natural sciences.

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