Abs

Notations

Traditional name

Traditional notation

Mathematica StandardForm notation

Primary definition

|z| is the absolute value of z. The absolute value (or modulus) of a complex number z is the Euclidean distance from z to the origin.

Specific values

Specialized values

Values at fixed points

Values at infinities

General characteristics

Domain and analyticity

|z| is nonanalytical function; it is a real-analytic function of the variable z for z≠0.

Symmetries and periodicities

Parity

|z| is an even function.

Mirror symmetry

Periodicity

No periodicity

Homogeneity

Scale symmetry

Sets of discontinuity

The function |z| is continuous function in ℂ.

Series representations

Other series representations

Limit representations

Differential equations

Ordinary linear differential equations and wronskians

In a distributional sense:

Transformations

Transformations and argument simplifications

Argument involving basic arithmetic operations

Addition formulas

Multiple arguments

Power of arguments

Exponent of arguments

Products, sums, and powers of the direct function

Products of the direct function

Powers of the direct function

Sums of powers of the direct function

Complex characteristics

Real part

Imaginary part

Absolute value

Argument

Conjugate value

Signum value

Differentiation

Low-order differentiation

In a distributional sense, for x∈ℝ :

Fractional integro-differentiation

Integration

Indefinite integration

Involving only one direct function

For x∈ℝ:

Definite integration

For the direct function itself

Involving the direct function

Contour integration

Integral transforms

Fourier exp transforms

Inverse Fourier exp transforms

Fourier cos transforms

Fourier sin transforms

Laplace transforms

Representations through more general functions

Through Meijer G

Classical cases involving Cosh

Classical cases involving Sinh

Generalized cases for powers of Abs

Representations through equivalent functions

With related functions

With Re

With Im

With Arg

With Conjugate

With Sign

Inequalities

Zeros

History

– J. R. Argand (1806, 1814) introduced the word "module" for absolute value
– K. Weierstrass (1841) introduced the notation |x|

Abs is encountered in mathematics and the natural sciences.

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