Abs

Abs[z]

gives the absolute value of the real or complex number z.

Details

Abs is also known as modulus.
Mathematical function, suitable for both symbolic and numerical manipulation.
For complex numbers z, Abs[z] gives the modulus .
Abs[z] is left unevaluated if z is not a numeric quantity.
Abs 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 (28)

Numerical Evaluation (6)

Evaluate numerically:

Complex number input:

Evaluate to high precision:

The precision of the output tracks the precision of the input:

Evaluate efficiently at high precision:

Abs threads elementwise over lists and matrices:

Abs can deal with real‐valued intervals:

Specific Values (6)

Values of Abs at fixed points:

Value at zero:

Values at infinity:

Evaluate symbolically:

Exact inputs:

Find real values of for which :

Substitute in the value of to create pairs:

Visualize the result:

Visualization (5)

Plot on the real axis:

Plot Abs in the complex plane:

Visualize Abs in three dimensions:

Use Abs to specify regions of the complex plane:

Function Properties (5)

Abs is defined for all real and complex inputs:

The range of Abs is the non-negative reals:

This is true even in the complex plane:

Abs is an even function:

Abs 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:

This result, restricted to real inputs, is the derivative of RealAbs:

TraditionalForm formatting:

Function Identities and Simplifications (6)

Expand assuming real variables x and y:

Simplify Abs using appropriate assumptions:

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

Express in terms of real and imaginary parts:

Abs commutes with real exponentiation:

This result is applied automatically for concrete powers:

Find the absolute value of a Root expression:

Applications (2)

Plot Abs over the complex plane:

Color plots according to Abs:

Properties & Relations (16)

Abs is idempotent:

Abs is defined for all complex numbers:

RealAbs is defined only for real numbers:

Simplify expressions containing Abs:

Simplification of some identities involving Abs may require explicit assumptions that variables are real:

The assumptions may not be needed if RealAbs is used instead:

Abs is not a differentiable function:

RealAbs is differentiable:

Use Abs as a target function in ComplexExpand:

Solve an equation involving Abs:

Prove an inequality containing Abs:

Definite integration:

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

Interpret as the indefinite integral for real arguments:

Integral transforms:

Obtain Abs from Limit:

Convert into Piecewise:

Denest:

ComplexPlot3D plots the magnitude of a function as height and colors using the phase:

Possible Issues (3)

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

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

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

Adding assumptions that the argument is real makes Abs differentiable:

Alternatively, use RealAbs, which assumes its argument is real:

Abs can stay unevaluated for some complicated numeric arguments:

No series can be formed from Abs for complex arguments:

For real arguments, a series can be found:

Neat Examples (2)

Form nested functions involving Abs:

Plot Abs at Gaussian integers:

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Abs

Details

Examples

Basic Examples (4)