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 realvalued 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 TemplateBox[{x}, Abs]=2:

Substitute in the value of to create pairs:

Visualize the result:

Visualization  (5)

Plot TemplateBox[{{1, +,  , x}}, Abs] on the real axis:

Plot TemplateBox[{{1, +, {ⅈ,  , x}}}, Abs] 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:

Introduced in 1988
 (1.0)