# Arg Arg[z]

gives the argument of the complex number z.

# Details • Mathematical function, suitable for both symbolic and numerical manipulation.
• Arg[z] is left unevaluated if z is not a numeric quantity.
• Arg[z] gives the phase angle of z in radians.
• The result from Arg[z] is always between and .
• Arg[z] has a branch cut discontinuity in the complex z plane running from to 0.
• Arg gives 0.
• Arg can be used with Interval and CenteredInterval objects. »
• Arg automatically threads over lists.

# Examples

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## Basic Examples(3)

The result is given in radians:

Plot over a subset of the reals:

Plot over a subset of the complexes:

## Scope(33)

### 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:

Arg threads elementwise over lists and matrices:

Arg can be used with Interval and CenteredInterval objects:

### Specific Values(6)

Values of Arg at fixed points:

Value at zero:

Values at infinity:

Exact inputs:

Evaluate for complex exponentials:

Find a value of x for which the Arg[I x]=π/2:

Visualize the result:

### Visualization(5)

Plot the on the real axis:

Plot on the reals:

Plot over the complex plane:

Visualize Arg in three dimensions:

Use Arg to specify regions of the complex plane:

### Function Properties(11)

Arg is defined for all real and complex inputs:

Function range of Arg for real inputs:

Except on the negative reals, :

Arg 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 ComplexExpand to get differentiable expressions for real-valued variables:

Arg is not an analytic function:

It has both singularities and discontinuities:

Over the complex plane, it is singular everywhere and discontinuous on the non-positive reals:

Arg is nonincreasing:

Arg is not injective:

Arg is not surjective:

Arg is non-negative:

Arg is neither convex nor concave:

### Function Identities and Simplifications(5)

Expand assuming real variables x and y:

Simplify Abs using appropriate assumptions:

Express a non-zero complex number in term of its Arg and Abs: is equal to :

Except for , :

## Applications(3)

Polar decomposition of a complex number:

Color a plot according to value of Arg:

Expand multivalued functions without making assumptions about variables:

## Properties & Relations(7)

Simplify expressions containing Arg:

Generate Arg from FullSimplify:

Use Arg as a target function in ComplexExpand:

Rescale Arg to run from 0 to 1:

Find the domain of positivity for a linear function:

Use Arg to specify assumptions about complex variables:

ComplexPlot plots the phase of a function using color and shades by the magnitude:

## Possible Issues(4)

Degenerate cases give intervals as results: Arg is a function of a complex variable and is therefore not differentiable:

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

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

Use ComplexExpand to get differentiable expressions for real-valued variables:

Numerical decision procedures with default settings cannot simplify this expression: The machine-precision result is incorrect:

The arbitrary-precision result indicates that the result may be incorrect: Using a larger setting for \$MaxExtraPrecision gives the correct result:

The input contains a hidden zero, and simplifying the argument gets the correct answer:

The argument principle of complex analysis cannot be used because Arg has range :