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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[0] gives 0.
  • Arg automatically threads over lists. »
  • Arg can be used with Interval and CenteredInterval objects. »

Examples

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Basic Examples  (3)Summary of the most common use cases

The result is given in radians:

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Plot over a subset of the reals:

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Plot over a subset of the complexes:

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Scope  (33)Survey of the scope of standard use cases

Numerical Evaluation  (6)

Evaluate numerically:

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Complex number inputs:

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Evaluate to high precision:

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For real inputs, the result is exact:

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For complex inputs, the precision of the output tracks the precision of the input:

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Evaluate efficiently at high precision:

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Compute the elementwise values of an array using automatic threading:

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Or compute the matrix Arg function using MatrixFunction:

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Arg can be used with Interval and CenteredInterval objects:

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Or compute average-case statistical intervals using Around:

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Specific Values  (6)

Values of Arg at fixed points:

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Value at zero:

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Values at infinity:

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Out[2]=2
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Exact inputs:

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Evaluate for complex exponentials:

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

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Visualize the result:

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Visualization  (5)

Plot the on the real axis:

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Plot on the reals:

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Plot over the complex plane:

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Visualize Arg in three dimensions:

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Use Arg to specify regions of the complex plane:

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Function Properties  (11)

Arg is defined for all real and complex inputs:

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Function range of Arg for real inputs:

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Except on the negative reals, arg(TemplateBox[{z}, Conjugate])=-arg(z):

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Arg is not a differentiable function:

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The difference quotient does not have a limit in the complex plane:

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There is only a limit in certain directions, for example, the real direction:

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

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Arg is not an analytic function:

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It has both singularities and discontinuities:

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

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Arg is nonincreasing:

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Arg is not injective:

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Arg is not surjective:

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Arg is non-negative:

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Arg is neither convex nor concave:

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TraditionalForm formatting:

Function Identities and Simplifications  (5)

Expand assuming real variables x and y:

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Simplify Abs using appropriate assumptions:

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

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is equal to :

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Except for , exp(ⅈ arg(z))=TemplateBox[{z}, Sign]):

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Applications  (3)Sample problems that can be solved with this function

Polar decomposition of a complex number:

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Color a plot according to value of Arg:

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Expand multivalued functions without making assumptions about variables:

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Properties & Relations  (7)Properties of the function, and connections to other functions

Simplify expressions containing Arg:

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Generate Arg from FullSimplify:

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Use Arg as a target function in ComplexExpand:

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Rescale Arg to run from 0 to 1:

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Find the domain of positivity for a linear function:

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Use Arg to specify assumptions about complex variables:

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

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Possible Issues  (4)Common pitfalls and unexpected behavior

Degenerate cases give intervals as results:

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

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

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

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

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Numerical decision procedures with default settings cannot simplify this expression:

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The machine-precision result is incorrect:

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The arbitrary-precision result indicates that the result may be incorrect:

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Using a larger setting for $MaxExtraPrecision gives the correct result:

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

Out[5]=5

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

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Neat Examples  (1)Surprising or curious use cases

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Wolfram Research (1988), Arg, Wolfram Language function, https://reference.wolfram.com/language/ref/Arg.html (updated 2021).
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Wolfram Research (1988), Arg, Wolfram Language function, https://reference.wolfram.com/language/ref/Arg.html (updated 2021).

Text

Wolfram Research (1988), Arg, Wolfram Language function, https://reference.wolfram.com/language/ref/Arg.html (updated 2021).

Copy to clipboard.
Wolfram Research (1988), Arg, Wolfram Language function, https://reference.wolfram.com/language/ref/Arg.html (updated 2021).

CMS

Wolfram Language. 1988. "Arg." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2021. https://reference.wolfram.com/language/ref/Arg.html.

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Wolfram Language. 1988. "Arg." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2021. https://reference.wolfram.com/language/ref/Arg.html.

APA

Wolfram Language. (1988). Arg. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Arg.html

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Wolfram Language. (1988). Arg. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Arg.html

BibTeX

@misc{reference.wolfram_2025_arg, author="Wolfram Research", title="{Arg}", year="2021", howpublished="\url{https://reference.wolfram.com/language/ref/Arg.html}", note=[Accessed: 16-March-2025 ]}

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@misc{reference.wolfram_2025_arg, author="Wolfram Research", title="{Arg}", year="2021", howpublished="\url{https://reference.wolfram.com/language/ref/Arg.html}", note=[Accessed: 16-March-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_arg, organization={Wolfram Research}, title={Arg}, year={2021}, url={https://reference.wolfram.com/language/ref/Arg.html}, note=[Accessed: 16-March-2025 ]}

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@online{reference.wolfram_2025_arg, organization={Wolfram Research}, title={Arg}, year={2021}, url={https://reference.wolfram.com/language/ref/Arg.html}, note=[Accessed: 16-March-2025 ]}