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Arg
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
open allclose allBasic Examples (3)Summary of the most common use cases
The result is given in radians:

https://wolfram.com/xid/0bue8r-ojq9ry


https://wolfram.com/xid/0bue8r-cb5l6f

Plot over a subset of the reals:

https://wolfram.com/xid/0bue8r-jv2byw

Plot over a subset of the complexes:

https://wolfram.com/xid/0bue8r-p3ah

Scope (33)Survey of the scope of standard use cases
Numerical Evaluation (6)

https://wolfram.com/xid/0bue8r-l274ju


https://wolfram.com/xid/0bue8r-hfml09


https://wolfram.com/xid/0bue8r-orsjwl

For real inputs, the result is exact:

https://wolfram.com/xid/0bue8r-202p7j

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

https://wolfram.com/xid/0bue8r-6i01jt

Evaluate efficiently at high precision:

https://wolfram.com/xid/0bue8r-di5gcr


https://wolfram.com/xid/0bue8r-bq2c6r

Compute the elementwise values of an array using automatic threading:

https://wolfram.com/xid/0bue8r-thgd2

Or compute the matrix Arg function using MatrixFunction:

https://wolfram.com/xid/0bue8r-o5jpo

Arg can be used with Interval and CenteredInterval objects:

https://wolfram.com/xid/0bue8r-g2g4a2


https://wolfram.com/xid/0bue8r-k7pzcv

Or compute average-case statistical intervals using Around:

https://wolfram.com/xid/0bue8r-cw18bq

Specific Values (6)
Values of Arg at fixed points:

https://wolfram.com/xid/0bue8r-nww7l


https://wolfram.com/xid/0bue8r-bmqd0y


https://wolfram.com/xid/0bue8r-ukawnk


https://wolfram.com/xid/0bue8r-4qp834


https://wolfram.com/xid/0bue8r-rqw7vw


https://wolfram.com/xid/0bue8r-6swnom

Evaluate for complex exponentials:

https://wolfram.com/xid/0bue8r-wnk6lp

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

https://wolfram.com/xid/0bue8r-f2hrld


https://wolfram.com/xid/0bue8r-g9p0xi

Visualization (5)

https://wolfram.com/xid/0bue8r-ecj8m7


https://wolfram.com/xid/0bue8r-eq46t4


https://wolfram.com/xid/0bue8r-i8jjg7

Visualize Arg in three dimensions:

https://wolfram.com/xid/0bue8r-i75zi3

Use Arg to specify regions of the complex plane:

https://wolfram.com/xid/0bue8r-uh9qgx

Function Properties (11)
Arg is defined for all real and complex inputs:

https://wolfram.com/xid/0bue8r-cl7ele


https://wolfram.com/xid/0bue8r-c4ycek

Function range of Arg for real inputs:

https://wolfram.com/xid/0bue8r-evf2yr

Except on the negative reals, :

https://wolfram.com/xid/0bue8r-zgiad8

Arg is not a differentiable function:

https://wolfram.com/xid/0bue8r-fb9jdx

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

https://wolfram.com/xid/0bue8r-fqx7yy

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

https://wolfram.com/xid/0bue8r-yvnsee

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

https://wolfram.com/xid/0bue8r-m494ns

Arg is not an analytic function:

https://wolfram.com/xid/0bue8r-h5x4l2

It has both singularities and discontinuities:

https://wolfram.com/xid/0bue8r-mdtl3h


https://wolfram.com/xid/0bue8r-mn5jws

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

https://wolfram.com/xid/0bue8r-190em8


https://wolfram.com/xid/0bue8r-gcytbr

Arg is nonincreasing:

https://wolfram.com/xid/0bue8r-nlz7s

Arg is not injective:

https://wolfram.com/xid/0bue8r-poz8g


https://wolfram.com/xid/0bue8r-ctca0g

Arg is not surjective:

https://wolfram.com/xid/0bue8r-cxk3a6


https://wolfram.com/xid/0bue8r-frlnsr

Arg is non-negative:

https://wolfram.com/xid/0bue8r-84dui

Arg is neither convex nor concave:

https://wolfram.com/xid/0bue8r-8kku21

TraditionalForm formatting:

https://wolfram.com/xid/0bue8r-hnfc67

Function Identities and Simplifications (5)
Expand assuming real variables x and y:

https://wolfram.com/xid/0bue8r-wunchk

Simplify Abs using appropriate assumptions:

https://wolfram.com/xid/0bue8r-bg5s5j

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

https://wolfram.com/xid/0bue8r-2z0il8


https://wolfram.com/xid/0bue8r-zj3thl


https://wolfram.com/xid/0bue8r-gi0gfa

Applications (3)Sample problems that can be solved with this function
Polar decomposition of a complex number:

https://wolfram.com/xid/0bue8r

Color a plot according to value of Arg:

https://wolfram.com/xid/0bue8r

Expand multivalued functions without making assumptions about variables:

https://wolfram.com/xid/0bue8r


https://wolfram.com/xid/0bue8r

Properties & Relations (7)Properties of the function, and connections to other functions
Simplify expressions containing Arg:

https://wolfram.com/xid/0bue8r


https://wolfram.com/xid/0bue8r

Generate Arg from FullSimplify:

https://wolfram.com/xid/0bue8r-ewnips

Use Arg as a target function in ComplexExpand:

https://wolfram.com/xid/0bue8r


https://wolfram.com/xid/0bue8r

Rescale Arg to run from 0 to 1:

https://wolfram.com/xid/0bue8r-nyt


https://wolfram.com/xid/0bue8r-pzc

Find the domain of positivity for a linear function:

https://wolfram.com/xid/0bue8r

Use Arg to specify assumptions about complex variables:

https://wolfram.com/xid/0bue8r-j30xgz


https://wolfram.com/xid/0bue8r-jba5p7

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

https://wolfram.com/xid/0bue8r-bf0664


https://wolfram.com/xid/0bue8r-vzzqj4

Possible Issues (4)Common pitfalls and unexpected behavior
Degenerate cases give intervals as results:

https://wolfram.com/xid/0bue8r-b22t2v


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

https://wolfram.com/xid/0bue8r-ygmemv

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

https://wolfram.com/xid/0bue8r-ff0qfc

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

https://wolfram.com/xid/0bue8r-2pb7j7


https://wolfram.com/xid/0bue8r-xvhk4r

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

https://wolfram.com/xid/0bue8r-zvk3va

Numerical decision procedures with default settings cannot simplify this expression:

https://wolfram.com/xid/0bue8r-nkews


The machine-precision result is incorrect:

https://wolfram.com/xid/0bue8r-0oo87

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

https://wolfram.com/xid/0bue8r-vn5sc


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

https://wolfram.com/xid/0bue8r-evk5q3

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

https://wolfram.com/xid/0bue8r-cw6o9q

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

https://wolfram.com/xid/0bue8r

Neat Examples (1)Surprising or curious use cases

https://wolfram.com/xid/0bue8r

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).
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.
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
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
]}
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
]}