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Argument (complex analysis)

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</style><div role="note" class="hatnote navigation-not-searchable">Not to be confused with <a href="Argument_of_a_function" title="Argument of a function">Argument of a function</a>.</div>

<div role="note" class="hatnote navigation-not-searchable">For other uses, see <a href="Argument_(disambiguation)" class="mw-disambig" title="Argument (disambiguation)">Argument (disambiguation)</a>.</div>

In <a href="Mathematics" title="Mathematics">mathematics</a> (particularly in <a href="Complex_analysis" title="Complex analysis">complex analysis</a>), the argument of a complex number z, denoted arg(z), is the <a href="Angle" title="Angle">angle</a> between the positive <a href="Real_number" title="Real number">real</a> <a href="Cartesian_coordinate_system" title="Cartesian coordinate system">axis</a> and the line joining the origin and z, represented as a point in the <a href="Complex_plane" title="Complex plane">complex plane</a>, shown as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }">

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<annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation>

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</math><img src="./33ee699558d09cf9d653f6351f9fda0b2f4aaa3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }" loading="lazy"> in Figure 1. By convention the positive real axis is drawn pointing rightward, the positive <a href="Imaginary_number" title="Imaginary number">imaginary</a> axis is drawn pointing upward, and complex numbers with positive real part are considered to have an <a href="Anticlockwise" class="mw-redirect" title="Anticlockwise">anticlockwise</a> argument with positive sign.

When any real-valued angle is considered, the argument is a <a href="Multivalued_function" title="Multivalued function">multivalued function</a> operating on the nonzero <a href="Complex_number" title="Complex number">complex numbers</a>. The <a href="Principal_value" title="Principal value">principal value</a> of this function is single-valued, typically chosen to be the unique value of the argument that lies within the interval (−π, π].<a href="#cite_note-:1-1">[1]</a><a href="#cite_note-2">[2]</a> In this article the multi-valued function will be denoted arg(z) and its principal value will be denoted Arg(z), but in some sources the capitalization of these symbols is exchanged.

In some older mathematical texts, the term "amplitude" was used interchangeably with argument to denote the angle of a complex number. This usage is seen in older references such as <a href="Lars_Ahlfors" title="Lars Ahlfors">Lars Ahlfors</a>' Complex Analysis: An introduction to the theory of analytic functions of one complex variable (1979), where amplitude referred to the argument of a complex number. While this term is largely outdated in modern texts, it still appears in some regional educational resources, where it is sometimes used in introductory-level textbooks.<a href="#cite_note-3">[3]</a>

<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>

An argument of the nonzero complex number z = x + iy, denoted arg(z), is defined in two equivalent ways:

<ol><li>Geometrically, in the <a href="Complex_plane" title="Complex plane">complex plane</a>, as the <a href="2D_polar_angle" class="mw-redirect" title="2D polar angle">2D polar angle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }">

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<annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation>

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</math><img src="./33ee699558d09cf9d653f6351f9fda0b2f4aaa3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }" loading="lazy"> from the positive real axis to the vector representing z. The numeric value is given by the angle in <a href="Radian" title="Radian">radians</a>, and is positive if measured counterclockwise.</li>

<li>Algebraically, as any real quantity <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }">

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</math><img src="./33ee699558d09cf9d653f6351f9fda0b2f4aaa3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }" loading="lazy"> such that <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=r(\cos \varphi +i\sin \varphi )=re^{i\varphi }}">

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<annotation encoding="application/x-tex">{\displaystyle z=r(\cos \varphi +i\sin \varphi )=re^{i\varphi }}</annotation>

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</math> for some positive real r (see <a href="Euler's_formula" title="Euler's formula">Euler's formula</a>). The quantity r is the <a href="Absolute_value" title="Absolute value">modulus</a> (or absolute value) of z, denoted |z|: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r={\sqrt {x^{2}+y^{2}}}.}">

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<annotation encoding="application/x-tex">{\displaystyle r={\sqrt {x^{2}+y^{2}}}.}</annotation>

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The argument of zero is usually left undefined. The names <a href="Magnitude_(mathematics)" title="Magnitude (mathematics)">magnitude</a>, for the modulus, and <a href="Phase_(waves)" title="Phase (waves)">phase</a>,<a href="#cite_note-phase-4">[4]</a><a href="#cite_note-:1-1">[1]</a> for the argument, are sometimes used equivalently.

Under both definitions, it can be seen that the argument of any non-zero complex number has many possible values: firstly, as a geometrical angle, it is clear that whole circle rotations do not change the point, so angles differing by an integer multiple of 2π <a href="Radian" title="Radian">radians</a> (a complete <a href="Turn_(angle)" title="Turn (angle)">turn</a>) are the same, as reflected by figure 2 on the right. Similarly, from the <a href="Periodic_function" title="Periodic function">periodicity</a> of <a href="Sine" class="mw-redirect" title="Sine">sin

</a> and <a href="Cosine" class="mw-redirect" title="Cosine">cos</a>, the second definition also has this property.

<div class="mw-heading mw-heading2"><h2 id="Principal_value">Principal value</h2></div>

Because a complete rotation around the origin leaves a complex number unchanged, there are many choices which could be made for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }">

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<annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation>

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</math><img src="./33ee699558d09cf9d653f6351f9fda0b2f4aaa3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }" loading="lazy"> by circling the origin any number of times. This is shown in figure 2, a representation of the <a href="Multi-valued" class="mw-redirect" title="Multi-valued">multi-valued</a> (set-valued) function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x,y)=\arg(x+iy)}">

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<annotation encoding="application/x-tex">{\displaystyle f(x,y)=\arg(x+iy)}</annotation>

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</math><img src="./6049f730c95e6590d293e888ac49a0ab74e8c063.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.879ex; height:2.843ex;" alt="{\displaystyle f(x,y)=\arg(x+iy)}" loading="lazy">, where a vertical line (not shown in the figure) cuts the surface at heights representing all the possible choices of angle for that point.

When a <a href="Well-defined" class="mw-redirect" title="Well-defined">well-defined</a> function is required, then the usual choice, known as the <a href="Principal_value" title="Principal value">principal value</a>, is the value in the open-closed <a href="Interval_(mathematics)" title="Interval (mathematics)">interval</a> (−π, π] radians, that is from −π to π <a href="Radian" title="Radian">radians</a> excluding −π radians itself (equiv., from −180 to +180 <a href="Degree_(angle)" title="Degree (angle)">degrees</a>, excluding −180° itself). This represents an angle of up to half a complete circle from the positive real axis in either direction.

Some authors define the range of the principal value as being in the closed-open interval [0, 2π).

<div class="mw-heading mw-heading3"><h3 id="Notation">Notation</h3></div>

The principal value sometimes has the initial letter capitalized, as in Arg z, especially when a general version of the argument is also being considered. Note that notation varies, so arg and Arg may be interchanged in different texts.

The set of all possible values of the argument can be written in terms of Arg as:

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<annotation encoding="application/x-tex">{\displaystyle \arg(z)=\{\operatorname {Arg} (z)+2\pi n\mid n\in \mathbb {Z} \}.}</annotation>

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</math><img src="./82d2b27912d8fadb28dfaa5d9861672948d071a5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.371ex; height:2.843ex;" alt="{\displaystyle \arg(z)=\{\operatorname {Arg} (z)+2\pi n\mid n\in \mathbb {Z} \}.}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Computing_from_the_real_and_imaginary_part">Computing from the real and imaginary part</h2></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Atan2" title="Atan2">atan2</a></div>

If a complex number is known in terms of its real and imaginary parts, then the function that calculates the principal value Arg is called the <a href="Atan2" title="Atan2">two-argument arctangent function, atan2</a>:

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {Arg} (x+iy)=\operatorname {atan2} (y,\,x)}</annotation>

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</math><img src="./615606c03da439a0de75ffcd3cbfc5aed549cb8c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.253ex; height:2.843ex;" alt="{\displaystyle \operatorname {Arg} (x+iy)=\operatorname {atan2} (y,\,x)}" loading="lazy">.</dd></dl>

The atan2 function is available in the math libraries of many programming languages, sometimes under a different name, and usually returns a value in the range (−π, π].<a href="#cite_note-:1-1">[1]</a>

In some sources the argument is defined as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Arg} (x+iy)=\arctan(y/x),}">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {Arg} (x+iy)=\arctan(y/x),}</annotation>

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</math><img src="./73b1002a1c281cbe01847170a09ffb33ffe61271.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.423ex; height:2.843ex;" alt="{\displaystyle \operatorname {Arg} (x+iy)=\arctan(y/x),}" loading="lazy"> however this is correct only when x > 0, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y/x}">

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<annotation encoding="application/x-tex">{\displaystyle y/x}</annotation>

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</math><img src="./c6ed5f4b5088118c3e87c080b87fd36b749d4902.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.648ex; height:2.843ex;" alt="{\displaystyle y/x}" loading="lazy"> is well-defined and the angle lies between <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -{\tfrac {\pi }{2}}}">

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<annotation encoding="application/x-tex">{\displaystyle -{\tfrac {\pi }{2}}}</annotation>

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</math><img src="./67a19acfb376df2a06f33d08ca28752fd43cb686.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:3.586ex; height:3.176ex;" alt="{\displaystyle -{\tfrac {\pi }{2}}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {\pi }{2}}.}">

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<annotation encoding="application/x-tex">{\displaystyle {\tfrac {\pi }{2}}.}</annotation>

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</math><img src="./4ab0b2f1f3a87ca54207a40e27b3c196b5a1b89f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:2.425ex; height:3.176ex;" alt="{\displaystyle {\tfrac {\pi }{2}}.}" loading="lazy"> Extending this definition to cases where x is not positive is relatively involved. Specifically, one may define the principal value of the argument separately on the half-plane x > 0 and the two quadrants with x < 0, and then patch the definitions together:

<dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Arg} (x+iy)=\operatorname {atan2} (y,\,x)={\begin{cases}\arctan \left({\frac {y}{x}}\right)&{\text{if }}x>0,\\[5mu]\arctan \left({\frac {y}{x}}\right)+\pi &{\text{if }}x<0{\text{ and }}y\geq 0,\\[5mu]\arctan \left({\frac {y}{x}}\right)-\pi &{\text{if }}x<0{\text{ and }}y<0,\\[5mu]+{\frac {\pi }{2}}&{\text{if }}x=0{\text{ and }}y>0,\\[5mu]-{\frac {\pi }{2}}&{\text{if }}x=0{\text{ and }}y<0,\\[5mu]{\text{undefined}}&{\text{if }}x=0{\text{ and }}y=0.\end{cases}}}">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {Arg} (x+iy)=\operatorname {atan2} (y,\,x)={\begin{cases}\arctan \left({\frac {y}{x}}\right)&{\text{if }}x>0,\\[5mu]\arctan \left({\frac {y}{x}}\right)+\pi &{\text{if }}x<0{\text{ and }}y\geq 0,\\[5mu]\arctan \left({\frac {y}{x}}\right)-\pi &{\text{if }}x<0{\text{ and }}y<0,\\[5mu]+{\frac {\pi }{2}}&{\text{if }}x=0{\text{ and }}y>0,\\[5mu]-{\frac {\pi }{2}}&{\text{if }}x=0{\text{ and }}y<0,\\[5mu]{\text{undefined}}&{\text{if }}x=0{\text{ and }}y=0.\end{cases}}}</annotation>

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</math><img src="./73bfbb35f671d62261506c3d87c7f98f36e0cfeb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -10.838ex; width:67.538ex; height:22.843ex;" alt="{\displaystyle \operatorname {Arg} (x+iy)=\operatorname {atan2} (y,\,x)={\begin{cases}\arctan \left({\frac {y}{x}}\right)&{\text{if }}x>0,\\[5mu]\arctan \left({\frac {y}{x}}\right)+\pi &{\text{if }}x<0{\text{ and }}y\geq 0,\\[5mu]\arctan \left({\frac {y}{x}}\right)-\pi &{\text{if }}x<0{\text{ and }}y<0,\\[5mu]+{\frac {\pi }{2}}&{\text{if }}x=0{\text{ and }}y>0,\\[5mu]-{\frac {\pi }{2}}&{\text{if }}x=0{\text{ and }}y<0,\\[5mu]{\text{undefined}}&{\text{if }}x=0{\text{ and }}y=0.\end{cases}}}" loading="lazy"></dd></dl>

See <a href="Atan2" title="Atan2">atan2</a> for further detail and alternative implementations.

<div class="mw-heading mw-heading2"><h2 id="Realizations_of_the_function_in_computer_languages">Realizations of the function in computer languages</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Wolfram_language_(Mathematica)">Wolfram language (Mathematica)</h3></div>

In Wolfram language, there's <code>Arg[z]</code>:<a href="#cite_note-5">[5]</a>

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<annotation encoding="application/x-tex">{\displaystyle ={\begin{cases}{\text{undefined}}&{\text{if }}|x|=\infty {\text{ and }}|y|=\infty ,\\[5mu]0&{\text{if }}x=0{\text{ and }}y=0,\\[5mu]0&{\text{if }}x=\infty ,\\[5mu]\pi &{\text{if }}x=-\infty ,\\[5mu]\pm {\frac {\pi }{2}}&{\text{if }}y=\pm \infty ,\\[5mu]\operatorname {Arg} (x+yi)&{\text{otherwise}}.\end{cases}}}</annotation>

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</math><img src="./3312c0b6fd0323590ac8284777895c42ca060d62.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -9.838ex; width:42.76ex; height:20.843ex;" alt="{\displaystyle ={\begin{cases}{\text{undefined}}&{\text{if }}|x|=\infty {\text{ and }}|y|=\infty ,\\[5mu]0&{\text{if }}x=0{\text{ and }}y=0,\\[5mu]0&{\text{if }}x=\infty ,\\[5mu]\pi &{\text{if }}x=-\infty ,\\[5mu]\pm {\frac {\pi }{2}}&{\text{if }}y=\pm \infty ,\\[5mu]\operatorname {Arg} (x+yi)&{\text{otherwise}}.\end{cases}}}" loading="lazy">

or using the language's <code>ArcTan</code>:

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<annotation encoding="application/x-tex">{\displaystyle ={\begin{cases}0&{\text{if }}x=0{\text{ and }}y=0,\\[5mu]{\text{ArcTan[x, y]}}&{\text{otherwise}}.\end{cases}}}</annotation>

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</math><img src="./417935b528c9e6995c670956eb27b01108161b0c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:38.569ex; height:6.509ex;" alt="{\displaystyle ={\begin{cases}0&{\text{if }}x=0{\text{ and }}y=0,\\[5mu]{\text{ArcTan[x, y]}}&{\text{otherwise}}.\end{cases}}}" loading="lazy">

<code>ArcTan[x, y]</code> is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {atan2} (y,x)}">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {atan2} (y,x)}</annotation>

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</math><img src="./cdf144c1f1dccdc6c0949ece514a77f3790530f7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.013ex; height:2.843ex;" alt="{\displaystyle \operatorname {atan2} (y,x)}" loading="lazy"> extended to work with infinities. <code>ArcTan[0, 0]</code> is <code>Indeterminate</code> (i.e. it's <a href="Indeterminate_form" title="Indeterminate form">still</a> defined), while <code>ArcTan[Infinity, -Infinity]</code> doesn't return anything (i.e. it's <a href="Undefined_(mathematics)" title="Undefined (mathematics)">undefined</a>).

<div class="mw-heading mw-heading3"><h3 id="Maple">Maple</h3></div>

<a href="Maple_(software)" title="Maple (software)">Maple</a>'s <code>argument(z)</code> behaves the same as <code>Arg[z]</code> in Wolfram language, except that <code>argument(z)</code> also returns <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi }">

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<annotation encoding="application/x-tex">{\displaystyle \pi }</annotation>

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</math><img src="./9be4ba0bb8df3af72e90a0535fabcc17431e540a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\displaystyle \pi }" loading="lazy"> if <code>z</code> is the special floating-point value <code>−0.</code>.<a href="#cite_note-6">[6]</a>

Also, Maple doesn't have <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {atan2} }">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {atan2} }</annotation>

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</math><img src="./474078eadf601ac09d580227d85df6f4e16e747f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.685ex; height:2.176ex;" alt="{\displaystyle \operatorname {atan2} }" loading="lazy">.

<div class="mw-heading mw-heading3"><h3 id="MATLAB">MATLAB</h3></div>

<a href="MATLAB" title="MATLAB">MATLAB</a>'s <code>angle(z)</code> behaves<a href="#cite_note-7">[7]</a><a href="#cite_note-8">[8]</a> the same as <code>Arg[z]</code> in Wolfram language, except that it is

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<annotation encoding="application/x-tex">{\displaystyle {\begin{cases}{\frac {1\pi }{4}}&{\text{if }}x=\infty {\text{ and }}y=\infty ,\\[5mu]-{\frac {1\pi }{4}}&{\text{if }}x=\infty {\text{ and }}y=-\infty ,\\[5mu]{\frac {3\pi }{4}}&{\text{if }}x=-\infty {\text{ and }}y=\infty ,\\[5mu]-{\frac {3\pi }{4}}&{\text{if }}x=-\infty {\text{ and }}y=-\infty .\end{cases}}}</annotation>

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</math><img src="./9a307c8b014a10f796f164859b966cebd0878694.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.671ex; width:33.989ex; height:16.509ex;" alt="{\displaystyle {\begin{cases}{\frac {1\pi }{4}}&{\text{if }}x=\infty {\text{ and }}y=\infty ,\\[5mu]-{\frac {1\pi }{4}}&{\text{if }}x=\infty {\text{ and }}y=-\infty ,\\[5mu]{\frac {3\pi }{4}}&{\text{if }}x=-\infty {\text{ and }}y=\infty ,\\[5mu]-{\frac {3\pi }{4}}&{\text{if }}x=-\infty {\text{ and }}y=-\infty .\end{cases}}}" loading="lazy">

Unlike in Maple and Wolfram language, MATLAB's <code>atan2(y, x)</code> is equivalent to <code>angle(x + y*1i)</code>. That is, <code>atan2(0, 0)</code> is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}">

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<annotation encoding="application/x-tex">{\displaystyle 0}</annotation>

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</math><img src="./2aae8864a3c1fec9585261791a809ddec1489950.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}" loading="lazy">.

<div class="mw-heading mw-heading2"><h2 id="Identities">Identities</h2></div>

One of the main motivations for defining the principal value Arg is to be able to write complex numbers in modulus-argument form. Hence for any complex number z,

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<annotation encoding="application/x-tex">{\displaystyle z=\left|z\right|e^{i\operatorname {Arg} z}.}</annotation>

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</math><img src="./487f2f489ffaa83e9545f23ff27eee6e58efd86a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.728ex; height:3.176ex;" alt="{\displaystyle z=\left|z\right|e^{i\operatorname {Arg} z}.}" loading="lazy"></dd></dl>

This is only really valid if z is non-zero, but can be considered valid for z = 0 if Arg(0) is considered as an <a href="Indeterminate_form" title="Indeterminate form">indeterminate form</a>—rather than as being undefined.

Some further identities follow. If z1 and z2 are two non-zero complex numbers, then

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\operatorname {Arg} (z_{1}z_{2})&\equiv \operatorname {Arg} (z_{1})+\operatorname {Arg} (z_{2}){\pmod {\mathbb {R} /2\pi \mathbb {Z} }},\\\operatorname {Arg} \left({\frac {z_{1}}{z_{2}}}\right)&\equiv \operatorname {Arg} (z_{1})-\operatorname {Arg} (z_{2}){\pmod {\mathbb {R} /2\pi \mathbb {Z} }}.\end{aligned}}}</annotation>

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</math><img src="./5869cd72017513b0bf5468b9008fc9275f9bb534.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:49.64ex; height:9.176ex;" alt="{\displaystyle {\begin{aligned}\operatorname {Arg} (z_{1}z_{2})&\equiv \operatorname {Arg} (z_{1})+\operatorname {Arg} (z_{2}){\pmod {\mathbb {R} /2\pi \mathbb {Z} }},\\\operatorname {Arg} \left({\frac {z_{1}}{z_{2}}}\right)&\equiv \operatorname {Arg} (z_{1})-\operatorname {Arg} (z_{2}){\pmod {\mathbb {R} /2\pi \mathbb {Z} }}.\end{aligned}}}" loading="lazy"></dd></dl>

If z ≠ 0 and n is any integer, then<a href="#cite_note-:1-1">[1]</a>

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {Arg} \left(z^{n}\right)\equiv n\operatorname {Arg} (z){\pmod {\mathbb {R} /2\pi \mathbb {Z} }}.}</annotation>

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</math><img src="./fffb4e908c318516e46321ea5e363d0e68bf34cd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.746ex; height:2.843ex;" alt="{\displaystyle \operatorname {Arg} \left(z^{n}\right)\equiv n\operatorname {Arg} (z){\pmod {\mathbb {R} /2\pi \mathbb {Z} }}.}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Example">Example</h3></div>

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {Arg} {\biggl (}{\frac {-1-i}{i}}{\biggr )}=\operatorname {Arg} (-1-i)-\operatorname {Arg} (i)=-{\frac {3\pi }{4}}-{\frac {\pi }{2}}=-{\frac {5\pi }{4}}}</annotation>

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</math><img src="./2e12244732fbc962ec086a87d93a51b3be724743.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:61.166ex; height:6.176ex;" alt="{\displaystyle \operatorname {Arg} {\biggl (}{\frac {-1-i}{i}}{\biggr )}=\operatorname {Arg} (-1-i)-\operatorname {Arg} (i)=-{\frac {3\pi }{4}}-{\frac {\pi }{2}}=-{\frac {5\pi }{4}}}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Using_the_complex_logarithm">Using the complex logarithm</h3></div>

From <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=|z|e^{i\operatorname {Arg} (z)}}">

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<annotation encoding="application/x-tex">{\displaystyle z=|z|e^{i\operatorname {Arg} (z)}}</annotation>

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</math><img src="./bf533ccaaf75e063203aea2642b8f179ca188584.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.587ex; height:3.343ex;" alt="{\displaystyle z=|z|e^{i\operatorname {Arg} (z)}}" loading="lazy">, we get <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i\operatorname {Arg} (z)=\ln {\frac {z}{|z|}}}">

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<annotation encoding="application/x-tex">{\displaystyle i\operatorname {Arg} (z)=\ln {\frac {z}{|z|}}}</annotation>

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</math><img src="./85197ca04913e8f76e8cc284d882c0e0c44870ca.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:16.547ex; height:5.509ex;" alt="{\displaystyle i\operatorname {Arg} (z)=\ln {\frac {z}{|z|}}}" loading="lazy">, alternatively <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Arg} (z)=\operatorname {Im} (\ln {\frac {z}{|z|}})=\operatorname {Im} (\ln z)}">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {Arg} (z)=\operatorname {Im} (\ln {\frac {z}{|z|}})=\operatorname {Im} (\ln z)}</annotation>

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</math><img src="./1404c3f57d4066485ee01a55d89fb5f7fc6820e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:31.04ex; height:5.509ex;" alt="{\displaystyle \operatorname {Arg} (z)=\operatorname {Im} (\ln {\frac {z}{|z|}})=\operatorname {Im} (\ln z)}" loading="lazy">. As we are taking the imaginary part, any normalisation by a real scalar will not affect the result. This is useful when one has the <a href="Complex_logarithm" title="Complex logarithm">complex logarithm</a> available.

<div class="mw-heading mw-heading2"><h2 id="Extended_argument">Extended argument</h2></div>

The extended argument of a number z (denoted as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {\arg }}(z)}">

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<annotation encoding="application/x-tex">{\displaystyle {\overline {\arg }}(z)}</annotation>

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</math><img src="./aea67ef662802c5b14aab3072336d2fce4bae769.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.249ex; height:2.843ex;" alt="{\displaystyle {\overline {\arg }}(z)}" loading="lazy">) is the set of all real numbers congruent to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \arg(z)}">

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<annotation encoding="application/x-tex">{\displaystyle \arg(z)}</annotation>

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</math><img src="./5d8bb9d3cf9c1a82f1666e4e72b6702b0cee4b21.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.134ex; height:2.843ex;" alt="{\displaystyle \arg(z)}" loading="lazy"> modulo 2<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi }">

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<annotation encoding="application/x-tex">{\displaystyle \pi }</annotation>

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</math><img src="./9be4ba0bb8df3af72e90a0535fabcc17431e540a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\displaystyle \pi }" loading="lazy">.<a href="#cite_note-9">[9]</a><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {\arg }}(z)=\arg(z)+2k\pi ,\forall k\in \mathbb {Z} }">

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<annotation encoding="application/x-tex">{\displaystyle {\overline {\arg }}(z)=\arg(z)+2k\pi ,\forall k\in \mathbb {Z} }</annotation>

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<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("./mw/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("./mw/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("./mw/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("./mw/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}

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</style><cite id="CITEREFWeisstein" class="citation web cs1">Weisstein, Eric W. <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/ComplexArgument.html">"Complex Argument"</a>. mathworld.wolfram.com. Retrieved 2020-08-31.</cite>

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<li id="cite_note-2"><a href="#cite_ref-2">^</a> <cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://mas-coursebuild.ncl.ac.uk/lti/content/ASK/default/core_mathematics/pure_maths/algebra/complex_numbers/modulus_and_argument/index.html">"Modulus and Argument"</a>. mas-coursebuild.ncl.ac.uk. <a href="Newcastle_University" title="Newcastle University">Newcastle University</a>. Retrieved 2025-01-05.</cite>

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<li id="cite_note-3"><a href="#cite_ref-3">^</a> <cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://byjus.com/jee/complex-numbers/#modulus-and-argument-of-a-complex-number">"Modulus and Argument of a Complex Number"</a>. Byju's. Retrieved 18 January 2025.</cite>

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<li id="cite_note-phase-4"><a href="#cite_ref-phase_4-0">^</a> Dictionary of Mathematics (2002). phase.

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<li id="cite_note-5"><a href="#cite_ref-5">^</a> <cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://reference.wolfram.com/language/ref/Arg.html">"Arg"</a>. Wolfram Language Documentation. Retrieved 2024-08-30.</cite>

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<li id="cite_note-6"><a href="#cite_ref-6">^</a> <cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.maplesoft.com/support/help/Maple/view.aspx?path=argument">"Argument - Maple Help"</a>.</cite>

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<li id="cite_note-7"><a href="#cite_ref-7">^</a> <cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.mathworks.com/help/matlab/ref/angle.html">"Phase angle - MATLAB angle"</a>.</cite>

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<li id="cite_note-8"><a href="#cite_ref-8">^</a> <cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.mathworks.com/help/matlab/ref/atan2.html">"Four-quadrant inverse tangent - MATLAB atan2"</a>.</cite>

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<li id="cite_note-9"><a href="#cite_ref-9">^</a> <cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.cut-the-knot.org/arithmetic/algebra/ComplexNumbers.shtml">"Algebraic Structure of Complex Numbers"</a>. www.cut-the-knot.org. Retrieved 2021-08-29.</cite>

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<div class="mw-heading mw-heading2"><h2 id="Bibliography">Bibliography</h2></div>

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<ul><li><cite id="CITEREFAhlfors1979" class="citation book cs1">Ahlfors, Lars (1979). Complex Analysis: An Introduction to the Theory of Analytic Functions of One Complex Variable (3rd ed.). New York;London: McGraw-Hill. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-07-000657-1</bdi>.</cite></li>

<li><cite id="CITEREFPonnuswamy2005" class="citation book cs1">Ponnuswamy, S. (2005). Foundations of Complex Analysis (2nd ed.). New Delhi;Mumbai: Narosa. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-81-7319-629-4</bdi>.</cite></li>

<li><cite id="CITEREFBeardon1979" class="citation book cs1">Beardon, Alan (1979). Complex Analysis: The Argument Principle in Analysis and Topology. Chichester: Wiley. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-471-99671-8</bdi>.</cite></li>

<li><cite id="CITEREFBorowskiBorwein2002" class="citation book cs1">Borowski, Ephraim; Borwein, Jonathan (2002) [1st ed. 1989 as Dictionary of Mathematics]. Mathematics. Collins Dictionary (2nd ed.). Glasgow: <a href="HarperCollins" title="HarperCollins">HarperCollins</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-00-710295-X</bdi>.</cite></li></ul>

</div>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php/Argument">Argument</a> at <a href="Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a>.</li></ul></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2025-08-03" href="https://en.wikipedia.org/wiki/?title=Argument_(complex_analysis)&oldid=1304064700">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

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