Inmathematics (particularly incomplex analysis), theargument of a complex numberz, denotedarg(z), is theangle between the positiverealaxis and the line joining the origin andz, represented as a point in thecomplex plane, shown as${\displaystyle \phi }$ in Figure 1. By convention thepositive real axis is drawn pointing rightward, the positiveimaginary axis is drawn pointing upward, and complex numbers with positive real part are considered to have ananticlockwise argument with positive sign.

When any real-valued angle is considered, the argument is amultivalued function operating on the nonzerocomplex numbers. Theprincipal value of this function is single-valued, typically chosen to be the unique value of the argument that lies within the interval(−π,π].^[1][2] In this article the multi-valued function will be denotedarg(z) and its principal value will be denotedArg(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 asLars Ahlfors'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 While this term is largely outdated in modern texts, it still appears in some regional educational resources, where it is sometimes used in introductory-level textbooks.^[3]

Anargument of the nonzero complex numberz =x +iy, denotedarg(z), is defined in two equivalent ways:

1. Geometrically, in the complex plane, as the2D polar angle${\displaystyle \phi }$ from the positive real axis to the vector representingz. The numeric value is given by the angle inradians, and is positive if measured counterclockwise.
2. Algebraically, as any real quantity${\displaystyle \phi }$ such that

$${\displaystyle z=r(\cos \phi +i\sin \phi )=r{e}^{i\phi }}$$

for some positive realr (seeEuler's formula). The quantityr is themodulus (or absolute value) ofz, denoted |z|:$${\displaystyle r=\sqrt{x^2+y^2}.}$$

The argument of zero is usually left undefined. The namesmagnitude, for the modulus, andphase,^[4][1] 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 of2πradians (a completeturn) are the same, as reflected by figure 2 on the right. Similarly, from theperiodicity ofsin andcos, the second definition also has this property.

Because a complete rotation around the origin leaves a complex number unchanged, there are many choices which could be made for${\displaystyle \phi }$ by circling the origin any number of times. This is shown in figure 2, a representation of themulti-valued (set-valued) function${\displaystyle f(x,y)=\arg (x+iy)}$, 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 awell-defined function is required, then the usual choice, known as theprincipal value, is the value in the open-closedinterval(−π,π] radians, that is from−π toπradians excluding−π radians itself (equiv., from −180 to +180degrees, 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).

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

The set of all possible values of the argument can be written in terms ofArg as:$${\displaystyle \arg (z)=\{\mathrm{Arg}(z)+2\pi n\mid n\in \mathbb{Z}\}.}$$

If a complex number is known in terms of its real and imaginary parts, then the function that calculates the principal valueArg is called thetwo-argument arctangent function,atan2:$${\displaystyle \mathrm{Arg}(x+iy)=\mathrm{atan2}(y,x)}$$Theatan2 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].^[1]Seeatan2 for further detail and alternative implementations, such as$${\displaystyle \mathrm{atan2}(y,x)=2\mathrm{atan}\frac{y}{\sqrt{x^2+y^2}+x},}$$which works except when⁠${\displaystyle y=0}$⁠ and⁠${\displaystyle x\le 0}$⁠, in which case the value is insteadπ if⁠⁠ or undefined when⁠${\displaystyle x=0}$⁠.

In Wolfram language, there'sArg[z]:^[5]

Arg[x + y I]

or using the language'sArcTan:

Arg[x + y I]

ArcTan[x, y] is${\displaystyle \mathrm{atan2}(y,x)}$ extended to work with infinities.ArcTan[0, 0] isIndeterminate (i.e. it'sstill defined), whileArcTan[Infinity, -Infinity] doesn't return anything (i.e. it'sundefined).

Maple'sargument(z) behaves the same asArg[z] in Wolfram language, except thatargument(z) also returns${\displaystyle \pi }$ ifz is the special floating-point value−0..^[6]Also, Maple doesn't have${\displaystyle \mathrm{atan2}}$.

MATLAB'sangle(z) behaves^[7][8] the same asArg[z] in Wolfram language, except that it is

Unlike in Maple and Wolfram language, MATLAB'satan2(y, x) is equivalent toangle(x + y*1i). That is,atan2(0, 0) is${\displaystyle 0}$.

One of the main motivations for defining the principal valueArg is to be able to write complex numbers in modulus-argument form. Hence for any complex numberz,$${\displaystyle z=\left|z\right|e^{i\mathrm{Arg}z}.}$$

This is only really valid ifz is non-zero, but can be considered valid forz = 0 ifArg(0) is considered as anindeterminate form—rather than as being undefined.

Some further identities follow. Ifz₁ andz₂ are two non-zero complex numbers, thenwhere${\displaystyle (\mathrm{m}\mathrm{o}\mathrm{d}2\pi \mathbb{Z})}$ means to add or subtract any integer multiple of2π if necessary to bring the value into the interval of(−π,π] radians.

Ifz ≠ 0 andn is any integer, then^[1]$${\displaystyle \mathrm{Arg}\left(z^n\right)\equiv n\mathrm{Arg}(z)(\mathrm{mod}2\pi \mathbb{Z}).}$$

$${\displaystyle \mathrm{Arg}(\frac{-1-i}{i})=\mathrm{Arg}(-1-i)-\mathrm{Arg}(i)=-\frac{3\pi }{4}-\frac{\pi }{2}=-\frac{5\pi }{4}}$$

From${\displaystyle z=|z|e^{i\mathrm{Arg}(z)}}$, we get${\displaystyle i\mathrm{Arg}(z)=\ln \frac{z}{|z|}}$, alternatively${\displaystyle \mathrm{Arg}(z)=\mathrm{Im}(\ln \frac{z}{|z|})=\mathrm{Im}(\ln z)}$. As we are taking the imaginary part, any normalisation by a real scalar will not affect the result. This is useful when one has thecomplex logarithm available.

The argument function is not complex differentiable, as the limit

$${\displaystyle \underset{z\to z_0}{lim}\frac{\arg (z)-\arg (z_0)}{z-z_0}}$$

does not exist for any${\displaystyle z_0\in \mathbb{C}}$ and on any branch (indeed, for any chosen branch the numerator takes the same value). However, theWirtinger derivatives may be applied to it. Begin from the logarithmic identity.

$${\displaystyle \ln (z)=\mathrm{Ln}|z|+i\arg (z)}$$

Use that${\displaystyle |z{|}^2=z\overline{z}}$.

$${\displaystyle \ln (z)=\frac{1}{2}\mathrm{Ln}(z)+\frac{1}{2}\mathrm{Ln}(\overline{z})+i\arg (z)}$$

Performing different branch cuts lets one apply the derivatives to all points of the principal logarithm in two goes. First, apply${\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }z}}$.

$${\displaystyle \frac{1}{z}=\frac{1}{2}\frac{1}{z}+0+i\frac{\mathrm{\partial }}{\mathrm{\partial }z}\arg (z)}$$

Rearranging gives${\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }z}\arg (z)=\frac{-i}{2z}}$. Now apply${\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }\overline{z}}}$.

$${\displaystyle 0=0+\frac{1}{2}\frac{1}{\overline{z}}+i\frac{\mathrm{\partial }}{\mathrm{\partial }\overline{z}}\arg (z)}$$

This shows that${\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }\overline{z}}\arg (z)=\frac{i}{2\overline{z}}}$

As both Wirtinger derivatives exist, one can then apply the Dolbeault derivatives.

$${\displaystyle \mathrm{\partial }\arg (z)=\frac{-i}{2z}dz}$$

$${\displaystyle \overline{\mathrm{\partial }}\arg (z)=\frac{i}{2\overline{z}}d\overline{z}}$$

This then means theexterior derivative exists, given by their sum.

$${\displaystyle d\arg (z)=\frac{-i}{2z}dz+\frac{i}{2\overline{z}}d\overline{z}}$$

This is an important differential form, as it is precisely the generator of the first de Rham cohomology of the circle,${\displaystyle H_{dR}^1(S^1)}$. In other words,

$${\displaystyle d\arg (z)=\frac{-i}{2z}dz+\frac{i}{2\overline{z}}d\overline{z}=\frac{xdy-ydx}{x^2+y^2}}$$

making this form closed but not exact in the usual sense (a byproduct of being multivalued).

$${\displaystyle \underset{S^1}{\oint }d\arg (z)=2\pi \ne 0}$$

with the standard orientation.

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