Argument (complex analysis) explained

Argument (complex analysis) should not be confused with Argument of a function.

In mathematics (particularly in complex analysis), the argument of a complex number, denoted, is the angle between the positive real axis and the line joining the origin and, represented as a point in the complex plane, shown as

\varphi

in Figure 1. By convention the positive real axis is drawn pointing rightward, the positive imaginary axis is drawn pointing upward, and complex numbers with positive real part are considered to have an anticlockwise argument with positive sign.

When any real-valued angle is considered, the argument is a multivalued function operating on the nonzero complex numbers. The principal value of this function is single-valued, typically chosen to be the unique value of the argument that lies within the interval .^[1] ^[2] In this article the multi-valued function will be denoted and its principal value will be denoted, 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 Lars 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 While this term is largely outdated in modern texts, it still appears in some regional educational resources, where it is sometimes used in introductory-level textbooks.^[3]

Definition

An argument of the nonzero complex number, denoted, is defined in two equivalent ways:

\varphi

from the positive real axis to the vector representing . The numeric value is given by the angle in radians, and is positive if measured counterclockwise.

Algebraically, as any real quantity

\varphi

such that

$z = r (\cos \varphi + i \sin \varphi) = r e^$ for some positive real (see Euler's formula). The quantity is the modulus (or absolute value) of, denoted ||: $r = \sqrt.$

The argument of zero is usually left undefined. The names magnitude, for the modulus, and phase,^[4] 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 radians (a complete turn) are the same, as reflected by figure 2 on the right. Similarly, from the periodicity of and , the second definition also has this property.

Principal value

Because a complete rotation around the origin leaves a complex number unchanged, there are many choices which could be made for

\varphi

by circling the origin any number of times. This is shown in figure 2, a representation of the multi-valued (set-valued) function

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 a well-defined function is required, then the usual choice, known as the principal value, is the value in the open-closed interval radians, that is from to radians excluding radians itself (equiv., from −180 to +180 degrees, 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 .

Notation

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

The set of all possible values of the argument can be written in terms of as: $\arg(z) = \.$

Computing from the real and imaginary part

See main article: atan2. If a complex number is known in terms of its real and imaginary parts, then the function that calculates the principal value is called the two-argument arctangent function, : $\operatorname(x + iy) = \operatorname(y,\, x)$ The function is available in the math libraries of many programming languages, sometimes under a different name, and usually returns a value in the range .See atan2 for further detail and alternative implementations, such as $\operatorname(y, x) = 2 \operatorname \frac\,,$ which works except when and, in which case the value is instead if or undefined when .

Realizations of the function in computer languages

Wolfram language (Mathematica)

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

Arg[x + y I]

= \begin{cases} 0&ifx=0andy=0,\\[5mu] undefined&if|x|=inftyand|y|=infty,\\[5mu] 0&ifx=inftyand|y|\neinfty,\\[5mu] \pi&ifx=-inftyand|y|\neinfty,\\[5mu] \pm

	\pi
	2

&ify=\pminftyand|x|\neinfty,\\[5mu] \operatorname{Arg}(x+yi)&otherwise. \end{cases}

or using the language's ArcTan:

Arg[x + y I]

= \begin{cases} 0&ifx=0andy=0,\\[5mu] ArcTan[x,y]&otherwise. \end{cases}

ArcTan[x, y] is

\operatorname{atan2}(y,x)

extended to work with infinities. ArcTan[0, 0] is Indeterminate (i.e. it's still defined), while ArcTan[Infinity, -Infinity] doesn't return anything (i.e. it's undefined).

Maple

Maple's argument(z) behaves the same as Arg[z] in Wolfram language, except that argument(z) also returns

\pi

if z is the special floating-point value −0..^[6] Also, Maple doesn't have

\operatorname{atan2}

MATLAB

MATLAB's angle(z) behaves^[7] ^[8] the same as Arg[z] in Wolfram language, except that it is

$\begin \frac &\text x = \infty \text y = \infty, \\[5mu] -\frac &\text x = \infty \text y = -\infty, \\[5mu] \frac &\text x = -\infty \text y = \infty, \\[5mu] -\frac &\text x = -\infty \text y = -\infty.\end$

Unlike in Maple and Wolfram language, MATLAB's atan2(y, x) is equivalent to angle(x + y*1i). That is, atan2(0, 0) is

Identities

One of the main motivations for defining the principal value is to be able to write complex numbers in modulus-argument form. Hence for any complex number, $z = \left| z \right| e^.$

This is only really valid if is non-zero, but can be considered valid for if is considered as an indeterminate form—rather than as being undefined.

Some further identities follow. If and are two non-zero complex numbers, then $\begin \operatorname(z_1 z_2) &\equiv \operatorname(z_1) + \operatorname(z_2) \pmod, \\ \operatorname\left(\frac\right) &\equiv \operatorname(z_1) - \operatorname(z_2) \pmod,\end$ where

(mod2\piZ)

means to add or subtract any integer multiple of if necessary to bring the value into the interval of radians.

If and is any integer, then $\operatorname\left(z^n\right) \equiv n \operatorname(z) \pmod.$

Example

$\operatorname\biggl(\frac\biggr) = \operatorname(-1 - i) - \operatorname(i) = -\frac - \frac = -\frac$

Using the complex logarithm

From

z=|z|eⁱ(z)}

, we get

i\operatorname{Arg}(z)=ln

	z
	\|z\|

, alternatively

\operatorname{Arg}(z)=\operatorname{Im}(ln

	z
	\|z\|

)=\operatorname{Im}(lnz)

. 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 complex logarithm available.

Differentiation

The argument function is not complex differentiable, as the limit

$\lim\limits_\frac$

does not exist for any

z₀\inC

and on any branch (indeed, for any chosen branch the numerator takes the same value). However, the Wirtinger derivatives may be applied to it. Begin from the logarithmic identity.

$\ln(z)=\operatorname|z|+i\arg(z)$

Use that

|z|²=z\bar{z}

$\ln(z)=\frac\operatorname(z)+\frac\operatorname(\bar)+i\arg(z)$

Performing different branch cuts lets one apply the derivatives to all points of the principal logarithm in two goes. First, apply

	\partial
	\partialz

$\frac=\frac\frac+0+i\frac\arg(z)$

Rearranging gives

	\partial	\arg(z)=
	\partialz

	-i
	2z

. Now apply

	\partial
	\partial\bar{z

$0=0+\frac\frac+i\frac\arg(z)$

This shows that

	\partial
	\partial\bar{z

}\arg(z)=\frac

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

$\partial\arg(z)=\frac\,dz$

$\bar\arg(z)=\frac\,d\bar$

This then means the exterior derivative exists, given by their sum.

$d\arg(z)=\frac\,dz+\frac\,d\bar$

This is an important differential form, as it is precisely the generator of the first de Rham cohomology of the circle,

	1
H
	dR

(S¹)

. In other words,

$d\arg(z)=\frac\,dz+\frac\,d\bar=\frac$

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

$\oint\limits_d\arg(z)=2\pi\neq0$

with the standard orientation.

Bibliography

Book: Ahlfors, Lars. Complex Analysis: An Introduction to the Theory of Analytic Functions of One Complex Variable . 3rd. New York;London. McGraw-Hill. 1979 . 0-07-000657-1.
Book: Ponnuswamy, S.. Foundations of Complex Analysis . 2nd. New Delhi;Mumbai. Narosa. 2005 . 978-81-7319-629-4.
Book: Beardon, Alan . Complex Analysis: The Argument Principle in Analysis and Topology . Chichester. Wiley. 1979 . 0-471-99671-8.
Book: Borowski. Ephraim . Borwein. Jonathan . Mathematics . Collins Dictionary . 2002 . 1st ed. 1989 as Dictionary of Mathematics . 0-00-710295-X . 2nd . . Glasgow.

External links

Argument at Encyclopedia of Mathematics.

Notes and References

Web site: Weisstein. Eric W.. Complex Argument. 2020-08-31. mathworld.wolfram.com. en.
Web site: Modulus and Argument . 2025-01-05 . mas-coursebuild.ncl.ac.uk . Newcastle University.
Web site: Modulus and Argument of a Complex Number . Byju's . 18 January 2025.
Dictionary of Mathematics (2002). phase.
Web site: Arg. Wolfram Language Documentation. 2024-08-30.
Web site: Argument - Maple Help .
Web site: Phase angle - MATLAB angle .
Web site: Four-quadrant inverse tangent - MATLAB atan2 .