Argument (complex analysis)

Figure 1. This Argand diagram represents the complex number lying on a plane. For each point on the plane, Template:Math is the function which returns the angle $\phi$.

In mathematics (particularly in complex analysis), the argument of a complex number Template:Mvar, denoted Template:Math, is the angle between the positive real axis and the line joining the origin and Template:Mvar, represented as a point in the complex plane, shown as $\phi$ 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 Template:Open-closed. In this article the multi-valued function will be denoted Template:Math and its principal value will be denoted Template:Math, 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 While this term is largely outdated in modern texts, it still appears in some regional educational resources, where it is sometimes used in introductory-level textbooks.

Definition

Figure 2. Two choices for the argument $\phi$

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

1. Geometrically, in the complex plane, as the 2D polar angle $\phi$ from the positive real axis to the vector representing Template:Mvar. The numeric value is given by the angle in radians, and is positive if measured counterclockwise.
2. Algebraically, as any real quantity $\phi$ such that $$z=r(\cos \phi +i\sin \phi )=r{e}^{i\phi }$$ for some positive real Template:Mvar (see Euler's formula). The quantity Template:Mvar is the modulus (or absolute value) of Template:Mvar, denoted |Template:Mvar|: $$r=\sqrt{x^2+y^2}.$$

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

Principal value

Figure 3. The principal value Template:Math of the blue point at Template:Math is Template:Math. The red line here is the branch cut and corresponds to the two red lines in figure 4 seen vertically above each other).

Because a complete rotation around the origin leaves a complex number unchanged, there are many choices which could be made for $\phi$ 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 Template:Open-closed radians, that is from Template:Math to Template:Math radians excluding Template:Math 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 Template:Closed-open.

Notation

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

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

$\arg (z)=\{\mathrm{Arg}(z)+2\pi n\mid n\in \mathbb{Z}\}.$

Computing from the real and imaginary part

Template:Main If a complex number is known in terms of its real and imaginary parts, then the function that calculates the principal value Template:Math is called the [[atan2|two-argument arctangent function, Template:Math]]:

$\mathrm{Arg}(x+iy)=\mathrm{atan2}(y,x)$.

The Template:Math function is available in the math libraries of many programming languages, sometimes under a different name, and usually returns a value in the range Template:Open-closed.

In some sources the argument is defined as $\mathrm{Arg}(x+iy)=\arctan (y/x),$ however this is correct only when Template:Math, where $y/x$ is well-defined and the angle lies between $-\frac{\pi }{2}$ and $\frac{\pi }{2}.$ Extending this definition to cases where Template:Math is not positive is relatively involved. Specifically, one may define the principal value of the argument separately on the half-plane Template:Math and the two quadrants with Template:Math, and then patch the definitions together:

$\mathrm{Arg}(x+iy)=\mathrm{atan2}(y,x)=\{\begin{array}{cc}\arctan \left(\frac{y}{x}\right)& \text{if }x>0,\\ \arctan \left(\frac{y}{x}\right)+\pi & \text{if }x<0\text{ and }y\ge 0,\\ \arctan \left(\frac{y}{x}\right)-\pi & \text{if }x<0\text{ and }y<0,\\ +\frac{\pi }{2}& \text{if }x=0\text{ and }y>0,\\ -\frac{\pi }{2}& \text{if }x=0\text{ and }y<0,\\ \text{undefined}& \text{if }x=0\text{ and }y=0.\end{array}$

See atan2 for further detail and alternative implementations.

Realizations of the function in computer languages

Wolfram language (Mathematica)

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

Arg[x + y I] $=\{\begin{array}{cc}\text{undefined}& \text{if }|x|=\mathrm{\infty }\text{ and }|y|=\mathrm{\infty },\\ 0& \text{if }x=0\text{ and }y=0,\\ 0& \text{if }x=\mathrm{\infty },\\ \pi & \text{if }x=-\mathrm{\infty },\\ \pm \frac{\pi }{2}& \text{if }y=\pm \mathrm{\infty },\\ \mathrm{Arg}(x+yi)& \text{otherwise}.\end{array}$

or using the language's ArcTan:

Arg[x + y I] $=\{\begin{array}{cc}0& \text{if }x=0\text{ and }y=0,\\ \text{ArcTan[x, y]}& \text{otherwise}.\end{array}$

ArcTan[x, y] is $\mathrm{atan2}(y,x)$ 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.. Also, Maple doesn't have $\mathrm{atan2}$.

MATLAB

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

$\{\begin{array}{cc}\frac{1\pi }{4}& \text{if }x=\mathrm{\infty }\text{ and }y=\mathrm{\infty },\\ -\frac{1\pi }{4}& \text{if }x=\mathrm{\infty }\text{ and }y=-\mathrm{\infty },\\ \frac{3\pi }{4}& \text{if }x=-\mathrm{\infty }\text{ and }y=\mathrm{\infty },\\ -\frac{3\pi }{4}& \text{if }x=-\mathrm{\infty }\text{ and }y=-\mathrm{\infty }.\end{array}$

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

Identities

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

$z=\left|z\right|e^{i\mathrm{Arg}z}.$

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

Some further identities follow. If Template:Math and Template:Math are two non-zero complex numbers, then

$\begin{array}{cc}\mathrm{Arg}(z_1{z}_2)& \equiv \mathrm{Arg}(z_1)+\mathrm{Arg}(z_2)(\mathrm{mod}ℝ/2\pi \mathbb{Z}),\\ \mathrm{Arg}\left(\frac{z_1}{z_2}\right)& \equiv \mathrm{Arg}(z_1)-\mathrm{Arg}(z_2)(\mathrm{mod}ℝ/2\pi \mathbb{Z}).\end{array}$

If Template:Math and Template:Mvar is any integer, then

$\mathrm{Arg}\left(z^n\right)\equiv n\mathrm{Arg}(z)(\mathrm{mod}ℝ/2\pi \mathbb{Z}).$

Example

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

Using the complex logarithm

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

Extended argument

The extended argument of a number z (denoted as $\bar{\arg }(z)$) is the set of all real numbers congruent to $\arg (z)$ modulo 2$\pi$.$$\bar{\arg }(z)=\arg (z)+2k\pi ,\mathrm{\forall }k\in \mathbb{Z}$$

Bibliography

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External links

• Argument at Encyclopedia of Mathematics.