Argument (complex analysis)

In mathematics (particularly in complex analysis), the argument of a complex number z, denoted arg(z), is the angle between the positive real axis and the line joining the origin and z, represented as a point in the complex plane, shown as $\varphi$ in Figure 1. It is a multivalued function operating on the nonzero complex numbers. To define a single-valued function, the principal value of the argument (sometimes denoted Arg z) is used. It is often chosen to be the unique value of the argument that lies within the interval $(- π, π]$ .^[1]^[2]

Definition

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

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

The names magnitude, for the modulus, and phase,^[3]^[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 of $2π$ radians (a complete circle) are the same, as reflected by figure 2 on the right. Similarly, from the periodicity of $sin$ and $cos$ , the second definition also has this property. The argument of zero is usually left undefined.

Alternative definition

The complex argument can also be defined algebraically in terms of complex roots as: $\arg(z)=\lim _{n\to \infty }n\cdot \operatorname {Im} {\sqrt[{n}]{z/|z|}}$ This definition removes reliance on other difficult-to-compute functions such as arctangent as well as eliminating the need for the piecewise definition. Because it's defined in terms of roots, it also inherits the principal branch of square root as its own principal branch. The normalization of $z$ by dividing by $|z|$ isn't necessary for convergence to the correct value, but it does speed up convergence and ensures that $\arg(0)$ is left undefined.

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 $(- π rad, π rad]$ , that is from $- π$ to $π$ radians, excluding $- π$ rad 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 $[0, 2 π)$ .

Notation

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:

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

Computing from the real and imaginary part

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 two-argument arctangent function atan2:

\operatorname {Arg} (x+iy)=\operatorname {atan2} (y,\,x)

.

The atan2 function (also called arctan2 or other synonyms) is available in the math libraries of many programming languages, and usually returns a value in the range $(-π, π]$ .^[1]

Many texts say the value is given by $arctan(y / x)$ , as $y / x$ is slope, and $arctan$ converts slope to angle. This is correct only when $x > 0$ , so the quotient is defined and the angle lies between $- π /2$ and $π /2$ , but 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 two half-planes $x > 0$ and $x < 0$ (separated into two quadrants if one wishes a branch cut on the negative $x$ -axis), $y > 0$ , $y < 0$ , and then patch together.

\operatorname {Arg} (x+iy)=\operatorname {atan2} (y,\,x)={\begin{cases}\arctan \left({\frac {y}{x}}\right)&{\text{if }}x>0,\\\arctan \left({\frac {y}{x}}\right)+\pi &{\text{if }}x<0{\text{ and }}y\geq 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{cases}}

A compact expression with 4 overlapping half-planes is

\operatorname {Arg} (x+iy)=\operatorname {atan2} (y,\,x)={\begin{cases}\arctan \left({\frac {y}{x}}\right)&{\text{if }}x>0,\\{\frac {\pi }{2}}-\arctan \left({\frac {x}{y}}\right)&{\text{if }}y>0,\\-{\frac {\pi }{2}}-\arctan \left({\frac {x}{y}}\right)&{\text{if }}y<0,\\\arctan \left({\frac {y}{x}}\right)\pm \pi &{\text{if }}x<0,\\{\text{undefined}}&{\text{if }}x=0{\text{ and }}y=0.\end{cases}}

It's also possible to use arccotangent for the definition:

\operatorname {Arg} (x+iy)={\begin{cases}\operatorname {arccot} \left({\frac {x}{y}}\right)&{\text{if }}y>0,\\\operatorname {arccot} \left({\frac {x}{y}}\right)-\pi &{\text{if }}y<0,\\0&{\text{if }}y=0{\text{ and }}x>0\\\pi &{\text{if }}y=0{\text{ and }}x<0\\{\text{undefined}}&{\text{if }}x=0{\text{ and }}y=0.\end{cases}}

For the variant where $Arg$ is defined to lie in the interval $[0, 2π)$ , the value can be found by adding $2π$ to the value above when it is negative (when $y<0$ ).

Alternatively, the principal value can be calculated in a uniform way using the tangent half-angle formula, the function being defined over the complex plane but excluding the origin:

\operatorname {Arg} (x+iy)={\begin{cases}\displaystyle 2\arctan \left({\frac {y}{{\sqrt {x^{2}+y^{2}}}+x}}\right)&{\text{if }}x>0{\text{ or }}y\neq 0,\\\pi &{\text{if }}x<0{\text{ and }}y=0,\\{\text{undefined}}&{\text{if }}x=0{\text{ and }}y=0.\end{cases}}

This is based on a parametrization of the circle (except for the negative $x$ -axis) by rational functions. This version of $Arg$ is not stable enough for floating point computational use (as it may overflow near the region $x < 0, y = 0$ ), but can be used in symbolic calculation.

A variant of the last formula which avoids overflow is sometimes used in high precision computation:

\operatorname {Arg} (x+iy)={\begin{cases}\displaystyle 2\arctan \left({\frac {{\sqrt {x^{2}+y^{2}}}-x}{y}}\right)&{\text{if }}y\neq 0,\\0&{\text{if }}x>0{\text{ and }}y=0,\\\pi &{\text{if }}x<0{\text{ and }}y=0,\\{\text{undefined}}&{\text{if }}x=0{\text{ and }}y=0.\end{cases}}

Identities

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 $z$ ,

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

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 indeterminate form—rather than as being undefined.

Some further identities follow. If $z 1$ and $z 2$ are two non-zero complex numbers, then

{\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}}

If $z \neq 0$ and $n$ is any integer, then^[1]

\operatorname {Arg} \left(z^{n}\right)\equiv n\operatorname {Arg} (z){\pmod {\mathbb {R} /2\pi \mathbb {Z} }}.

Example

\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}}

Using the complex logarithm

From $z=|z|e^{i\operatorname {Arg} (z)}$ , it easily follows that $\operatorname {Arg} (z)=-i\ln {\frac {z}{|z|}}$ . This is useful when one has the complex logarithm available.

Extended argument

The extended argument of a number z (denoted as ${\overline {\arg }}(z)$ ) is the set of all real numbers congruent to $\arg(z)$ modulo 2 $\pi$ .^[4] ${\overline {\arg }}(z)=\arg(z)+2k\pi ,\forall k\in \mathbb {Z}$

References

^ ^a ^b ^c ^d Weisstein, Eric W. "Complex Argument". mathworld.wolfram.com. Retrieved 2020-08-31.
^ "Pure Maths". internal.ncl.ac.uk. Retrieved 2020-08-31.
^ Dictionary of Mathematics (2002). phase.
^ "Algebraic Structure of Complex Numbers". www.cut-the-knot.org. Retrieved 2021-08-29.

Bibliography

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

External links

Argument at Encyclopedia of Mathematics.

[:1-1] Weisstein, Eric W. "Complex Argument". mathworld.wolfram.com. Retrieved 2020-08-31.

[2] "Pure Maths". internal.ncl.ac.uk. Retrieved 2020-08-31.

[phase-3] Dictionary of Mathematics (2002). phase.

[4] "Algebraic Structure of Complex Numbers". www.cut-the-knot.org. Retrieved 2021-08-29.

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