Argument (complex analysis)

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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 ${\displaystyle \varphi }$ in Figure 1. [1] It is a multi-valued 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]. [2] [3]

Definition

An argument of the complex number z = x + iy, denoted arg(z), [1] is defined in two equivalent ways:

1. Geometrically, in the complex plane, as the 2D polar angle ${\displaystyle \varphi }$ 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.
2. Algebraically, as any real quantity ${\displaystyle \varphi }$ such that
${\displaystyle z=r(\cos \varphi +i\sin \varphi )=re^{i\varphi }}$
for some positive real r (see Euler's formula). The quantity r is the modulus (or absolute value) of z, denoted |z|: [1]
${\displaystyle r={\sqrt {x^{2}+y^{2}}}.}$

The names magnitude, for the modulus, and phase , [4] [2] 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 circle) are the same, as reflected by figure 2 on the right. Similarly, from the periodicity of [[sine|sin]] and cos, the second definition also has this property. The argument of zero is usually 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 ${\displaystyle \varphi }$ by circling the origin any number of times. This is shown in figure 2, a representation of the multi-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 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, 2π).

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:

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

Likewise

${\displaystyle \operatorname {Arg} (z)=\{\arg(z)-2\pi n\mid n\in \mathbb {Z} \land -\pi <\arg(z)-2\pi n\leq \pi \}.}$

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:

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

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]. [2]

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.

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

A compact expression with 4 overlapping half-planes is

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

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

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:

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

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:

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

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,

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

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 z1 and z2 are two non-zero complex numbers, then

{\displaystyle {\begin{aligned}\operatorname {Arg} (z_{1}z_{2})&\equiv \operatorname {Arg} (z_{1})+\operatorname {Arg} (z_{2}){\pmod {(-\pi ,\pi ]}},\\\operatorname {Arg} \left({\frac {z_{1}}{z_{2}}}\right)&\equiv \operatorname {Arg} (z_{1})-\operatorname {Arg} (z_{2}){\pmod {(-\pi ,\pi ]}}.\end{aligned}}}

If z ≠ 0 and n is any integer, then [2]

${\displaystyle \operatorname {Arg} \left(z^{n}\right)\equiv n\operatorname {Arg} (z){\pmod {(-\pi ,\pi ]}}.}$

Example

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

Using the complex logarithm

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

Extended Argument

Extended argument of a number z (denoted as ${\displaystyle {\overline {\arg }}(z)}$) is the set of all real numbers congruent to ${\displaystyle \arg(z)}$ modulo 2${\displaystyle \pi }$. [5]

${\displaystyle {\overline {\arg }}(z)=\arg(z)+2k\pi ,\forall k\in \mathbb {N} }$

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References

1. "Comprehensive List of Algebra Symbols". Math Vault. 2020-03-25. Retrieved 2020-08-31.
2. Weisstein, Eric W. "Complex Argument". mathworld.wolfram.com. Retrieved 2020-08-31.
3. "Pure Maths". internal.ncl.ac.uk. Retrieved 2020-08-31.
4. Dictionary of Mathematics (2002). phase.
5. "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.