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*argument (complex analysis)*

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argument (complex analysis)

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**argument**is a multi-valued function operating on the nonzero complex numbers. With complex number

*z*visualized as a point in the complex plane, the argument of

*z*is the angle between the positive real axis and the line joining the point to the origin, shown as {{math|

*Ï†*}} in figure 1 and denoted arg

*z*. To define a single-valued function, the principal value of the argument (sometimes denoted Arg

*z*) is used. It is chosen to be the unique value of the argument that lies within the interval (â€“Ï€, Ï€].

## Definition

(File:Complex number illustration multiarg.svg|thumb|Figure 2. Two choices for the argument {{mvar|Ï†}})An**argument**of the complex number {{math|1=

*z*=

*x*+

*iy*}}, denoted {{math|arg(

*z*)}}, is defined in two equivalent ways:

- Geometrically, in the complex plane, as the angle {{mvar|Ï†}} from the positive real axis to the vector representing {{mvar|z}}. The numeric value is given by the angle in radians and is positive if measured counterclockwise.
- Algebraically, as any real quantity {{mvar|Ï†}} such that

z = r (cos varphi + i sin varphi) = r e^{ivarphi}

for some positive real {{mvar|r}} (see Euler's formula). The quantity {{mvar|r}} is the *modulus*of {{mvar|z}}, denoted |{{mvar|z}}|:

r = sqrt{x^2 + y^2}.

*magnitude,*for the modulus, and

*phase*,Dictionary of Mathematics (2002).

*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 {{math|2Ï€}} radians (a complete circle) are the same, as reflected by figure 2 on the right. Similarly, from the periodicity of {{math|sin}} and {{math|cos}}, the second definition also has this property. The argument of zero is usually left undefined.

## Principal value

(File:Principal value of arg.svg|thumb|275px|Figure 3. The principal value {{math|Arg}} of the blue point at {{math|1 +*i*}} is {{math|Ï€/4}}. 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 {{mvar|Ï†}} 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 {{open-closed|âˆ’Ï€ rad, Ï€ rad}}, that is from {{math|âˆ’Ï€}} to {{math|Ï€}} radians, excluding {{math|âˆ’Ï€}} rad itself (equivalently from âˆ’180 to +180 degrees, excluding âˆ’180Â° 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 {{closed-open|0, 2Ï€}}.

### Notation

The principal value sometimes has the initial letter capitalized as in {{math|Arg*z*}}, especially when a general version of the argument is also being considered. Note that notation varies, so {{math|arg}} and {{math|Arg}} may be interchanged in different texts.The set of all possible values of the argument can be written in terms of {{math|Arg}} as:

operatorname{arg}(z) in {operatorname{Arg}(z) + 2pi n;|; n in mathbb Z}.

Likewise
operatorname{Arg}(z) = operatorname{arg}(z) - 2pi n;|; n in mathbb Z land -pi < operatorname{arg}(z) - 2pi n le 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 {{math|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 {{open-closed|âˆ’Ï€, Ï€}}.Many texts say the value is given by {{math|arctan(*y*/

*x*)}}, as {{math|

*y*/

*x*}} is slope, and {{math|arctan}} converts slope to angle. This is correct only when {{math|

*x*> 0}}, so the quotient is defined and the angle lies between {{math|âˆ’

*Ï€*/2}} and {{math|

*Ï€*/2}}, but extending this definition to cases where {{math|

*x*}} is not positive is relatively involved. Specifically, one may define the principal value of the argument separately on the two half-planes {{math|

*x*> 0}} and {{math|

*x*< 0}} (separated into two quadrants if one wishes a branch cut on the negative {{math|

*x*}}-axis), {{math|

*y*> 0}}, {{math|

*y*< 0}}, and then patch together.

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

begin{cases}arctan(frac y x) &text{if } x > 0, arctan(frac y x) + pi &text{if } x < 0 text{ and } y ge 0, arctan(frac y x) - 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}arctanleft(frac{y}{x}right) &text{if } x > 0, frac{pi}{2} - arctanleft(frac{x}{y}right) &text{if } y > 0, -frac{pi}{2} - arctanleft(frac{x}{y}right) &text{if } y < 0, arctanleft(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 {{math|Arg}} is defined to lie in the interval {{closed-open|0, 2Ï€}}, the value can be found by adding {{math|2Ï€}} 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:
operatorname{Arg}(x + iy) =

begin{cases}2 arctanleft(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 {{mvar|x}}-axis) by rational functions. This version of {{math|Arg}} is not stable enough for floating point computational use (it may overflow near the region {{math|1=*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}2 arctanleft(frac{sqrt{x^2 + y^2} - x}{y}right) &text{if } y neq 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 {{math|Arg}} is to be able to write complex numbers in modulus-argument form. Hence for any complex number {{mvar|z}},
z = left| z right| e^{i operatorname{Arg} z}.

This is only really valid if {{mvar|z}} is non-zero but can be considered as valid also for {{math|1=*z*= 0}} if {{math|Arg(0)}} is considered as being an indeterminate form rather than as being undefined.Some further identities follow. If {{math|

*z*1}} and {{math|

*z*2}} are two non-zero complex numbers, then

operatorname{Arg}(z_1 z_2) equiv operatorname{Arg}(z_1) + operatorname{Arg}(z_2) pmod{(-pi,pi]},
operatorname{Arg}biggl(frac{z_1}{z_2}biggr) equiv operatorname{Arg}(z_1) - operatorname{Arg}(z_2) pmod{(-pi,pi]}.

If {{math|*z*â‰ 0}} and {{mvar|n}} is any integer, then

operatorname{Arg}left(z^nright) equiv n operatorname{Arg}(z) pmod {(-pi,pi]}.

### Example

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

### Using the complex logarithm

From z = |z| e^{i theta}, it easily follows that operatorname{Arg}(z) = -i lnleft(frac{z}{|z|}right). This is useful when one has the complex logarithm available.## References

### Notes

{{reflist}}### 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

, , 0-07-000657-1

- BOOK, Ponnuswamy, S., Foundations of Complex Analysis

, 2nd, New Delhi;Mumbai, Narosa, 2005

, 978-81-7319-629-4

, , 978-81-7319-629-4

- BOOK, Beardon, Alan

, Complex Analysis: The Argument Principle in Analysis and Topology

, Chichester, Wiley, 1979

, 0-471-99671-8

, , Chichester, Wiley, 1979

, 0-471-99671-8

- BOOK, Borowski, Ephraim, Borwein, Jonathan

, Mathematics, Collins Dictionary, 2002

, 1st ed. 1989 as

, 2nd, HarperCollins, Glasgow,

, 1st ed. 1989 as

*Dictionary of Mathematics*, 0-00-710295-X, 2nd, HarperCollins, Glasgow,

## External links

- {{MathWorld|title=Complex Argument|urlname=ComplexArgument}}

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