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{{redirect|Arg (mathematics)|argument of a function|Argument of a function}}
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[[File:Complex_number_illustration_modarg.svg|thumb|right|Figure 1. This [[Argand diagram]] represents the [[complex number]]s lying on a [[Plane (geometry)|plane]]. For each point on the plane, {{math|arg}} is the function which returns the angle {{math|''φ''}}.]]
 
In [[mathematics]], '''arg''' is a [[function (mathematics)|function]] operating on [[complex number]]s (visualized in a [[complex plane]]). It gives the [[angle]] between the line joining the point to the origin and the positive [[real number|real]] [[Cartesian coordinate system|axis]], shown as {{math|''φ''}} in figure&nbsp;1 opposite, known as an '''argument''' of the point (that is, the angle between the half-lines of the [[position vector]] representing the number and the positive real axis).
 
==Definition==
[[File:Complex number illustration multiarg.svg|thumb|right|Figure 3. Two choices for the argument {{math|''φ''}}]]
 
Arguments are defined in two equivalent ways:
*Geometrically, in relation to an [[Argand diagram]], {{math|arg ''z''}} is the [[angle]] {{mvar|φ}} from the positive real axis to the vector representing {{mvar|z}}. The numeric value is given by the angle in [[radian]]s and is positive if measured counter-clockwise.
*Algebraically, an argument of the complex number {{math||1=''z'' = ''x'' + ''iy''}} is any real quantity <math>\phi</math> such that
::<math> z = x + i y = r \cos \phi + i r \sin \phi \ </math>
:for some positive real {{mvar|r}}. The quantity {{mvar|r}} is the ''[[Absolute value|modulus]]'' of {{mvar|z}}, written
::<math> r = |z| = \sqrt{x^2 + y^2} \ . </math>
 
The names '''amplitude'''<ref>{{Cite book | author1=Knopp, Konrad |author2= Bagemihl, Frederick | authorlink1=Konrad Knopp | title=Theory of Functions Parts I and II | date=1996 | publisher=Dover Publications | isbn=0-486-69219-1 | page=3}}</ref> or '''phase'''<ref name="phase">Dictionary of Mathematics (2002). ''phase''.</ref> 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|2π}} [[radian]]s (a complete circle) are the same. Similarly, from the [[periodic function|periodicity]] of [[sine|{{math|sin}}]] and [[cosine|{{math|cos}}]], the second definition also has this property.
 
== Principal value ==
[[File:Principal value of arg.svg|thumb|275px|right|Figure 4. The principal value {{math|Arg}} of the blue point at {{math|1+''i''}} is {{math|π/4}}. The red line here is the branch cut and corresponds to the two red lines in figure&nbsp;2 seen vertically above each other).]]
 
Because a complete rotation around 0 leaves a complex number unchanged, there are many choices which could be made for {{mvar|φ}} by circling the origin any number of times. This is shown in figure&nbsp;3, a representation of the [[multi-valued]] (set-valued) function, where a vertical line 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 (mathematics)|interval]] {{open-closed|−π, π}}, that is from {{math|−π}} to {{math|π}} radians, excluding {{math|−π}} itself (−180 to +180 [[degree (angle)|degree]]s). This represents an angle of up to half a complete circle from the positive real axis in either direction, the angle {{mvar|φ}} is constrained to lie between  {{math|−π}} and {{math|π}} [[radian]]s. This portion of the surface is shown hatched in red in figure&nbsp;2, and projected onto the plane in figure&nbsp;4.
 
=== 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.
 
Some authors define the range of the principal value as being in the closed-open interval {{closed-open|0, 2π}}.
 
The set of all possible values of the argument can be written in terms of {{math|Arg}} as:
:<math>\arg{z}=\{\operatorname{Arg} z+2\pi n:n\in \mathbb Z\}</math>.
 
== Covering space ==
[[File:Principle branch of arg on Riemann (small).png|thumb|300px|right|Figure&nbsp;2. The arguments of the complex plane are plotted vertically. {{math|arg}} measures the angle of points, shown by the fact that the outward radial lines, which have constant angle to the real axis, lie on the surface. The layered structure shows that each point has infinitely many arguments, each one corresponding an intersection between a vertical line through the point and the sheet. The red hatching indicates the surface corresponding to the principal value. ''See [[:Image:Principle branch of arg on Riemann.png|larger version]]''.]]
 
In informal situations, {{math|arg}} may be left not well-defined, for instance {{math|arg ''z''(''t'')}} where {{math|''z''}} depends on a parameter {{mvar|t}} may change by {{math|2π}} every time {{mvar|z}} goes around the origin. This idea can be made more precise by considering {{math|''z''(''t'')}} as being defined not on the complex plane but on a [[covering space]]. [[Polar coordinates]] excluding the origin and with an unconstrained angle provide such a space, in this case arg is defined by:
:<math>\begin{align}
\arg \colon \mathbb{R}^+\times\mathbb{R} &\to \mathbb{R} \\
(r,\ \phi) &\mapsto  \phi
\end{align}</math>
 
The covering space has as base space the punctured complex plane. This is equivalent to the product of a positive non-zero radius and an angle on a [[unit circle]] that is:
:<math>\mathbb{C}\smallsetminus\{0\} = \mathbb{R}^+ \times \mathbb{S}^1</math>
The principal value {{math|Arg}} then maps the unit circle component of this representation to the interval {{open-closed|−π, π}}.
 
== Computation ==
The principal value {{math|Arg}} of a complex number given as {{math|''x''+''iy''}} is normally available in programming languages using the function [[atan2]](&nbsp;) or some language specific variant. The value of {{math|atan2(''y'', ''x'')}} is the principal value in the range {{open-closed|−π, π}}.
 
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|−''π''/2}} and {{math|''π''/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 four half-planes {{math|''x'' > 0}}, {{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.
 
:<math> \operatorname{Arg}: \mathbb{C} \smallsetminus \{0\} \to (-\pi,\pi] </math>
 
:<math> \operatorname{Arg}(x + iy) = \operatorname{atan2}(y,x) =
  \begin{cases}
  \arctan \left( y/x \right) & \qquad x > 0 \\
  \pi/2 - \arctan \left( x/y \right) & \qquad y > 0 \\
  -\pi/2 - \arctan \left( x/y \right) & \qquad y < 0 \\
  \pi + \arctan \left( y/x \right) & \qquad x < 0, y \ge 0 \\
  -\pi + \arctan \left( y/x \right) & \qquad x < 0, y < 0 \\
  \text{undefined} & \qquad x = 0, y = 0
  \end{cases} </math>
 
For the variant where {{math|Arg}} is defined to lie in the interval {{closed-open|0, 2π}}, the value can be found by adding {{math|2π}} 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:
 
:<math>
\operatorname{Arg}(x + iy) =
\begin{cases}
2 \arctan \left( \frac{y}{\sqrt{x^2+y^2}+x} \right) & \qquad x > 0 \text{ or } y \ne 0  \\
\pi & \qquad x < 0 \text{ and } y = 0 \\
\text{undefined} & \qquad  x = 0 \text{ and } y = 0
\end{cases}
</math>
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 numerical use but can be used in symbolic calculation. In many programming libraries there is a function called '''atan2''' which performs an equivalent computation.
 
==Identities==
 
One of the main motivations for defining the principal value {{math|Arg}} is to be able to write complex numbers in [[absolute value|modulus]]-argument form (the ''modulus'' of {{math|1=''z'' = ''x'' + ''iy''}} is {{math|1={{mabs|''z''}} = √(''x''<sup>2</sup> + ''y''<sup>2</sup>)}}, the length of the vector on the Argand diagram). Hence for any complex number {{mvar|z}},
:<math> z = \left| z \right| e ^ { i \operatorname{Arg} \left( z \right) } </math>.
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''<sub>1</sub>}} and {{math|''z''<sub>2</sub>}} are two non-zero complex numbers then
:<math>\operatorname{Arg}\left(z_1  z_2\right) \equiv \operatorname{Arg}(z_1) + \operatorname{Arg}(z_2) \pmod {(-\pi,\pi]}</math>
:<math>\operatorname{Arg}\left(\frac{z_1}{z_2}\right) \equiv \operatorname{Arg}(z_1) - \operatorname{Arg}(z_2) \pmod {(-\pi,\pi]}</math>
 
If {{math|''z'' ≠ 0}} and {{mvar|n}} is any integer then
 
:<math>\operatorname{Arg}\left(z^n\right) \equiv n \operatorname{Arg}(z) \pmod {(-\pi,\pi]} .</math>
 
===Example===
:<math>
\operatorname{Arg}\left(\frac{-1- i}{i}\right) \equiv \operatorname{Arg}(-1-i) - \operatorname{Arg}(i) = -\frac{3\pi}{4} - \frac{\pi}{2} = -\frac{5\pi}{4} \equiv \frac{3\pi}{4} \pmod {(-\pi,\pi]}
</math>
 
==References==
===Notes===
{{reflist}}
 
===Bibliography===
{{refbegin}}
*{{cite book|last=Ahlfors|first=Lars|title=Complex Analysis: An Introduction to the Theory of Analytic Functions of One Complex Variable
    |edition=3rd|location=New York;London|publisher=McGraw-Hill|date=1979
    |isbn=0-07-000657-1
}}
*{{cite book|last=Beardon| first=Alan
    |title=Complex Analysis: The Argument Principle in Analysis and Topology
    |location=Chichester|publisher=Wiley|year=1979
    |isbn=0-471-99671-8
}}
*{{cite book|last1=Borowski|first1=Ephraim |last2=Borwein|first2=Jonathan
    |title=Mathematics |series=Collins Dictionary |date=2002
    |origyear=1st ed. 1989 as ''Dictionary of Mathematics'' |isbn=0-00-710295-X
    |edition=2nd |publisher=[[HarperCollins]] |location=Glasgow}}
{{refend}}
 
==External links==
*{{MathWorld|title=Complex Argument|urlname=ComplexArgument}}
 
[[Category:Trigonometry]]
[[Category:Complex analysis]]
[[Category:Signal processing]]

Latest revision as of 16:53, 8 May 2014

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