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 φ {displaystyle varphi } 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 (−π, π].[1][2] In this article the multi-valued function will be denoted arg(z) and its principal value will be denoted Arg(z), 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 While this term is largely outdated in modern texts, it still appears in some regional educational resources, where it is sometimes used in introductory-level textbooks.[3]

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

1. Geometrically, in the complex plane, as the 2D polar angle φ {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 φ {displaystyle varphi } such that

z = r ( cos ⁡ φ + i sin ⁡ φ ) = r e i φ {displaystyle z=r(cos varphi +isin varphi )=re^{ivarphi }}

for some positive real r (see Euler's formula). The quantity r is the modulus (or absolute value) of z, denoted |z|: r = x 2 + y 2 . {displaystyle r={sqrt {x^{2}+y^{2}}}.}

The argument of zero is usually left undefined. The names magnitude, for the modulus, and phase,[4][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 turn) 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.

Because a complete rotation around the origin leaves a complex number unchanged, there are many choices which could be made for φ {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 f ( x , y ) = arg ⁡ ( x + i y ) {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, π] radians, that is from −π to π radians excluding −π 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 [0, 2π).

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 ) = { Arg ⁡ ( z ) + 2 π n ∣ n ∈ Z } . {displaystyle arg(z)={operatorname {Arg} (z)+2pi nmid nin mathbb {Z} }.}

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: Arg ⁡ ( x + i y ) = atan2 ⁡ ( y , x ) {displaystyle operatorname {Arg} (x+iy)=operatorname {atan2} (y,,x)} The atan2 function is available in the math libraries of many programming languages, sometimes under a different name, and usually returns a value in the range (−π, π].[1] See atan2 for further detail and alternative implementations, such as atan2 ⁡ ( y , x ) = 2 atan ⁡ y x 2 + y 2 + x , {displaystyle operatorname {atan2} (y,x)=2operatorname {atan} {frac {y}{{sqrt {x^{2}+y^{2}}}+x}},,} which works except when ⁠ y = 0 {displaystyle y=0} ⁠ and ⁠ x ≤ 0 {displaystyle xleq 0} ⁠, in which case the value is instead π if ⁠ x < 0 {displaystyle x<0} ⁠ or undefined when ⁠ x = 0 {displaystyle x=0} ⁠.

In Wolfram language, there's Arg[z]:[5]

Arg[x + y I] = { 0 if x = 0 and y = 0 , undefined if | x | = ∞ and | y | = ∞ , 0 if x = ∞ and | y | ≠ ∞ , π if x = − ∞ and | y | ≠ ∞ , ± π 2 if y = ± ∞ and | x | ≠ ∞ , Arg ⁡ ( x + y i ) otherwise . {displaystyle ={begin{cases}0&{text{if }}x=0{text{ and }}y=0,[5mu]{text{undefined}}&{text{if }}|x|=infty {text{ and }}|y|=infty ,[5mu]0&{text{if }}x=infty {text{ and }}|y|neq infty ,[5mu]pi &{text{if }}x=-infty {text{ and }}|y|neq infty ,[5mu]pm {frac {pi }{2}}&{text{if }}y=pm infty {text{ and }}|x|neq infty ,[5mu]operatorname {Arg} (x+yi)&{text{otherwise}}.end{cases}}}

or using the language's ArcTan:

Arg[x + y I] = { 0 if x = 0 and y = 0 , ArcTan[x, y] otherwise . {displaystyle ={begin{cases}0&{text{if }}x=0{text{ and }}y=0,[5mu]{text{ArcTan[x, y]}}&{text{otherwise}}.end{cases}}}

ArcTan[x, y] is atan2 ⁡ ( y , x ) {displaystyle operatorname {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's argument(z) behaves the same as Arg[z] in Wolfram language, except that argument(z) also returns π {displaystyle pi } if z is the special floating-point value −0..[6] Also, Maple doesn't have atan2 {displaystyle operatorname {atan2} } .

MATLAB's angle(z) behaves[7][8] the same as Arg[z] in Wolfram language, except that it is

{ 1 π 4 if x = ∞ and y = ∞ , − 1 π 4 if x = ∞ and y = − ∞ , 3 π 4 if x = − ∞ and y = ∞ , − 3 π 4 if x = − ∞ and y = − ∞ . {displaystyle {begin{cases}{frac {1pi }{4}}&{text{if }}x=infty {text{ and }}y=infty ,[5mu]-{frac {1pi }{4}}&{text{if }}x=infty {text{ and }}y=-infty ,[5mu]{frac {3pi }{4}}&{text{if }}x=-infty {text{ and }}y=infty ,[5mu]-{frac {3pi }{4}}&{text{if }}x=-infty {text{ and }}y=-infty .end{cases}}}

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 {displaystyle 0} .

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 = | z | e i Arg ⁡ z . {displaystyle z=left|zright|e^{ioperatorname {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 Arg ⁡ ( z 1 z 2 ) ≡ Arg ⁡ ( z 1 ) + Arg ⁡ ( z 2 ) ( mod 2 π Z ) , Arg ⁡ ( z 1 z 2 ) ≡ Arg ⁡ ( z 1 ) − Arg ⁡ ( z 2 ) ( mod 2 π Z ) , {displaystyle {begin{aligned}operatorname {Arg} (z_{1}z_{2})&equiv operatorname {Arg} (z_{1})+operatorname {Arg} (z_{2}){pmod {2pi mathbb {Z} }},operatorname {Arg} left({frac {z_{1}}{z_{2}}}right)&equiv operatorname {Arg} (z_{1})-operatorname {Arg} (z_{2}){pmod {2pi mathbb {Z} }},end{aligned}}} where ( m o d 2 π Z ) {displaystyle (mathrm {mod} ,,2pi mathbb {Z} )} means to add or subtract any integer multiple of 2π if necessary to bring the value into the interval of (−π, π] radians.

If z ≠ 0 and n is any integer, then[1] Arg ⁡ ( z n ) ≡ n Arg ⁡ ( z ) ( mod 2 π Z ) . {displaystyle operatorname {Arg} left(z^{n}right)equiv noperatorname {Arg} (z){pmod {2pi mathbb {Z} }}.}

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

From z = | z | e i Arg ⁡ ( z ) {displaystyle z=|z|e^{ioperatorname {Arg} (z)}} , we get i Arg ⁡ ( z ) = ln ⁡ z | z | {displaystyle ioperatorname {Arg} (z)=ln {frac {z}{|z|}}} , alternatively Arg ⁡ ( z ) = Im ⁡ ( ln ⁡ z | z | ) = Im ⁡ ( ln ⁡ z ) {displaystyle operatorname {Arg} (z)=operatorname {Im} (ln {frac {z}{|z|}})=operatorname {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.

The argument function is not complex differentiable, as the limit

lim z → z 0 arg ⁡ ( z ) − arg ⁡ ( z 0 ) z − z 0 {displaystyle lim limits _{zto z_{0}}{frac {arg(z)-arg(z_{0})}{z-z_{0}}}}

does not exist for any z 0 ∈ C {displaystyle z_{0}in mathbb {C} } and on any branch (indeed, for any chosen branch the numerator takes the same value). However, the Wirtinger derivatives may be applied to it. Begin from the logarithmic identity.

ln ⁡ ( z ) = Ln ⁡ | z | + i arg ⁡ ( z ) {displaystyle ln(z)=operatorname {Ln} |z|+iarg(z)}

Use that | z | 2 = z z ¯ {displaystyle |z|^{2}=z{bar {z}}} .

ln ⁡ ( z ) = 1 2 Ln ⁡ ( z ) + 1 2 Ln ⁡ ( z ¯ ) + i arg ⁡ ( z ) {displaystyle ln(z)={frac {1}{2}}operatorname {Ln} (z)+{frac {1}{2}}operatorname {Ln} ({bar {z}})+iarg(z)}

Performing different branch cuts lets one apply the derivatives to all points of the principal logarithm in two goes. First, apply ∂ ∂ z {displaystyle {frac {partial }{partial z}}} .

1 z = 1 2 1 z + 0 + i ∂ ∂ z arg ⁡ ( z ) {displaystyle {frac {1}{z}}={frac {1}{2}}{frac {1}{z}}+0+i{frac {partial }{partial z}}arg(z)}

Rearranging gives ∂ ∂ z arg ⁡ ( z ) = − i 2 z {displaystyle {frac {partial }{partial z}}arg(z)={frac {-i}{2z}}} . Now apply ∂ ∂ z ¯ {displaystyle {frac {partial }{partial {bar {z}}}}} .

0 = 0 + 1 2 1 z ¯ + i ∂ ∂ z ¯ arg ⁡ ( z ) {displaystyle 0=0+{frac {1}{2}}{frac {1}{bar {z}}}+i{frac {partial }{partial {bar {z}}}}arg(z)}

This shows that ∂ ∂ z ¯ arg ⁡ ( z ) = i 2 z ¯ {displaystyle {frac {partial }{partial {bar {z}}}}arg(z)={frac {i}{2{bar {z}}}}}

As both Wirtinger derivatives exist, one can then apply the Dolbeault derivatives.

∂ arg ⁡ ( z ) = − i 2 z d z {displaystyle partial arg(z)={frac {-i}{2z}},dz}

∂ ¯ arg ⁡ ( z ) = i 2 z ¯ d z ¯ {displaystyle {bar {partial }}arg(z)={frac {i}{2{bar {z}}}},d{bar {z}}}

This then means the exterior derivative exists, given by their sum.

d arg ⁡ ( z ) = − i 2 z d z + i 2 z ¯ d z ¯ {displaystyle darg(z)={frac {-i}{2z}},dz+{frac {i}{2{bar {z}}}},d{bar {z}}}

This is an important differential form, as it is precisely the generator of the first de Rham cohomology of the circle, H d R 1 ( S 1 ) {displaystyle H_{dR}^{1}(S^{1})} . In other words,

d arg ⁡ ( z ) = − i 2 z d z + i 2 z ¯ d z ¯ = x d y − y d x x 2 + y 2 {displaystyle darg(z)={frac {-i}{2z}},dz+{frac {i}{2{bar {z}}}},d{bar {z}}={frac {x,dy-y,dx}{x^{2}+y^{2}}}}

making this form closed but not exact in the usual sense (a byproduct of being multivalued).

∮ S 1 d arg ⁡ ( z ) = 2 π ≠ 0 {displaystyle oint limits _{S^{1}}darg(z)=2pi neq 0}

with the standard orientation.

• Argument at Encyclopedia of Mathematics.