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]. 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.
An argument of the complex number z = x + iy, denoted arg(z), is defined in two equivalent ways:
The names magnitude, for the modulus, and 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 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.
The complex argument can also be defined algebraically in terms of complex roots as:
arg ( z ) = lim n → ∞ n ⋅ Im z / | z | n {\displaystyle \arg(z)=\lim _{n\to \infty }n\cdot \operatorname {Im} {\sqrt{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 {\displaystyle z} by dividing by | z | {\displaystyle |z|} isn't necessary for convergence to the correct value, but it does speed up convergence and ensures that arg ( 0 ) {\displaystyle \arg(0)} is left undefined.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 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 .
In some sources the argument is defined as Arg ( x + i y ) = arctan ( y / x ) , {\displaystyle \operatorname {Arg} (x+iy)=\arctan(y/x),} however this is correct only when x > 0, where y / x {\displaystyle y/x} is well-defined and the angle lies between − π 2 {\displaystyle -{\tfrac {\pi }{2}}} and π 2 . {\displaystyle {\tfrac {\pi }{2}}.} 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 half-plane x > 0 and the two quadrants with x < 0, and then patch the definitions together:
Arg ( x + i y ) = atan2 ( y , x ) = { arctan ( y x ) if x > 0 , arctan ( y x ) + π if x < 0 and y ≥ 0 , arctan ( y x ) − π if x < 0 and y < 0 , + π 2 if x = 0 and y > 0 , − π 2 if x = 0 and y < 0 , undefined if x = 0 and y = 0. {\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}}}See atan2 for further detail and alternative implementations.
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|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
Arg ( z 1 z 2 ) ≡ Arg ( z 1 ) + Arg ( z 2 ) ( mod R / 2 π Z ) , Arg ( z 1 z 2 ) ≡ Arg ( z 1 ) − Arg ( z 2 ) ( mod R / 2 π Z ) . {\displaystyle {\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 ≠ 0 and n is any integer, then
Arg ( z n ) ≡ n Arg ( z ) ( mod R / 2 π Z ) . {\displaystyle \operatorname {Arg} \left(z^{n}\right)\equiv n\operatorname {Arg} (z){\pmod {\mathbb {R} /2\pi \mathbb {Z} }}.}From z = | z | e i Arg ( z ) {\displaystyle z=|z|e^{i\operatorname {Arg} (z)}} , we get i Arg ( z ) = ln z | z | {\displaystyle i\operatorname {Arg} (z)=\ln {\frac {z}{|z|}}} , alternatively Arg ( z ) = I m = I m {\displaystyle \operatorname {Arg} (z)=Im=Im} . 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 extended argument of a number z (denoted as arg ¯ ( z ) {\displaystyle {\overline {\arg }}(z)} ) is the set of all real numbers congruent to arg ( z ) {\displaystyle \arg(z)} modulo 2 π {\displaystyle \pi } .
arg ¯ ( z ) = arg ( z ) + 2 k π , ∀ k ∈ Z {\displaystyle {\overline {\arg }}(z)=\arg(z)+2k\pi ,\forall k\in \mathbb {Z} }