Argument (complex analysis)

In mathematics, the 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 φ 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 (–π, π]. In mathematics, the 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 φ 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 (–π, π]. 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. Because a complete rotation around the origin leaves a complex number unchanged, there are many choices which could be made for φ 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 (equivalently 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: