Argument (complex analysis)

In mathematics, arg is a function operating on complex numbers (visualized in a complex plane). It gives the angle between the positive real axis to the line joining the point to the origin, shown as φ in figure 1, known as an argument of the point.

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 (or amplitude) 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. 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 by circling the origin any number of times. This is shown in figure 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 (−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.

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