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Concept
Argument (complex analysis)
Summary
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 in Figure 1.
It is a multivalued function operating on the nonzero complex numbers.
To define a single-valued function, the principal value of the argument (sometimes denoted Arg z) is used. It is often chosen to be the unique value of the argument that lies within the interval .
An argument of the complex number z = x + iy, denoted arg(z), is defined in two equivalent ways:
Geometrically, in the complex plane, as the 2D polar angle 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.
Algebraically, as any real quantity such that for some positive real r (see Euler's formula). The quantity r is the modulus (or absolute value) of z, denoted |z|:
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 [[sine|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:
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 by dividing by isn't necessary for convergence to the correct value, but it does speed up convergence and ensures that is left undefined.
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