Argument (complex analysis)


In mathematics,[sine|] the argument of a complex number, denoted, is the angle between the positive real axis and the line joining the origin and, represented as a point in the complex plane, shown as 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. In this article the multi-valued function will be denoted and its principal value will be denoted, 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, 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 While this term is largely outdated in modern texts, it still appears in some regional educational resources, where it is sometimes used in introductory-level textbooks.

Definition

An argument of the nonzero complex number, denoted, is defined in two equivalent ways:
  1. Geometrically, in the complex plane, as the 2D polar angle from the positive real axis to the vector representing. The numeric value is given by the angle in radians, and is positive if measured counterclockwise.
  2. Algebraically, as any real quantity such that
for some positive real . The quantity is the modulus of, denoted ||:
The argument of zero is usually left undefined. 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 radians are the same, as reflected by figure 2 on the right. Similarly, from the periodicity of
and cosine|, the second definition also has this property.

Principal value

Because a complete rotation around the origin leaves a complex number unchanged, there are many choices which could be made for by circling the origin any number of times. This is shown in figure 2, a representation of the multi-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 radians, that is from to radians excluding radians 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.

Notation

The principal value sometimes has the initial letter capitalized, as in, especially when a general version of the argument is also being considered. Note that notation varies, so and may be interchanged in different texts.
The set of all possible values of the argument can be written in terms of as:

Computing from the real and imaginary part

If a complex number is known in terms of its real and imaginary parts, then the function that calculates the principal value is called the two-argument arctangent function, :
The function is available in the math libraries of many programming languages, sometimes under a different name, and usually returns a value in the range.
See atan2 for further detail and alternative implementations, such as
which works except when and, in which case the value is instead if or undefined when.

Realizations of the function in computer languages

Wolfram language (Mathematica)

In Wolfram language, there's Arg:
or using the language's ArcTan:
ArcTan is extended to work with infinities. ArcTan is Indeterminate, while ArcTan doesn't return anything.

Maple

's argument behaves the same as Arg in Wolfram language, except that argument also returns if z is the special floating-point value −0..
Also, Maple doesn't have.

MATLAB

's angle behaves the same as Arg in Wolfram language, except that it is
Unlike in Maple and Wolfram language, MATLAB's atan2 is equivalent to angle. That is, atan2 is.

Identities

One of the main motivations for defining the principal value is to be able to write complex numbers in modulus-argument form. Hence for any complex number,
This is only really valid if is non-zero, but can be considered valid for if is considered as an indeterminate form—rather than as being undefined.
Some further identities follow. If and are two non-zero complex numbers, then
where means to add or subtract any integer multiple of if necessary to bring the value into the interval of radians.
If and is any integer, then

Example

Using the complex logarithm

From, we get, alternatively. 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.

Differentiation

The argument function is not complex differentiable, as the limit
does not exist for any and on any branch. However, the Wirtinger derivatives may be applied to it. Begin from the logarithmic identity.
Use that.
Performing different branch cuts lets one apply the derivatives to all points of the principal logarithm in two goes. First, apply.
Rearranging gives. Now apply.
This shows that
As both Wirtinger derivatives exist, one can then apply the Dolbeault derivatives.
This then means the exterior derivative exists, given by their sum.
This is an important differential form, as it is precisely the generator of the first de Rham cohomology of the circle,. In other words,
making this form closed but not exact in the usual sense.
with the standard orientation.