In mathematics (particularly in complex analysis), 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 in Figure1. 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.

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. such that

for some positive real (see Euler's formula). The quantity is the modulus (or absolute value) 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. In some older mathematical texts, the term "amplitude" was used interchangeably with argument to denote the angle 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 While this term is largely outdated in modern texts, it still appears in some regional educational resources, where it is sometimes used in introductory-level textbooks.

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 (a complete turn) 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.

Principal value

by circling the origin any number of times. This is shown in figure2, a representation of the multi-valued, 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 radians, that is from to radians excluding radians itself (equiv., from −180 to +180 degreess, 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 .

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

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, atan2 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 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[z]:

1. Arg[x + y I]

or using the language's ArcTan:

1. Arg[x + y I]

ArcTan[x, y] extended to work with infinities. ArcTan[0, 0] is Indeterminate (i.e. it is still defined), while ArcTan[Infinity, -Infinity] doesn't return anything (i.e. it is undefined).

Maple

Maple's argument(z) behaves the same as Arg[z] in Wolfram language, except that argument(z) if z is the special floating-point value −0...

MATLAB

MATLAB's angle(z) behaves the same as Arg[z] in Wolfram language, except that it is

Unlike in Maple and Wolfram language, MATLAB's atan2(y, x) is equivalent to angle(x + y*1i). That is, atan2(0, 0).

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 is considered as an indeterminate form—rather than as being undefined.

Some further identities follow. If and means to add or subtract any integer multiple of if necessary to bring the value into the interval of

If and is any integer, then

Example

Using the complex logarithm

. As we are taking the imaginary part, any normalization by a real scalar will not affect the result. This is useful when one has the complex logarithm available.

Differentiation

and on any branch (indeed, for any chosen branch the numerator takes the same value). However, the Wirtinger derivatives This then means the exterior derivative This is an important differential form, as it is precisely the generator of the first de Rham cohomology with the standard orientation.

Notes

References

Bibliography

External links

• at Encyclopedia of Mathematics.