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. The names magnitudefor the modulus, and phase[3] [1] 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 costhe second definition also has this property. The argument of zero is usually left undefined. Because it's defined in terms of rootsit also inherits the principal branch of square root as its own principal branch.
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, 2 π. The principal value sometimes has the initial letter capitalized, as in Arg zespecially 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:. If a complex number is known in terms of its real and imaginary parts, then the function that calculates the principal value Arg is called the two-argument arctangent function atan2 :.
Alternatively, the principal value can be calculated in a uniform way using the tangent half-angle formulathe function being defined over Blandat och udda complex plane but excluding the origin:. This is based on a parametrization of the circle except for the negative x -axis by rational functions.
A variant of the last formula which avoids overflow is sometimes used in high precision computation:. One of the main motivations for defining the principal value Arg is to be able to write complex numbers in modulus-argument form.
Hence for any complex number z. Some further identities follow. If z 1 and z 2 are two non-zero complex numbers, then. This is useful when one has the complex logarithm available. Contents move to sidebar hide. Article Talk. Read Edit View history. Tools Tools.