Arg

Notations

Traditional name

Traditional notation

Mathematica StandardForm notation

Primary definition

Arg(z) is the argument of z, such that z==|z| (e^(i Arg(z))). The argument of a complex number z is the phase angle (in radians) that the line from 0 to z makes with the positive real axis.

Specific values

Specialized values

Values at fixed points

Values at infinities

General characteristics

Domain and analyticity

Arg(z) is a nonanalytical function; it is a real-analytic function of the complex variable z for z≠0.

Symmetries and periodicities

Parity

Arg(z) is an odd function for almost all z.

Mirror symmetry

Periodicity

No periodicity

Homogeneity

Sets of discontinuity

The function Arg(z) is a single-valued, continuous function on the z-plane cut along the interval (-∞, 0), where it is continuous from above.

Transformations

Transformations and argument simplifications

Argument involving complex characteristics

Argument involving basic arithmetic operations

Addition formulas

Multiple arguments

For products

For quotients

Power of arguments

Exponent of arguments

Some functions of arguments

Products, sums, and powers of the direct function

Sums of the direct function

Differences of the direct function

Linear combinations of the direct function

Related transformations

Complex characteristics

Real part

Imaginary part

Absolute value

Argument

Conjugate value

Signum value

Representations through equivalent functions

With related functions

With Re

With Im

With Abs

With Conjugate

With Sign

With inverse trigonometric functions

With ArcSin

With ArcCos

With ArcTan

With inverse hyperbolic functions

Inequalities

Pavlyk O. (2006)

Zeros

History

Arg is encountered often in mathematics and the natural sciences.

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