Conjugate

Notations

Traditional name

Traditional notation

Mathematica StandardForm notation

Primary definition

Specific values

Specialized values

Values at fixed points

Values at infinities

General characteristics

Domain and analyticity

(OverBar[z]) is a nonanalytical function. The real and the imaginary parts of (OverBar[z]) are real-analytic functions of the variable z.

Symmetries and periodicities

Parity

(OverBar[z]) is an odd function.

Mirror symmetry

Periodicity

No periodicity

Homogeneity

Sets of discontinuity

The function (OverBar[z]) is continuous function in ℂ.

Transformations

Transformations and argument simplifications

Argument involving basic arithmetic operations

Addition formulas

Multiple arguments

Ratio of arguments

Power of arguments

Exponent of arguments

Products, sums, and powers of the direct function

Products of the direct function

Sums of the direct function

Powers of the direct function

Complex characteristics

Real part

Imaginary part

Absolute value

Argument

Conjugate value

Signum value

Differentiation

Low-order differentiation

Fractional integro-differentiation

Representations through equivalent functions

With related functions

With Re

With Im

With Abs

With Arg

With Sign

Zeros

History

– A. L. Cauchy (1821) (used the word "conjugate" the first time in the current sense)

The function (OverBar[z]) is encountered often in mathematics and the natural sciences.

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