Hermitian function

Hermitian function

In mathematical analysis, a Hermitian function is a complex function with the property that its complex conjugate is equal to the original function with the variable changed in sign:

:f(-x) = overline{f(x)}

for all x in the domain of of f.

This definition extends also to functions of two or more variables, e.g., in the case that f is a function of two variables it is Hermitian if

:f(-x_1, -x_2) = overline{f(x_1, x_2)}

for all pairs (x_1, x_2) in the domain of f.

From this definition it follows immediately that, if f is a Hermitian function, then

* the real part of f is an even function
* the imaginary part of f is an odd function

Motivation

Hermitian functions appear frequently in mathematics and signal processing. As an example, the following statements are important when dealing with Fourier transforms:

* The function f is real-valued if and only if the Fourier transform of f is Hermitian.

* The function f is Hermitian if and only if the Fourier transform of f is real-valued.

See also

* Even function
* Odd function


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