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