Hermitian function

From Wikipedia, the free encyclopedia

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

[edit] Motivation

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