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) = f(x)^{*} \$

for all $x \$ in the domain of definition of $f \$ .

$f(x)^{*} \$ is the complex conjugate of $f(x) \$ .

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) = f(x_1, x_2)^{*} \$

for all $x_1, x_2 \$ in the domain of definition of $f \$ .

From this definition follows immediately that $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: