Hermitian function

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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( − x₁, − x₂) = f(x₁,x₂) ^*

for all x₁,x₂ 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:

• The function f is real-valued the Fourier transform of f is Hermitian.

• The function f is Hermitian the Fourier transform of f is real-valued.