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:

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

(where the ${\displaystyle ^{*}}$ indicates the complex conjugate) for all ${\displaystyle x}$ in the domain of ${\displaystyle f}$. In physics, this property is referred to as PT symmetry.

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

${\displaystyle f^{*}(x_{1},x_{2})=f(-x_{1},-x_{2})}$

for all pairs ${\displaystyle (x_{1},x_{2})}$ in the domain of ${\displaystyle f}$.

From this definition it follows immediately that: ${\displaystyle f}$ is a Hermitian function if and only if

• the real part of ${\displaystyle f}$ is an even function,
• the imaginary part of ${\displaystyle f}$ is an odd function.