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:

(where the  indicates the complex conjugate) for all  in the domain of . 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  is a function of two variables it is Hermitian if

for all pairs  in the domain of . From this definition it follows immediately that:  is a Hermitian function if and only if

 the real part of  is an even function, 
 the imaginary part of  is an odd function. Motivation 
Hermitian functions appear frequently in mathematics, physics, and signal processing. For example, the following two statements follow from basic properties of the Fourier transform:

 The function  is real-valued if and only if the Fourier transform of  is Hermitian. The function  is Hermitian if and only if the Fourier transform of  is real-valued. Since the Fourier transform of a real signal is guaranteed to be Hermitian, it can be compressed using the Hermitian even/odd symmetry. This, for example, allows the discrete Fourier transform of  a signal (which is in general complex) to be stored in the same space as the original real signal. If f is Hermitian, then . Where the  is cross-correlation, and  is convolution. If both f and g are Hermitian, then . See also 

 
 

Types of functions
Calculus