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mathematical analysis > complex analysis > complex function > Hermitian function

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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 ${\\displaystyle ^{*}}$ ${\\displaystyle ^{*}}$ indicates the complex conjugate) for all ${\\displaystyle x}$ ${\\displaystyle x}$ in the domain of ${\\displaystyle f}$ ${\\displaystyle f}$ . 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 ${\\displaystyle f}$ ${\\displaystyle f}$ is a function of two variables it is Hermitian if
${\\displaystyle f^{*}(x_{1},x_{2})=f(-x_{1},-x_{2})}$
for all pairs ${\\displaystyle (x_{1},x_{2})}$ ${\\displaystyle (x_{1},x_{2})}$ in the domain of ${\\displaystyle f}$ ${\\displaystyle f}$ . From this definition it follows immediately that: ${\\displaystyle f}$ ${\\displaystyle f}$ 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.

complex function

fonction hermitienne

francés

URI

http://data.loterre.fr/ark:/67375/PSR-BXW2PWFS-H

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