As of November 19, 2023, the Wikipedia page for even and odd functions defines odd/even symmetric complex-valued functions as below:
even: $f(x)=\overline{f(-x)}$,
odd: $f(x)=-\overline{f(-x)}$.
What is the motivation for this definition? It is kind of strange to me as it says that for a complex-valued function to be even, its real and imaginary parts must be even and odd, respectively. Similarly, for a complex-valued function to be odd, its real and imaginary parts must be odd and even, respectively.
In the Gaussian plane, complex conjugation is reflection $P_1: e_i \to -e_{-i} =-e_i$ wrt to the real axis, changing the class holomorphic to antiholmorphic, while $R: z\to -z$ is a rotation by $\pi$ that does not change the class.
The combination $ P_i = R P_1 = P_1 R$ is the reflection wrt to the imaginary axis $e_1\to e_{-1}$, that defines the real classes of of even and odd functions with class change holomorphic to antiholomorphic.
$$\text{evenQ}(f) : f(z)=\overline{f(-z)} =( P_1\ f\ R)\ z : \quad P_1\ f \ R = f; \quad f \ R = P_1 \ f $$ $$\text{oddQ}(f) : f(z)=-\overline{f(-z)} =( P_i\ f \ R)\ z : \quad P_i\ f \ R = f; \quad P_i\ f = f \ R $$