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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.

Robert
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  • This question is related, but I still can't realize the motivation for this definition though. https://math.stackexchange.com/questions/3178678/generalization-of-even-odd-functions?rq=1 – Robert Nov 19 '23 at 07:30
  • The definition is certainly non standard (at least for most of physics/maths). Usually even and odd functions (complex valued or not) are defined exactly the same.With this definition even/odd functions would be the eigenfunctions of a conjugate linear operator instead of a linear one (which would usually be more desireable). The involution $I$ in the answer of the linked post is the linear map $f\mapsto (x \mapsto f(-x))$ in this case. If we would add complex conjugation it would not be linear anymore. – jd27 Nov 19 '23 at 08:05

1 Answers1

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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.

$$\text{evenQ}(f) : f(z)=\overline{f(-z)} =( P_1\ f\ R)\ z : \quad P_1\ f \ R = f; \quad f \ R = P_1 \ f $$ $$\text{oddQ}(f) : f(z)=-\overline{f(-z)} =( P_i\ f \ R)\ z : \quad P_i\ f \ R = f; \quad P_i\ f = f \ R $$

Roland F
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