Considering scalar complex-valued functions $f(\theta)\in\mathbb C$ for $\theta\in[0,2\pi)$ on a circle, what is the name (if there is one) for symmetry after rotating $\theta$ by $\pi$ and conjugation?

\begin{equation} f(θ) = \operatorname{conj}[f(\theta+\pi)] \label{eq1} \qquad\qquad\qquad(1) \end{equation}

Is it a symmetry group with a common name and notation? ("no" is a fine answer too!)

Edit: I ran into this when considering the symmetry of Fourier transforms of 2D real-valued signals, evaluated along a ring in the frequency domain with a constant wavelength. So, it's another flavor of the more familiar conjugate-symmetry for 1D Fourier transforms of real-valued signals. I don't know it's name, though.

Maybe another way to ask this is: What notations are available for talking about the set of functions satisfying $(1)$ abstractly? For recognizing other algebraic structures that may share similar properties?

  • 1
    $\begingroup$ Thinking of $[0,2\pi)$ as the unit circle $S^1\subset \mathbb{C}$, this is doing a reflection of it with respect to the $Y$-axis. $\endgroup$
    – plop
    Commented Sep 6, 2022 at 13:59
  • 1
    $\begingroup$ @user85667, the line of antisymmetry is arbitrary. For example, $f = i$ for the part of the circle above $\theta = \pi/4$, $-i$ for part below $\theta = \pi/4$. The line of reflection symmetry will be the perpendicular bisector of this line. The example of the $y$-axis is one possibility, and corresponds to the case when $\theta = 0$ is line of antisymmetry. $\endgroup$
    – Doug
    Commented Sep 6, 2022 at 14:05


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