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Let $\Gamma \subset \mathbb{C}$ be a smooth curve such that monomials are orthogonal on it, i.e. with $n,m \in \mathbb{N} \cup\{0\}$

$$\int_{\Gamma} z^n \overline{z^m} |dz| = 0, \qquad \qquad \forall n \neq m$$

An example of such curve is a circle around origin. Are there any other examples?

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  • $\begingroup$ Suppose that the curve is parameterized by $r(t) e^{i \theta(t)}$, $0 \leq t \leq 1$. If you plug this into your equations, you get a system of integral / differential equations. Maybe you can mess around with those to show that $r$ must be constant - I don't know if that will be the case or not, as I didn't get anywhere with this approach. $\endgroup$ – Lorenzo May 11 at 23:59
  • $\begingroup$ At least when $\theta(t)=2\pi t$, @Lorenzo's approach should amount to $r$ having all Fourier coefficients (except the constant term) $=0$ ... $\endgroup$ – Hagen von Eitzen May 12 at 9:12

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