# When is a complex function orthogonal to its derivative

Consider the complex valued function $$f(t)$$ where $$t \in [0, 2\pi)$$.

Under what conditions are $$f(t)$$ and $$f'(t)$$ orthogonal to each other? I'm defining orthogonal here to be

$$\int_0^{2\pi} \overline{f(t)} f'(t) dt = 0$$

If $$f(t)$$ is real valued then this old question suggests a method of proof involving a Fourier decomposition, but I don't fully follow the proof and it's not clear to me if it's applicable to complex functions.

• Essentially, you’re dealing with integral of the conjugate, that is $\int\overline{z}dz$, which depending on the values assumed at $0$ and $2\pi$, will be probably a change in argument times an area estimation of the interior components of the image curve. See the following post for an example: math.stackexchange.com/questions/445781/… – Jack LeGrüß Oct 7 '20 at 17:01