The eigenvectors of the derivative operator are the exponential function. The derivative operator is not Hermitian, so we get complex eigenvalues.
$\frac{d}{dx} f(x) = \lambda f(x) \Rightarrow f(x)=Ce^{\lambda x}$
what is the solution of the eigenvalue problem: $e^{zcos(x)}$
$\frac{d}{dx} e^{zcosx} = -ze^{zcosx}sin(x)$
$-\frac{1}{sinx}\frac{d}{dx} e^{zcosx} = ze^{zcosx}$
seeking for the eigenvectors. I don't see the solution.
I am very lost. Thank you.