I have encountered the following integral:

\begin{equation} \int_0^{1} e^{-i F(x)} dx, \quad F(x) = \sum_{k=1}^L a_k \sin(2\pi k x) + b_k \cos(2\pi k x). \end{equation} I have found several great sources such that this one, which provided a satisfactory closed-form solution for the integral when $L = 1$.

I have spent a considerable amount of time trying to generalize the approach in that post to the case for arbitrary natural numbers $L > 1$, but have not succeeded so far. I would appreciate it if anyone could give me advice/guidance. Are there closed-form (or compact) solutions for the integral when $L = 2, 3, \ldots$?

  • 1
    $\begingroup$ I would say it’s unlikely for there to be a closed form for generic $F$. What applications would a closed form help you with, if any? $\endgroup$
    – Diffusion
    Commented Jun 5 at 1:35


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