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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 ...

Consider the following sum (with $a$ being a real number and $N$ an even integer) $$S(a, N) = \sum_{m=1}^{N/2} \frac{4}{N+1} \sin^2\left( \frac{2\pi m}{N+1} \right)\sin \left( 2 a \cos \left( \frac{2\...

I would like to know if the following integrals are known, or in case that aren't in the literature we can calculate these in closed-form (in terms of elementary and standard functions). I wondered ...