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Is there an easy way to compute the definite integral \begin{align*} \int_{-\pi}^{\pi} x^{n}e^{\mathrm{i}a x^2-\mathrm{i}b x}\,dx, \end{align*} where $ n\in \mathbb{N} $ and $ a,b\in \mathbb{R} ?$ I found some formulas for $\int_{-\infty}^{\infty}$ or $\int_{0}^{\infty}$ but I want to solve for finite limits.

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I don't think it's doable, but perhaps it makes your life a bit simpler to realize that \begin{align*} I&=\int_{-\pi}^{\pi}e^{\mathrm{i}a x^2-\mathrm{i}b x}\,dx \end{align*}

So that $$\frac{\partial^nI}{\partial b^n}=(-i)^n\int_{-\pi}^{\pi}x^ne^{\mathrm{i}a x^2-\mathrm{i}b x}\,dx.$$

This identity should at least make it easier to get $I$ in terms of special functions.

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