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Approximating integral of product of gaussian and cosines

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I am trying to evaluate the following integral\begin{equation}I=\int_{-\infty}^{\infty}\exp\left(-\frac{x^2}{b}\right)\prod_i^N\cos{\left(a_ix\right)}dx,\end{equation}

where $b>0$ and $a_i$ are integers between 1 and $A$ and $A,N$ are positive integers. I am looking for a way to approximate this integral. So far I have an approximation for small $b$. In this case the exponential dominates and the function is non-zero over a small window around zero. We Taylor expand the cosines to get

\begin{equation}\int_{-\infty}^{\infty}\exp\left(-\frac{x^2}{b}\right)\prod_i^N\left(1-a_i^2x^2\right)dx,\end{equation}then if we retain only terms up to $x^2$ we get\begin{equation}\int_{-\infty}^{\infty}\exp\left(-\frac{x^2}{b}\right)\left(1-\frac{x^2}{2}\left(\sum_ia_i^2\right)\right)dx.\end{equation}

Evaluating this integral gives us the approximation\begin{equation}I\approx\sqrt{\pi b}-\frac{\sqrt{\pi}b^{3/2}}{4}\left(\sum_ia_i^2\right).\end{equation}

Now I am at a loss for approximating this for large $b$. In this case the exponential will very slowly decay and getting rid of the product of cosines isn't as easy.


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