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I am trying to solve the following integral, which may be thought of as an integral of an exponentially-decaying distance from a point $r$ on the $x$-axis to a circumference of a circle of radius $s$.

$$\int_0^{2\pi} \exp\left(-\delta\sqrt{r^2-2\cos(\phi)rs+s^2}\right)\,d\phi$$

Does this even have a closed form?

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    No. But is $r \gg s$ then you can Taylor expand the square root. To first order in the small quantity $s/r$, the integral may be expressed as a Bessel function.2012-12-20

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