# integration – Gaussian integral with a sine in the exponential

This Gaussian integral came up while working on a likelihood analysis for pulsar timing arrays:

$$int_{-pi}^{pi} expleft( -(x-y sin{gamma} )^2 right) mathrm{d}gamma$$

I’ve tried everything I can think of, but I can’t get an analytic solution.

If you expand the square, the cross term can be written in terms of a cosh, but I don’t know where to go from there. That gives the integrand as $$exp{(-x^2)} exp{(-y^2 sin^2gamma)} cosh{(2 x y sin{gamma})}$$.

The substitution $$betaequivsin{gamma}$$ leads to
$$oint_0^0 frac{expleft( -(x-y beta )^2 right)}{sqrt{1-beta^2}} mathrm{d}beta$$

I also tried $$arctan alpha equiv gamma$$, which gives

$$int frac{expleft(-left( x- frac{alpha y}{sqrt{1+alpha^2}} right)^2right)}{1+alpha^2} mathrm{d}alpha$$

Neither Mathematica nor Rubi are able to evaluate any of these integrals. I was hoping the residue theorem might be applicable, or one of these substitutions might get into a form that Mathematica knows. Any help will be greatly appreciated and certainly land you in the acknowledgements of the paper my collaborators and I are working on:)

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