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In my research I have come across the following integral

\begin{equation} f = \int_0^{2\pi} \text{d}\theta \exp\left\{\frac{3}{2}(h_1 \cos^2\theta + h_2 \sin^2\theta + 2 h_{12} \sin\theta \cos\theta)\right\} \end{equation} where $h_1$, $h_2$ and $h_{12}$ are constants of order unity, but which can be either positive or negative.

Eventually, I will have to evaluate this integral numerically a large number of times. Hence, it would be quite interesting if it could be written in terms of special functions (if $h_2 = h_{12} = 0$, for instance, it can be written in terms of Bessel Functions).

However, I was unable to figure out how.

Any suggestions?

More generally, do you have any tips on how to implement it numerically efficiently?

Thank you very much in advance. Best regards,


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Just curious --- what research did you get this integral from? I am always interested how one obtains interesting analytical integrals in practice. – Ondřej Čertík May 20 '13 at 17:48
up vote 5 down vote accepted

Using the half angle formulas you can convert the exponent into the form $$a + b \cos 2\theta + c \sin 2\theta$$ which then integrates nicely into Bessel functions. Mathematica gives $$\int_0^{2\pi}e^{a + b \cos 2\theta + c \sin 2\theta} d\theta = e^a \pi I_0\left(\sqrt{b^2 + c^2}\right)$$

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This works. Awsome. Thanks! – Gabriel Landi Nov 12 '12 at 16:40

Do you get anywhere by noticing that $h_1\cos^2\theta + h_2\sin^2\theta +2h_{12}\sin\theta\cos\theta$ can be written in the form $(h_1\cos(\theta)+a\sin\theta)(\cos(\theta)+b\sin\theta))$? A second step would be to realize that $h_1 \cos(\theta)+a\sin\theta = \cos(\theta-\psi)$ for some $\psi$. This helps you because the integral is over $\theta=0\ldots 2\pi$, so you can shift the entire integration interval by $\psi$ and hopefully come out with something that has a better structured exponent.

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That's good advice. I'll work on it. Thank you very much. – Gabriel Landi Nov 12 '12 at 0:01

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