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I need help in solving the following definite integral. I could not find any example like this $\int_{0}^{2\pi}\int_{0}^{d}\exp\!\Big(\frac{-r^2 +2\alpha\; r\cos\theta}{4\;\sigma^2}\Big)r\; dr\; d\theta$

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    I have changed the above problem to the definite integral using cdf, here is the link. http://math.stackexchange.com/questions/61103/definite-integral-of-cdf-of-the-form-phi-alpha-sqrtd2-fracx22-sigma2. Help needed...2011-09-01

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Integration with respect to $\theta$ can be carried out explicitly in terms of modified Bessel function of the first kind.

$ \int_0^{2 \pi} \exp\left( \frac{ \alpha r}{2 \sigma^2} \cos \theta \right) \, \mathrm{d} \theta = 2 \pi I_0 \left( \frac{ \alpha r}{2 \sigma^2} \right) $

The remaining integral

$ \mathcal{I}_d = 2 \pi \int_0^d r \cdot \mathrm{e}^{-\frac{r^2}{4 \sigma^2}} \cdot I_0 \left( \frac{ \alpha r}{2 \sigma^2} \right) \mathrm{d} r = 4 \pi \sigma^2 \exp\left( \frac{\alpha^2}{4 \sigma^2} \right) \left(1 - Q_1\left( \frac{\alpha}{\sqrt{2} \sigma}, \frac{d}{\sqrt{2} \sigma} \right)\right) $ where $Q_1(a,b)$ is the Marcum Q-function.

Notice that $ \lim_{d \to \infty} \mathcal{I}_d = 4 \pi \sigma^2 \exp\left( \frac{\alpha^2}{4 \sigma^2} \right) $

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    Marcum Q-function is [supported](http://reference.wolfram.com/mathematica/ref/MarcumQ.html?q=MarcumQ) in _Mathematica_ if you need to work with it numerically. Otherwise use the mathematical properties it has.2011-09-01