Approximation for a Bessel function integral

Question

I'm trying to calculate hit probabilities on a dart board if the dart thrower has some Gaussian angle distribution function with width $\Delta$ and some systematic angle offsets $\phi_0, \theta_0$. This is a problem related to my work.

The distribution function is

$p(\theta, \phi) = \frac{1}{2 \pi \Delta^2} e^{-\frac{(\phi-\phi_0)^2}{2\Delta^2}}e^{-\frac{(\theta-\theta_0)^2}{2\Delta^2}}$

To get the hit probability on the dart boad, this distribution function must be integrated over a circle centered around 0 with some radius $R$ at a far away distance $D$.

So far I turned in into an integral in circle coordinates and now I'm stuck with this bad boy:

$p_{tot} = e^{-\rho_0^2} \int_0^{\frac{R}{\sqrt{2} D \Delta}} d\rho \rho e^{-\rho^2} I_0(2\rho \rho_0)$

where $\rho = \sqrt{\phi^2 + \theta^2}$ and $\rho_0$ equivalently. $I_0$ is the 0th modified Bessel function.

Any ideas on how to properly evaluate this? I'd be happy about a reasonable approximation. Taylor expansions don't really work because the function in the integral has a peak that shifts to the right with increasing $\rho_0$. The area under the peak will grow like $e^{\rho_0^2}$ which can lead to some numerical issues but I guess I can approximate it with a delta distribution for really large $\rho_0$.

Answer 1

Q: The OP seeks a "reasonable approximation" for large or small $\rho_0$ of the function $$P(x)= \int_0^{x} e^{-\rho^2-\rho_0^2} \rho I_0(2\rho \rho_0)\,d\rho,\;\;x\geq 0.$$

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Consider the kernel $$p_{\rm exact}(\rho)=e^{-\rho^2-\rho_0^2} \rho I_0(2\rho \rho_0).$$ For small $\rho_0$ a Taylor series in powers of $\rho_0$ is accurate, $$p_{\rm small}(\rho)=e^{-\rho^2} \rho (1+\rho^2 \rho_0^2-\rho_0^2).$$ For large $\rho_0$ an asymptotic expansion of the Bessel function gives $$p_{\rm large}(\rho)=\frac{1}{2\sqrt{\pi}}e^{-\rho^2-\rho_0^2}e^{2\rho\rho_0}.$$

In the plots below I compare $\int_0^x p(\rho)\,d\rho$ for the exact expression (blue) and the approximation (orange). It turns out the approximations $p_{\rm small}$ and $p_{\rm large}$ are already quite accurate for $\rho_0\lesssim 0.5$ and $\rho_0\gtrsim 3$, respectively.

Left plot: comparison of $\int_0^x p_{\rm exact}(\rho)\,d\rho$ and $\int_0^x p_{\rm small}(\rho)\,d\rho$ for $\rho_0=0.5$. Right plot: comparison of $\int_0^x p_{\rm exact}(\rho)\,d\rho$ and $\int_0^x p_{\rm large}(\rho)\,d\rho$ for $\rho_0=3$. The approximations (orange) are almost indistinguishable from the exact answer (blue).