Solving Bessel integration What would be the solution of the bessels equation,
$$b=k A(t)\int_0^{\infty} J_0 (k \rho) e^ \frac{-\rho^2}{R^2} \rho d \rho$$
Can I sove that by using this formulation?
$$c= \int_0^{\infty}j_0(t) e^{-pt} dt= \frac{1}{\sqrt{1+p^2}}$$

According to Gradshteyn and Ryzhik, we have:
$$\int_0^{\infty}x^{\mu}\exp(-\alpha x^2)J_{\nu}(\beta x)dx = \frac{\beta^{\nu}\Gamma\left(\frac{1}{2}\nu+\frac{1}{2}\mu+\frac{1}{2}\right)}{2^{\nu+1}\alpha^{\frac{1}{2}(\mu+\nu+1)}\Gamma(\nu+1)}\mbox{}_1 F_1\left(\frac{\nu+\mu+1}{2};\mbox{ }\nu+1;\mbox{ }-\frac{\beta^2}{4\alpha}\right).$$
Here $\mbox{}_1F_1$ is a hypergeometric function. Inputting the proper values gives
$$\int_0^{\infty}\rho\exp\left(-\frac{\rho^2}{R^2}\right)J_0(k\rho)d\rho = \frac{\Gamma(1)}{2\frac{1}{R^2}\Gamma(1)}\mbox{}_1F_1\left(1;1;-\frac{k^2R^2}{4}\right).$$
Using a property of the hypergeometric function ($_1F_1(a;a;x) = \exp(x)$) we get:
$$\frac{R^2}{2}\exp\left(-\frac{k^2R^2}{4}\right).$$
