# How to solve this integral involving a Bessel function and an exponential? [closed]

I have simplified a problem and now have to calculate the following integral:

$$E(x) = \int_0^R dr r e^{-r^2} J_0(x r c)$$

Here, $$J_0$$ is the Bessel function of the zeroth order and $$c$$ is a constant.

I know how to solve this without the exponential but not how to solve it with the exponential.

• If this is a question about how to solve it with Wolfram Mathematica software, please include the relevant code. Otherwise, this belongs to Math.SE. Apr 27 at 9:18
• Try:f[a_, R_, M_] := 1/2 E^(-(a^2/4)) - Sum[-((2^(-1 - m) (-1 + (-1)^m) (I a)^(-1 + m) Gamma[1/4 (-1)^m (-1 + (-1)^m) (1 + m), R^2])/ Gamma[(1 + m)/2]^2), {m, 0, M}] // N; g[a_, R_] := NIntegrate[r*Exp[-r^2] BesselJ[0, a*r], {r, 0, R}]; {f[2, 4, 100], g[2, 4]} ? May 5 at 10:38

Only product c x occurs, that's why it's useful to introduce a new variable cx
int[cx_?NumericQ, R_?NumericQ] :=NIntegrate[r Exp[-r^2] BesselJ[0, r cx],{r, 0, R}] 