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.

  • 6
    $\begingroup$ 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. $\endgroup$
    – Domen
    Apr 27 at 9:18
  • $\begingroup$ 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]} ? $\endgroup$ May 5 at 10:38

1 Answer 1


If you re looking for a numerical solution:

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}]

Plot3D[int[cx, R], {cx, 0, 5}, {R, 0, 10} ]

enter image description here


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