Integral over spherical Bessel function

[hmhm696]

Is there somebody who can help me how to solve this integral

$$\int_{0}^{+\infty} dr r^{^{n+1}} e^{-\alpha r} j_l(kr)$$

 

[dextercioby]

I have a saying for this type of questions. If it's not in one of the Gradshteyn-Rytzhik editions of their famous book, then it must be discovered. :)

 

[Dickfore]

This is the answer Mathematica gives:

If[Re[a] >
0 && (Abs[Im[k]] <
Re[a] || (Abs[Im[k]] == Re[a] && Re[n] < 0)) && (Re[k] >
0 || (Im[k] > 0 && Re[k] == 0)),
2^(-1 - l) a^(-2 - n) (k^2/a^2)^(l/2) Sqrt[\[Pi]]
Gamma[2 + l + n] Hypergeometric2F1Regularized[1/2 (2 + l + n),
1/2 (3 + l + n), 3/2 + l, -(k^2/a^2)],
Integrate[
E^(-a r) r^(1 + n) SphericalBesselJ[l, k r], {r, 0, \[Infinity]},
Assumptions ->
Abs[Im[k]] > Re[a] || (Abs[Im[k]] >= Re[a] && Re[n] >= 0) ||
Re[k] < 0 || (Re[k] <= 0 && Im[k] <= 0) || Re[a] <= 0]]

 

[dextercioby]

I think you can relate the spherical Besselfunction to the normal J Bessel function by the definition:

http://functions.wolfram.com/Bessel-TypeFunctions/SphericalBesselJ/02/

and then use formula attached below which is taken from G & R

 

[hmhm696]

Thanks guys.
I think that given formula has error in the part where it derivative by alpha instead of betha.
For me is important a process, how i can get it.