# Definite integral of Bessel function of the first kind times $x^{3/2}$

I am looking for preferably a closed form (or series solution if not possible) for the following integral:

$$\int_0^a x^{3/2} J_\nu (bx) dx$$

where $$\nu$$ is an integer. This 1D integral appears when taking the polar Fourier transform of a separable radially symmetric function in 2D that I would like to propagate using the angular spectrum method. I understand that I can express this as a finite Hankel transform of $$\sqrt{x}$$ but I was hoping there was an analytic solution for this simple case.

The integral requires $$\nu>-5/2$$ for convergence, and then becomes a hypergeometric function: $$\int_0^a x^{3/2} J_\nu (bx) dx=\frac{2^{1-\nu} a^{\nu+\frac{5}{2}} b^{\nu}}{(2 \nu+5) \Gamma (\nu+1)}\, _1F_2\left(\frac{\nu}{2}+\frac{5}{4};\frac{\nu}{2}+\frac{9}{4},\nu+1;-\frac{1}{4} a^2 b^2\right).$$

• Thank you Carlo! Is there any further simplification if $\nu$ can only be an integer?
– Alex
Jul 16 at 19:12
• I don't think you can get rid of the hypergeometric function if $\nu$ is an integer; if $\nu$ is half-integer the integral is obviously elementary. Jul 16 at 20:26
• Hi, what would the integral be in that case?
– Alex
Jul 16 at 20:46
• What if $\nu=1$?
– Alex
Jul 16 at 20:47
• for half-integer order you can use the recursion relation for spherical Bessel functions given in arxiv.org/abs/1703.06428 ; no simplification if $\nu=1$. Jul 16 at 20:59