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.


1 Answer 1


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

  • $\begingroup$ Thank you Carlo! Is there any further simplification if $\nu$ can only be an integer? $\endgroup$
    – Alex
    Jul 16 at 19:12
  • $\begingroup$ 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. $\endgroup$ Jul 16 at 20:26
  • $\begingroup$ Hi, what would the integral be in that case? $\endgroup$
    – Alex
    Jul 16 at 20:46
  • $\begingroup$ What if $\nu=1$? $\endgroup$
    – Alex
    Jul 16 at 20:47
  • 1
    $\begingroup$ for half-integer order you can use the recursion relation for spherical Bessel functions given in arxiv.org/abs/1703.06428 ; no simplification if $\nu=1$. $\endgroup$ Jul 16 at 20:59

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