Numerical evaluation of integrals containing the spherical Bessel function is of importance in many fields
of computational science and engineering since the spherical Bessel function is often used as the
eigenfunction for spherical coordinate systems. The integrals are known as the spherical Bessel
transform (SBT) which is classified into a more general family of the Hankel or Fourier-Bessel transforms.
One can find that SBT is involved in many physical problems such as the scattering in atomic or nuclear systems,
the simulation of the cosmic microwave background, and the interaction of electrons in molecules
and crystals.
We propose a new method for the numerical evaluation of the spherical Bessel transform whose
computational cost scales as O(*N log _{2}N*) [1].
A formula is derived for the transform by using an integral representation of the spherical
Bessel function and by changing the integration variable. The resultant algorithm consists
of a set of the Fourier transforms and numerical integrations over a linearly spaced grid
of variable k in Fourier space. Because the -dependence appears in the upper limit of the
integration range, the integrations can be performed effectively in a recurrence formula.
We expect that this numerical technique will be useful for a wide variety of problems
appearing in many scientific and engineering fields.

- "Fast spherical Bessel transform via fast Fourier transform and recurrence formula", M. Toyoda and T. Ozaki, Comp. Phys. Comm. 181, 277 (2010).
- "Numerical Fourier and Bessel transforms in logarithmic variables", J.D. Talman, J. Comput. Phys. 29, 35 (1978).