Editing Hankel transform (section)

== Numerical evaluation ==
A simple and efficient approach to the numerical evaluation of the Hankel transform is based on the observation that it can be cast in the form of a [[convolution]] by a logarithmic change of variables<ref>{{cite journal |last=Siegman |first=A.E. |date=1977-07-01 |title=Quasi fast Hankel transform |journal=Optics Letters |volume=1 |issue=1 |pages=13 |doi=10.1364/ol.1.000013 |pmid=19680315 |bibcode=1977OptL....1...13S |issn=0146-9592}}</ref>
<math display="block">r = r_0 e^{-\rho}, \quad k = k_0 \, e^{\kappa}.</math>
In these new variables, the Hankel transform reads
<math display="block">\tilde F_\nu(\kappa) = \int_{-\infty}^\infty \tilde f(\rho) \tilde J_\nu(\kappa - \rho) \,\mathrm{d}\rho,</math>
where
<math display="block">\tilde f(\rho) = \left(r_0 \, e^{-\rho} \right)^{1-n} \, f(r_0 e^{-\rho}),</math>
<math display="block">\tilde F_\nu(\kappa) = \left(k_0 \, e^{\kappa} \right)^{1+n} \, F_\nu(k_0 e^\kappa),</math>
<math display="block">\tilde J_\nu(\kappa-\rho) = \left(k_0 \, r_0 \, e^{\kappa-\rho} \right)^{1+n} \, J_\nu(k_0 r_0 e^{\kappa-\rho}).</math>

Now the integral can be calculated numerically with <math display="inline">O(N \log N)</math> [[Computational complexity|complexity]] using [[fast Fourier transform]]. The algorithm can be further simplified by using a known analytical expression for the Fourier transform of <math>\tilde J_\nu</math>:<ref>{{cite journal |last=Talman |first=James D. |date=October 1978 |title=Numerical Fourier and Bessel transforms in logarithmic variables |journal=Journal of Computational Physics |volume=29 |issue=1 |pages=35–48 |doi=10.1016/0021-9991(78)90107-9 |bibcode=1978JCoPh..29...35T |issn=0021-9991}}</ref>
<math display="block">
 \int_{-\infty}^{+\infty} \tilde J_\nu(x) e^{-i q x} \,\mathrm{d}x =
 \frac{\Gamma\left(\frac{\nu + 1 + n - iq}{2} \right)}{\Gamma\left(\frac{\nu + 1 - n + iq}{2}\right)} \, 2^{n - iq}e^{iq \ln(k_0 r_0)}.</math>
The optimal choice of parameters <math>r_0, k_0, n</math> depends on the properties of <math>f(r),</math> in particular its asymptotic behavior at <math>r \to 0</math> and <math>r \to \infty.</math>

This algorithm is known as the "quasi-fast Hankel transform", or simply "fast Hankel transform".

Since it is based on [[fast Fourier transform]] in logarithmic variables, <math>f(r)</math> has to be defined on a logarithmic grid. For functions defined on a uniform grid, a number of other algorithms exist, including straightforward [[Numerical integration|quadrature]], methods based on the [[projection-slice theorem]], and methods using the [[Bessel function#Asymptotic forms|asymptotic expansion]] of Bessel functions.<ref>{{Cite journal |last1=Cree |first1=M. J. |last2=Bones |first2=P. J. |date=July 1993 |title=Algorithms to numerically evaluate the Hankel transform |journal=Computers & Mathematics with Applications |volume=26 |issue=1 |pages=1–12 |doi=10.1016/0898-1221(93)90081-6 |doi-access=free |issn=0898-1221}}</ref>