transform of becomes where the last equality follows from §6.567.1 of. The expansion coefficients are accessible with discrete Fourier transform techniques: if the radial distance is scaled with the Fourier-Chebyshev series coefficients emerge as Using the re-expansion yields expressed as sums of . This is one flavor of fast Hankel transform techniques. Relation to the Fourier and Abel transforms The Hankel transform is one member of the FHA cycle of integral operators. In two dimensions, if we define as the Abel transform operator, as the Fourier transform operator, and as the zeroth-order Hankel transform operator, then the special case of the projection-slice theorem for circularly symmetric functions states that In other words, applying the Abel transform to a 1-dimensional function and then applying the Fourier transform to that result is the same as applying the Hankel transform to that function. This concept can be extended to higher dimensions. 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 In these new variables, the Hankel transform reads where Now the integral can be calculated numerically with complexity using fast Fourier transform. The algorithm can be further simplified by using a known analytical expression for the Fourier transform of : The optimal choice of parameters depends on the properties of in particular its asymptotic behavior at and 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, has to be defined on a logarithmic grid. For functions defined on a uniform grid, a number of other algorithms exist, including straightforward quadrature, methods based on the projection-slice theorem, and methods using the asymptotic expansion of Bessel functions. Some Hankel transform pairs is a modified Bessel function of the second kind. is the complete elliptic integral of the first kind. The expression coincides with the expression for the Laplace operator in polar coordinates applied to a spherically symmetric function The Hankel transform of Zernike polynomials are essentially Bessel Functions (Noll 1976): for even . See also Fourier transform Integral transform Abel transform Fourier–Bessel series Neumann polynomial Y and H transforms References Integral transforms