Hankel transformation

Weisstein EW 1999b Hankel transform From MathWorld - A Wolfram Web Resource, http // mathworld.wolfram.com/HankelTransform.html. 

The most common technique for the derivation of fundamental solutions is to use integral transforms, such as, Fourier, Laplace or Hankel transforms [29, 39]. For simple operators, such as the Laplacian, direct integration and the use of the properties of the Dirac delta are typically used to construct the fundamental solution. For the case of a two-dimensional Laplace equation we can use a two-dimensional Fourier transform, F, to get the fundamental solution as follows,... 

A final remark in this Section concerns axially symmetric problems. We usually treat these in radial coordinates and apply a numerical Hankel transform instead of the Fourier transform. This is a slow transform with a dense matrix, but due to the relatively small computational domain radially symmetric problems require, this is not a big problem. Alternatively, one could treat such situations by finite differencing in the radial dimension, but it would mean accepting additional (paraxial) approximation, and would introduce artificial numerical dispersion into the algorithm. 

It is then Hankel-transformed backwards to the direct space by... 

Lennard-Jones potential, the Hankel transform of the direct correlation function has the analytic form... 

Its Hankel transform has no singularity at p -> 0, and so the expansion of the DCF at p = 0 keeps the analytic form (44). Accordingly, the total correlation function keeps the asymptotics (43). However, the matrices of the expansion coefficients Co, C2, C4,... in (44) have other, modified values. Through Equation (40) this, in general, changes the profile p and hence results in a modified inverse decay length appearing in the asymptotics (42), (43) and (46). 

FIGURE 5.11. Sections of the inhomogeneous two-paiticle direct conelation function c(s, z, zi) of the Lennard-Jones fluid along the liquid/vapour interface (part a), and its Hankel transform c(p, zt,zz) (ptut b). Line nomenclature is the same as in Figure 5.10. Results of die lOZ-KHM/LMBW theory. The inset in part a zooms in the long-range tail of c(s, Zi,Z2) with the asynqitodcs (47). 

As mentioned above, for q >- 2ir / L the rod gives only a contribution to the measured intensity if q is perpendicular to the long axis [10], i. e., if a 0. Thus, the intensity measured at higher scattering angles is directly related to the Hankel-transform of the excess electron density Ap(rc) which means that [10],... 

Jfi is the Bessel function of the first kind of order 0, and the Hankel transformation of order zero and its inverse are given by... 

If objects with symmetries in higher dimensions are reduced to one dimension in a similar fashion, Hankel transforms in terms of higher-order Bessel functions result Bral 1. 

The Abel transformation relates the radial information Frir) of a circular object to the projection Fix) (Fig. 4.4.1). In NMR imaging, the projection P(x) s obtained by Fourier transformation of the FID signal measured in a constant magnetic field gradient G, and the radial information Frir) is the inverse Hankel transform of the FID Maj 11. 

For evaluation of radial NMR images Frir) of circular objects, processing of the FID in two steps by Fourier transformation and subsequent inverse Abel transformation is preferred over straight forward Hankel transformation, because established phase correction, baseline correction, and filter routines can be used in calculation of the projections P(jc) as intermediate results [Majl]. As an alternative to Hankel and Abel transformations, the back-projection technique (cf. Section 6.1) can be applied for radial evaluation of circular objects, using copies of just one projection for input. As opposed to the inverse Abel transformation, however, this provides the radial information with nonuniform spatial resolution. 

Different approaches can be taken to obtain radial images. Radial field gradients can be applied by the use of dedicated hardware [Hakl, Leel, Lee2]. Alternatively, a 2D image can be reconstructed from one projection by the backprojection technique, and a radial cross-section can be taken through it. The most direct way to access the radial image from a projection consists in computing the inverse Hankel transformation (cf. Section 4.4.2) of the FID measured in Cartesian k space (cf. Fig. 4.4.1) [Majl]. But in practice, the equivalent route via Fourier transformation of the FID and subsequent inverse Abel transformation (cf. Section 4.4.3) is preferred because established phase and baseline correction routines can be used in the calculation of the projection as an intermediate result. 

Here h,(k) and h k) are the usual Fourier transforms, while hp k) is a Hankel transform. 

The methods for solving a second-order partial differential equation are separation of variables, similarity variable, Laplace transform, Fourier transform, and Hankel transform. Each of the... 

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