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Hankel TransformIn mathematics, the Hankel transform of order ν of a function f(r) is given by: -
F_\nu(k) = \int_0^\infty f(r)J_\nu(kr)\,r\,dr where Jν is the Bessel function of the first kind of order ν with ν ≥ −1/2. The inverse Hankel transform of Fν(k) is defined as: -
f(r) =\int_0^\infty F_\nu(k)J_\nu(kr) k~dk which can be readily verified using the orthogonality relationship described below. The Hankel transform is an integral transform and was first developed by the mathematician Hermann Hankel. It is also known as the Fourier-Bessel transform. Domain of definition The Hankel transform of a function f(r) is valid at every point at which f(r) is continuous provided that the function is defined in (0, ∞), is piecewise continuous and of bounded variation in every finite subinterval in (0, ∞), and the integral -
\int_0^\infty |f(r)|\,r^{1/2}\,dr is finite. Orthogonality The Bessel functions form an orthogonal basis with respect to the weighting factor r: -
\int_0^\infty J_\nu(kr)J_\nu(k'r)r~dr = \frac{\delta (k-k')}{k} for k and k' greater than zero. The Plancherel theorem and Parseval's theorem If f(r) and g(r) are such that their Hankel transforms Fν(k) and Gν(k) are well defined, then the Plancherel theorem states -
\int_0^\infty f(r)g(r)r~dr = \int_0^\infty F_\nu(k)G_\nu(k) k~dk Parseval's theorem is a special case of the Plancherel theorem which states: -
\int_0^\infty |f(r)|^2r~dr = \int_0^\infty |F_\nu(k)|^2 k~dk These theorems can be proven using the orthogonality property. Relation to other functions Relation to the Fourier transform The Hankel transform of order zero is essentially the two dimensional continuous Fourier transform of a circularly symmetric function. Consider a two-dimensional function f(r) of the radius vector r. Its Fourier transform is: -
F(\mathbf{k})=\frac{1}{2\pi}\int_{-\infty}^\infty f(\mathbf{r}) e^{-i\mathbf{k}\cdot\mathbf{r}}\,d^2r With no loss of generality, we can pick a polar coordinate system (r, θ) such that the k vector lies on the θ = 0 axis. The Fourier transform is now written in these polar coordinates as: -
F(\mathbf{k})=\frac{1}{2\pi}\int_{r=0}^\infty \int_{\theta=0}^{2\pi}f(r,\theta)e^{-ikr\cos(\theta)}\,r\,dr\,d\theta where θ is the angle between the k and r vectors. If the function f happens to be circularly symmetric, it will have no dependence on the angular variable θ and may be written f(r). The integration over θ may be carried out, and the Fourier transform is now written: -
F(\mathbf{k})=F(k)= \int_0^\infty f(r) J_0(kr) r\,dr which is just the zero-order Hankel transform of f(r). 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 A as the Abel transform operator, F as the Fourier transform operator and H 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. See also References - Gaskill, Jack D., "Linear Systems, Fourier Transforms, and Optics", John Wiley & Sons, New York, 1978. ISBN 0-471-29288-5
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