Hankel transform

:"Besides the meaning discussed in this article, the "Hankel transform" may also refer to the determinant of the Hankel matrix of a sequence".

In mathematics, the Hankel transform of order ν of a function "f"("r") is given by:

:$F\_\; u(k)\; =\; int\_0^infty\; f(r)J\_\; u(kr),r,dr$

where "J"_ν is the Bessel function of the first kind of order ν with ν ≥ −1/2. The inverse Hankel transform of "F"_ν("k") is defined as:

:$f(r)\; =int\_0^infty\; F\_\; u(k)J\_\; u(kr)\; k~dk$

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. Just as the continuous Fourier transform for an infinite interval is related to the Fourier series over a finite interval, so the Hankel transform over an infinite interval is related to the Fourier–Bessel series over a finite interval.

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. However, like the Fourier Transform, the domain can be extended by a density argument to include some functions whose above integral is not finite, for example $f(r)\; =\; r$; this extension will not be discussed in this article.

Orthogonality

The Bessel functions form an orthogonal basis with respectto the weighting factor "r":

:$int\_0^infty\; J\_\; u(kr)J\_\; u(k\text{'}r)r~dr\; =\; frac\{delta\; (k-k\text{'})\}\{k\}$

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 "F"_ν("k") and "G"_ν("k") arewell defined, then the Plancherel theorem states

:$int\_0^infty\; f(r)g(r)r~dr\; =\; int\_0^infty\; F\_\; u(k)G\_\; u(k)\; k~dk.$

Parseval's theorem is a special case of the Plancherel theorem which states:

:$int\_0^infty\; |f(r)|^2r~dr\; =\; int\_0^infty\; |F\_\; u(k)|^2\; k~dk.$

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:

:$F(mathbf\{k\})=frac\{1\}\{2pi\}iint\; f(mathbf\{r\})e^\{-imathbf\{k\}cdotmathbf\{r,dmathbf\{r\}.$

With no loss of generality, we can pick a polar coordinate system ("r", θ) suchthat the k vector lies on the θ = 0 axis. The Fourier transform is now written in these polar coordinates as:

:$F(mathbf\{k\})=frac\{1\}\{2pi\}int\_\{r=0\}^inftyint\_\{\; heta=0\}^\{2pi\}f(r,\; heta)e^\{-ikrcos(\; heta)\},r,dr,d\; heta$

where θ is the angle between the k and r vectors. If thefunction f happens to be circularly symmetric, it will have no dependence onthe 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:

:$FA=H.,$

In other words, applying the Abel transform to a 1-dimensional function andthen applying the Fourier transform to that result is the same as applyingthe Hankel transform to that function. This concept can be extended to higherdimensions.

Some Hankel transform pairs

{pi kK_{-1/2}(k|z|),
-
$frac\{1\}\{r^2+z^2\},$
$K\_0(k|z|),$
-
$e^\{iar\}/r,$

-
$,$
$1/sqrt\{\; k^2\; -\; a^2\}\; quad\; (a>0,\; k>a)\; ,$
-
$e^\{a^2r^2/2\},$
$frac\{e^\{k^2/2a^2\{a^2\}$
-
$-r^2\; f(r),$
$frac\{d^2\; F\_0\}\{dk^2\}+frac\{1\}\{k\}frac\{d\; F\_0\}\{dk\}$$K\_n(z)$ is a modified Bessel function of the second kind.The expression$frac\{d^2\; F\_0\}\{dk^2\}+frac\{1\}\{k\}frac\{d\; F\_0\}\{dk\}$coincides with the expressionfor the Laplace operator in polar coordinates$(k,\; heta)$applied to a spherically symmetric function $F\_0(k)$.

ee also

* Continuous Fourier transform

* Discrete Hankel transform

References

* Gaskill, Jack D., "Linear Systems, Fourier Transforms, and Optics", John Wiley & Sons, New York, 1978. ISBN 0-471-29288-5
* Polyanin, A. D. and Manzhirov, A. V., "Handbook of Integral Equations", CRC Press, Boca Raton, 1998. ISBN 0-8493-2876-4
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