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Hankel Transform

Equivalent to a 2-D Fourier Transform with a radially symmetric Kernel, and also called the Fourier-Bessel Transform.

g(u,v)={\mathcal F}[f(r)] = \int_{-\infty}^\infty \int_{-\infty}^\infty f(r) e^{-2\pi i(ux+vy)}\,dx\,dy.
\end{displaymath} (1)

$\displaystyle x+iy$ $\textstyle =$ $\displaystyle re^{i\theta}$ (2)
$\displaystyle u+iv$ $\textstyle =$ $\displaystyle qe^{i\phi }$ (3)

so that
$\displaystyle x$ $\textstyle =$ $\displaystyle r\cos\theta$ (4)
$\displaystyle y$ $\textstyle =$ $\displaystyle r\sin\theta$ (5)
$\displaystyle r$ $\textstyle =$ $\displaystyle \sqrt{x^2+y^2}$ (6)

$\displaystyle u$ $\textstyle =$ $\displaystyle q\cos\phi$ (7)
$\displaystyle v$ $\textstyle =$ $\displaystyle q\sin\phi$ (8)
$\displaystyle q$ $\textstyle =$ $\displaystyle \sqrt{u^2+v^2}.$ (9)

$\displaystyle g(q)$ $\textstyle =$ $\displaystyle \int_0^\infty\int_0^{2\pi} f(r) e^{-2\pi i rq(\cos\phi\cos\theta+\sin\phi\sin\theta)} r\,dr\,d\theta$  
  $\textstyle =$ $\displaystyle \int_0^\infty\int_0^{2\pi} f(r)e^{-2\pi irq\cos (\theta-\phi)}r\,dr\,d\theta$  
  $\textstyle =$ $\displaystyle \int_0^\infty\int_{-\phi}^{2\pi-\phi} f(r)e^{-2\pi irq\cos \theta }r\,dr\,d\theta$  
  $\textstyle =$ $\displaystyle \int_0^\infty \int_0^{2\pi} f(r)e^{-2\pi irq\cos \theta }r\,dr\,d\theta$  
  $\textstyle =$ $\displaystyle \int_0^\infty f(r) \left[{\int_0^{2\pi} e^{-2\pi irq\cos \theta }\,d\theta }\right]r\,dr$  
  $\textstyle =$ $\displaystyle 2\pi\int_0^\infty f(r)J_0(2\pi qr)r\,dr,$ (10)

where $J_0(z)$ is a zeroth order Bessel Function of the First Kind. Therefore, the Hankel transform pairs are
g(k) = \int_0^\infty f(x)J_0(kx)x\, dx
\end{displaymath} (11)

f(x) = \int_0^\infty g(k)J_0(kx)k\, dk.
\end{displaymath} (12)

See also Bessel Function of the First Kind, Fourier Transform, Laplace Transform


Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, p. 795, 1985.

Bracewell, R. The Fourier Transform and Its Applications. New York: McGraw-Hill, pp. 244-250, 1965.

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© 1996-9 Eric W. Weisstein