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Cylinder Function

The cylinder function is defined as

C(x,y)\equiv \cases{
1 & for $\sqrt{x^2+y^2} \leq a$\cr
0 & for $\sqrt{x^2+y^2} > a$.\cr}
\end{displaymath} (1)

The Bessel Functions are sometimes also called cylinder functions. To find the Fourier Transform of the cylinder function, let
$\displaystyle k_x$ $\textstyle =$ $\displaystyle k\cos\alpha$ (2)
$\displaystyle k_y$ $\textstyle =$ $\displaystyle k\sin\alpha$ (3)

$\displaystyle x$ $\textstyle =$ $\displaystyle r\cos\theta$ (4)
$\displaystyle y$ $\textstyle =$ $\displaystyle r\sin\theta.$ (5)

$\displaystyle F(k,a)$ $\textstyle =$ $\displaystyle {\mathcal F}(C(x,y))$  
  $\textstyle =$ $\displaystyle \int_0^{2\pi} \int_0^a e^{i(k\cos \alpha r\cos \theta +k\sin \alpha r\sin\theta)}r\,dr\,d\theta$  
  $\textstyle =$ $\displaystyle \int_0^{2\pi}\int_0^a e^{ikr\cos (\theta -\alpha )}r\,dr\,d\theta.$ (6)

Let $b=\theta-\alpha$, so $db=d\theta$. Then
$\displaystyle F(k,a)$ $\textstyle =$ $\displaystyle \int_{-\alpha }^{2\pi-\alpha }\int_0^a e^{ikr\cos b}r\,dr\,d\theta$  
  $\textstyle =$ $\displaystyle \int_0^{2\pi} \int_0^a e^{ikr\cos b}r\,dr\,d\theta$  
  $\textstyle =$ $\displaystyle 2\pi \int_0^a J_0(kr)r\,dr,$ (7)

where $J_0$ is a zeroth order Bessel Function of the First Kind. Let $u\equiv kr$, so $du=k\,dr$, then
$\displaystyle F(k,a)$ $\textstyle =$ $\displaystyle {2\pi\over k^2} \int_0^{ka} J_0(u)u\,du = {2\pi\over k^2} [uJ_1(u)]_0^{ka}$  
  $\textstyle =$ $\displaystyle {2\pi a\over k}J_1(ka) = 2\pi a^2 {J_1(ka)\over ka}.$ (8)

As defined by Watson (1966), a ``cylinder function'' is any function which satisfies the Recurrence Relations

{\mathcal C}_{\nu-1}(z)+{\mathcal C}_{\nu+1}(z)={2\nu\over z}{\mathcal C}_\nu(z)
\end{displaymath} (9)

{\mathcal C}_{\nu-1}(z)-{\mathcal C}_{\nu+1}(z)=2{\mathcal C}_\nu'(z).
\end{displaymath} (10)

This class of functions can be expressed in terms of Bessel Functions.

See also Bessel Function of the First Kind, Cylinder Function, Cylindrical Function, Hemispherical Function


Watson, G. N. A Treatise on the Theory of Bessel Functions, 2nd ed. Cambridge, England: Cambridge University Press, 1966.

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© 1996-9 Eric W. Weisstein