Cylinder Function

The cylinder function is defined as

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

The Bessel Functions are sometimes also called cylinder functions. To find the Fourier Transform of the cylinder function, let

$$k_x = k\cos\alpha$$ (2)

$$k_y = k\sin\alpha$$ (3)

$$x = r\cos\theta$$ (4)

$$y = r\sin\theta.$$ (5)

Then

$$F(k,a) = {\mathcal F}(C(x,y))$$

$$= \int_0^{2\pi} \int_0^a e^{i(k\cos \alpha r\cos \theta +k\sin \alpha r\sin\theta)}r\,dr\,d\theta$$

$$= \int_0^{2\pi}\int_0^a e^{ikr\cos (\theta -\alpha )}r\,dr\,d\theta.$$ (6)

Let $b=\theta-\alpha$, so $db=d\theta$. Then

$$F(k,a) = \int_{-\alpha }^{2\pi-\alpha }\int_0^a e^{ikr\cos b}r\,dr\,d\theta$$

$$= \int_0^{2\pi} \int_0^a e^{ikr\cos b}r\,dr\,d\theta$$

$$= 2\pi \int_0^a J_0(kr)r\,dr,$$ (7)

where $J_0$ is a zeroth order Bessel Function of the First Kind. Let $u\equiv kr$, so $du=k\,dr$, then

$$F(k,a) = {2\pi\over k^2} \int_0^{ka} J_0(u)u\,du = {2\pi\over k^2} [uJ_1(u)]_0^{ka}$$

$$= {2\pi a\over k}J_1(ka) = 2\pi a^2 {J_1(ka)\over ka}.$$ (8)

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

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

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

References

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