I am currently trying to derive an analytical expression for the Fourier transform of the projection of a spherical cap of the unit sphere onto the xy-plane. Setting up the integration in cylindrical coordinates, and letting $0 < h < 1$ represent the height of the cap (see http://en.wikipedia.org/wiki/Spherical_cap), we have (letting $x = r\cos\theta$, $y=r\sin\theta$, $k_x = \kappa\cos\phi$, $k_y = \kappa\sin\phi$):
$$ \int_0^{2\pi} \int_0^R \int_{1-h}^{\sqrt{1-r^2}}re^{-2\pi i r\kappa\cos\theta} dz dr d\theta$$
where $R=\sqrt{1-(1-h)^2}$. The $\phi$-dependence vanishes due to symmetry of the integral over $\theta$. After performing the integration over $z$:
$$ \int_0^{2\pi} \int_0^R r \sqrt{1-r^2}e^{-2\pi i r\kappa \cos\theta} dr d\theta - \int_0^{2\pi} \int_0^R r(1-h) e^{-2\pi i r\kappa \cos\theta} dr d\theta $$
which reduces to: $$ 2\pi\int_0^R r\sqrt{1-r^2}J_0(2\pi r \kappa) dr - 2\pi\int_0^R r(1-h)J_0(2\pi r\kappa) dr $$
Due to the property that $\int_0^x uJ_0(u) du = xJ_1(x)$, the latter integral can easily be evaluated as:
$$ (1-h)R\frac{J_1(2\pi R\kappa)}{\kappa} $$
The first integral, however, is more troublesome, as the argument does not have an antiderivative. I imagine that it can be expressed in some (relatively) compact form in terms of Bessel functions of varying degrees or associated Legendre polynomials, but I am having trouble seeing it.