= Hankel representation of thin-disk gravity
{c}
{title2=$\Phi=\int_0^\infty S(k)J_0(kR)e^{-k|z|}\,dk$}
For an axisymmetric reflection-symmetric <thin disk>, the <Laplace equation in cylindrical coordinates> separates into regular radial <Bessel functions> and vertical decaying exponentials. The <divergence theorem> imposes $\Phi_z^+-\Phi_z^-=4\pi G\Sigma$. <Hankel transform> inversion gives $S(k)=-2\pi G\int_0^\infty\Sigma(R)J_0(kR)R\,dR$. Its radial derivative gives the <galaxy rotation curve>. Infinite scale-free models may require subtracting an additive divergent constant.
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