F3 - hankel transforms

expression of a function as the weighted sum of an infinite number of bessel functions of the first kind, $J_{ν} (k r)$
the bessel functions are all of the same order, $ν$ , but have different scaling factors, $k$ , along the $r$ axis

F_{ν} (k) = \int_{0}^{\infty} f (r) J_{ν} (k r) r d r

where, $ν \leq - 1 / 2$

the inverse henkel transform is defined as:

f (r) = \int_{0}^{\infty} F_{ν} (k) J_{ν} (k r) k d k

considering laplace's equation in cylindrical coordinates:

H_{0} {\frac{\partial^{2} u}{\partial r^{2}} + \frac{1}{r} \frac{\partial u}{\partial r} + \frac{\partial^{2} u}{\partial z^{2}}} = - k^{2} U + \frac{\partial^{2}}{\partial z^{2}} U

where, $U = H_{0} u$

this becomes an ODE in the transformed function, $U$
the bessel functions form an orthogonal basis with respect to the weighting factor, $r :$

\int_{0}^{\infty} J_{ν} (k r) J_{ν} (k^{'} r) r d r = \frac{δ (k - k^{'})}{k}, k, k^{'} > 0

hankel functions

\begin{aligned} H_{α}^{(1)} (x) & = J_{α} (x) + i Y_{α (x)} \\ H_{α}^{(2)} (x) & = J_{α} (x) - i Y_{α} (x) \end{aligned}