Editing Hankel transform (section)

== Transforming Laplace's equation ==
The Hankel transform can be used to transform and solve [[Laplace's equation#Forms in different coordinate systems|Laplace's equation]] expressed in cylindrical coordinates. Under the Hankel transform, the Bessel operator becomes a multiplication by <math>-k^2</math>.<ref>{{Cite book |title=The transforms and applications handbook |date=1996 |publisher=CRC Press |author=Poularikas, Alexander D. |isbn=0-8493-8342-0 |location=Boca Raton Fla. |oclc=32237017}}</ref> In the axisymmetric case, the [[partial differential equation]] is transformed as

: <math>\mathcal{H}_0 \left\{ \frac{\partial^2 u}{\partial r^2} + \frac 1 r \frac{\partial u}{\partial r} 
+ \frac{\partial ^2 u}{\partial z^2} \right\} = -k^2 U + \frac{\partial^2}{\partial z^2} U,</math>

where <math>U = \mathcal{H}_0 u</math>. Therefore, the Laplacian in cylindrical coordinates becomes an ordinary differential equation in the transformed function <math>U</math>.