# Hankel Transform.

1. Sep 14, 2006

### Clausius2

Hey guys,

Do you have any advice of a place for learning about Hankel's Transform and its application to Laplace Equation?.

There are a couple of lines of a paper in which I am stuck on, I don't know how do they do this stuff:

Defining the operator $$L_m^2=\frac{1}{r}\frac{\partial}{\partial r} \left(r\frac{\partial}{\partial r}\right)-\frac{m^2}{r^2}+\frac{\partial^2}{\partial z^2}$$

then the solution of $$L_m^2 f=0$$ under the bipolar change of variables $$r=2\eta/(\eta^2+\xi^2)$$ and $$z=2\xi/(\eta^2+\xi^2)$$ is given by:

$$f=(\xi^2+\eta^2)^{1/2}\int_0^\infty(A(s)sinh(s\xi)+B(s)cosh(s\xi))J_m(s\eta)ds$$

I have tried to perform the change of variables in the differential operator, but it turns out to be the Hell when doing that for the second derivative. Any advice?

Thanks.