- #1
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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 [tex]L_m^2=\frac{1}{r}\frac{\partial}{\partial r}<br /> \left(r\frac{\partial}{\partial r}\right)-\frac{m^2}{r^2}+\frac{\partial^2}{\partial z^2}[/tex]
then the solution of [tex]L_m^2 f=0[/tex] under the bipolar change of variables [tex]r=2\eta/(\eta^2+\xi^2)[/tex] and [tex]z=2\xi/(\eta^2+\xi^2)[/tex] is given by:
[tex]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[/tex]
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.
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 [tex]L_m^2=\frac{1}{r}\frac{\partial}{\partial r}<br /> \left(r\frac{\partial}{\partial r}\right)-\frac{m^2}{r^2}+\frac{\partial^2}{\partial z^2}[/tex]
then the solution of [tex]L_m^2 f=0[/tex] under the bipolar change of variables [tex]r=2\eta/(\eta^2+\xi^2)[/tex] and [tex]z=2\xi/(\eta^2+\xi^2)[/tex] is given by:
[tex]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[/tex]
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.