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## Main Question or Discussion Point

Hey!

I am trying to solve this quite nasty (as least I think so : - ) partial differential equation (the Helmholtz equation):

[tex]

\frac{1}{r}\frac{\partial}{\partial r} \left( r \frac{\partial\Psi}{\partial r}\right) + \frac{1}{r^2}\frac{\partial^2 \Psi}{\partial \phi^2} + \frac{\partial^2 \Psi}{\partial z^2} + m^2 k^2 \Psi = 0

[/tex]

I use separation of variables and write the unkown function [tex]\Psi(r,\phi,z)[/tex] as [tex]\Psi(r,\phi,z) = R(r)\Phi(\phi)Z(z)[/tex], insert this in the equation and divide by [tex]R(r)\Phi(\phi)Z(z)[/tex]. This gives me:

[tex]

\frac{1}{r R} \frac{d}{d r} \left( r \frac{d R}{d r}\right) + \frac{1}{r^2 \Phi} \frac{d^2

\Phi}{d \phi^2} + \frac{1}{Z}\frac{d^2 Z}{d z^2} + m^2 k^2 = 0

[/tex]

Now, I am not sure how to move on from here because I have 1/r^2 in the [tex]\Phi[/tex] term so that I cannot use the usual procedures for solving PDE (equating one term to minus the other terms and setting both equal to some constant). Could someone give me a hint on how to proceed from here?

Thanks in advance

I am trying to solve this quite nasty (as least I think so : - ) partial differential equation (the Helmholtz equation):

[tex]

\frac{1}{r}\frac{\partial}{\partial r} \left( r \frac{\partial\Psi}{\partial r}\right) + \frac{1}{r^2}\frac{\partial^2 \Psi}{\partial \phi^2} + \frac{\partial^2 \Psi}{\partial z^2} + m^2 k^2 \Psi = 0

[/tex]

I use separation of variables and write the unkown function [tex]\Psi(r,\phi,z)[/tex] as [tex]\Psi(r,\phi,z) = R(r)\Phi(\phi)Z(z)[/tex], insert this in the equation and divide by [tex]R(r)\Phi(\phi)Z(z)[/tex]. This gives me:

[tex]

\frac{1}{r R} \frac{d}{d r} \left( r \frac{d R}{d r}\right) + \frac{1}{r^2 \Phi} \frac{d^2

\Phi}{d \phi^2} + \frac{1}{Z}\frac{d^2 Z}{d z^2} + m^2 k^2 = 0

[/tex]

Now, I am not sure how to move on from here because I have 1/r^2 in the [tex]\Phi[/tex] term so that I cannot use the usual procedures for solving PDE (equating one term to minus the other terms and setting both equal to some constant). Could someone give me a hint on how to proceed from here?

Thanks in advance

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