Orthogonality Relationship for Legendre Polynomials in Cylindrical Coordinates

[Mike706]

Hello everyone,

Sorry if this is in the wrong sub-forum, I wasn't sure exactly where to place it.

I was wondering if there is an orthogonality relationship for the Legendre polynomials P$^{0}_{n}$(x) that have been converted to cylindrical coordinates from spherical coordinates, similar to the form where P$^{0}_{n}(x)$P$^{0}_{m}(x)$ is integrated from x = -1 to x = 1.

By converted to cylindrical coordinates from spherical, I mean that originally x is taken as:
cos($\varphi$)
($\varphi$ being the angle between the z axis and the position vector (from the solution to Laplace's equation)),
and cos($\varphi$) is replaced by $\frac{z}{\sqrt{r^{2}+z^{2}}}$.

Thanks, I appreciate the help.

 

[fresh_42]

The properties do not depend on the coordinates. If we change from Cartesian to polar or cylindrical coordinates, the scalar product gets another form, too.