Orthogonality Relationship for Legendre Polynomials in Cylindrical Coordinates

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SUMMARY

The discussion centers on the orthogonality relationship of Legendre polynomials P0n(x) when transformed from spherical to cylindrical coordinates. The transformation involves substituting cos(ϕ) with z/√(r²+z²), where ϕ is the angle between the z-axis and the position vector. It is established that the properties of these polynomials remain invariant under coordinate transformations, indicating that the scalar product adapts accordingly. This highlights the robustness of Legendre polynomials in various coordinate systems.

PREREQUISITES
  • Understanding of Legendre polynomials and their properties
  • Familiarity with coordinate transformations, specifically from spherical to cylindrical coordinates
  • Basic knowledge of Laplace's equation and its solutions
  • Concept of scalar products in different coordinate systems
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  • Research the orthogonality properties of Legendre polynomials in various coordinate systems
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  • Explore the implications of coordinate transformations on polynomial functions
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Mathematicians, physicists, and engineers interested in the properties of Legendre polynomials and their applications in solving differential equations in different coordinate systems.

Mike706
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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.
 
Last edited:
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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.
 

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