Legendre Functions – Page 2 – Numerical Explorations

The two primary references used to create my application were: “A Numerical Library in C for Scientists and Engineers” by H. T. Lau and the following website: https://en.wikipedia.org/wiki/Jacobi_polynomials.

Using the Jacobi parameters alpha = 0 and beta = 0, we have the Legendre polynomials for degrees 4 and 6 and their associated roots:

Using alpha = 0.5 and beta= 0.5 we obtain for degrees 4 and 6:

Electron Probability Distribution Function Etc. (c) James Pate Williams, Jr. December 2023

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Partial Solution to a Neat Problem from an Online Textbook

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Win32 C Orthogonal Polynomials

I learned about the Laguerre and Legendre polynomials when I first read “Introduction to Quantum Mechanics” by Pauling and Wilson way back in the early 1970s. I later learned about the Chebyshev and other orthogonal polynomials. Beginning on March 30, 2015, I created yet another application to graph various orthogonal polynomials in C#. A few days ago, I wrote a Win32 C application to graph the Chebyshev, Laguerre, and Legendre orthogonal polynomials. At a later date, I will probably add Hermite and Jacobi polynomials.