Legendre Polynomials: Beginner's Guide

[tardon007]

hi folks!

I have been trying to figure out some plausible geometric intrepretation to legendre polynomials and what are they meant to do.

I have come across the concept of orthogonal polynomials while working with some boundary value problems in solid mechanics and wasn't able to come to terms with that since then.

Can some one amongst you suggest a beginner's literature on this ?

It would be invalueble if someone can share ur insights on this ?

 

[BobG]

Try this link:

http://www.floridageomatics.com/publications/gfl/toc.htm

The section on Legendre polynomials, zonal harmonics, sectorial harmonics, and tesseral harmonics are in the appendix.

It's basically a sine regression. You have an actual shape of a non-perfect sphere that you want to model. Legendre polynomials give you a method of creating an equation that will model the irregularities in the object's shape.

 

[ravenprp]

Hi there,

Legendre polynomials are a type of orthogonal polynomial that are commonly used in mathematics and physics. They were first introduced by Adrien-Marie Legendre in the late 18th century and have since been widely studied and applied in various fields.

One way to understand Legendre polynomials is to think of them as a set of functions that can be used to represent any other function. They have a special property called orthogonality, which means that when multiplied together and integrated over a certain range, they result in zero. This property makes them useful in solving boundary value problems, as you mentioned in your post.

A geometric interpretation of Legendre polynomials can be seen in their graphical representation, which resembles a series of curves with peaks and valleys. Each peak and valley corresponds to a root of the polynomial, and together they form a pattern that can be used to approximate other functions.

In terms of beginner's literature, I would recommend checking out "A First Course in the Numerical Analysis of Differential Equations" by Arieh Iserles. It provides a good introduction to Legendre polynomials and their applications in solving differential equations.

I hope this helps! Feel free to share any insights or questions you may have.