Legendre Polynomial: Understanding the Basics

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    Legendre Polynomial
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Discussion Overview

The discussion revolves around the origins and properties of Legendre polynomials, including their derivation and applications in physics and mathematics. Participants explore various methods of generating these polynomials and their significance in different contexts.

Discussion Character

  • Exploratory, Technical explanation, Conceptual clarification

Main Points Raised

  • One participant requests a step-by-step explanation of why Legendre polynomials were developed.
  • Another participant notes that Legendre polynomials are related to the Legendre differential equation and can be derived using the Frobenius method, mentioning that there are multiple approaches to obtain them.
  • It is suggested that Legendre polynomials can be generated using the Gram-Schmidt orthogonalization process on the interval -1 to 1, starting with a constant 1 as L_0.
  • A similar point is reiterated regarding the generation of Legendre polynomials from a linearly independent set of functions.
  • One participant emphasizes the importance of mastering Legendre polynomials due to their frequent occurrence in physics, particularly in calculating moments of charge distributions.

Areas of Agreement / Disagreement

Participants express various methods of generating Legendre polynomials, but there is no consensus on a single approach or the necessity of one method over another. The discussion remains open-ended with multiple viewpoints presented.

Contextual Notes

The discussion does not resolve the specific assumptions or definitions related to the methods mentioned, nor does it clarify the mathematical steps involved in deriving Legendre polynomials.

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can someone explain step-by-step why the legendre polynomial came into being? I'm having one hard time understanding it...
 
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If memory serves me correctly, you can generate the Legendre polynomials using Gram-Schmidt orthogonalization on the interval [tex]-1 \leq x \leq 1[/tex] starting with a constant 1 as [tex]L_0[/tex].

The generating function method is used to calculate moments of charge distributions.

You have to master them to get anywhere in physics., they pop up all the time.
 
Dr Transport said:
If memory serves me correctly, you can generate the Legendre polynomials using Gram-Schmidt orthogonalization on the interval [tex]-1 \leq x \leq 1[/tex] starting with a constant 1 as [tex]L_0[/tex].

Yup, assuming you start with the linearly independent set [tex]\{ 1, \, x, \, x^2, \, x^3 \ldots \}[/tex].
 
thank you very much! :)
 

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