Table of Contents

# Math Bits - Legendre Polynomials

## Prerequisites

Students should be familiar with the series solutions to Legendre's equation.

## In-class Content

- Guessing the Legendre Polynomial Expansion of a Function (Optional Maple Activity, 10-15 minutes)
- Legendre Polynomial Properties (Lecture, 10 minutes)
- Legendre Series (Lecture, 20 minutes)
- Legendre Polynomial Series Coefficients (Maple Activity, 10-15 minutes)

## Ideas for new in-class activities

- Legendre Integrals (SGA - 20 min)
- An activity in parallel with the Harmonic Integrals activity from Waves and Oscillators - we did not have time to design or run such an activity.

## Homework for Central Forces

- (LegendreSine)
*Expanding the sine function in a Legendre series.*Use your favorite tool (\emph{e.g.} Maple, Mathematica, Matlab, pencil) to generate the Legendre polynomial expansion to the function $f(z)=\sin(\pi z)$. How many terms do you need to include in a partial sum to get a “good” approximation to $f(z)$ for $-1<z<1$? What do you mean by a “good” approximation? How about the interval $-2<z<2$? How good is your approximation? Discuss your answers. Answer the same set of questions for the function $g(z)=\sin(3\pi z)$