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
- 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 $g(z)=\sin(3\pi z)$