Math Bits - Legendre Polynomials

Prerequisites

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

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

1. (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 $g(z)=\sin(3\pi z)$

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