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Finding Legendre Coefficients: Instructor's Guide

Main Ideas

1. Maple is used to compute Legendre Coefficients
2. Students practice approximating a function by adding orthogonal functions
3. Students add many Legendre functions together and inspect the goodness of fit
4. Students inspect the region over which the Legendre functions are orthogonal

Estimated Time: 10-15 minutes

Students use maple to find the first few coefficients of a Legendre Series to approximate a function. They then explore how this approximations changes as the number of terms included in the Legendre Series is increased.

Prerequisite Knowledge

• Students should know the Legendre Polynomials.
• Students should have previously calculated coefficients for Fourier Series.

Activity: Introduction

This activity should follow a short lecture defining the Legendre Polynomial Series and relating that to other series approximations students have done, e.g., fourier series, linear combinations of spin states.

We find that students understand this activity better if we begin by making a rough plot of the function $f(x)$ to be approximated by the Legendre Series. Then draw plots of each of the first 3 or 4 Legendre polynomials and tell them that the purpose of the maple worksheet is to help them see how they will find out “How much of $P_1 (z)$ is in $f(x)$, and how much of $P_2 (z)$ is in $f(x)$, and how much of $P_3 (z)$ is in $f(x)$, …” This helps them to get a qualitative grasp of what they will do in the worksheet before they begin.

Activity: Student Conversations

Note: When more than about 50 terms are used, the resulting plot may show anomalous behavior due to numerical artifacts.

• How many terms does it take to get a good fit?
• How does the function fit outside of the range -1 to 1?
• Are the nonzero coefficients for mostly even powers or odd powers?
• How many odd power coefficients are there?
• Why do you think $c_1=1/2$ and all the other odd coefficients are zero?

Activity: Wrap-up

It is useful to reemphasize the form of the Legendre Series and how one goes about finding the coefficients.

$$f(z) = \sum_k a_k P_k(z)$$

$$a_k = (k + \frac{1}{2})\int_{-1}^1 P_k(z)^* f(z) dz$$

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