MATH3271

FW02

Assignment 5 Due date: Monday, Dec. 2

1. Legendre polynomials

(a) Use Rodrigues’ formula to find the Legendre polynomials P₀, P₁ and P₂.

(b) Using Bonnet’s recurrence relation and the results from part (a) find the Legendre polynomials P₃, P₄ and P₅.

(c) Show that P₂ and P₄ are orthogonal over the interval [-1, 1].

(d) Show that P₃ satisfies the normalization condition that

(e) Expand the generating function using the expansion

with t = h² – 2xh. Gather up the coefficients of 1, h and h² and show that these are P₀, P₁ and P₂.

(f) Use MAPLE to plot the first four Legendre polynomials on the interval [-1, 1] on the same graph.

2. (a) Expand x⁴ in a Legendre series in P_n(x).

(b) Expand cos(2θ) in a Legendre series in P_n(cosθ).

3. Associated Legendre functions

(a) Use Rodrigues’ formula to show that

and P_n^m(x) = 0 if m > n.

(b) Use the recurrence relation

and the results of part (a) to find P₃²(x) and P₄²(x).

(c) Show that P₂²(x) and P₄²(x) are orthogonal on the interval [-1, 1].

(d) Show that P₃²(x) satisfies the normalization condition

4. Use MAPLE to produce a three-dimensional plot of |Y_2,0(θ,φ)| and |Y_4,2(θ,φ)|.

5. MAPLE does not have a specific function for the spherical Bessel functions but since

we can use BesselJ(n+1/2,x) to evaluate these functions.

(a) Plot j₀ and j₁ over the interval [0, 20] on the same graph. What can you say about the zeros of j₁ relative to the zeros of j₀.

(b) Find the first ten zeros of j_0. Note that j₀ has the same zeros as J_1/2. Identify these zeros.

6. Solve the symmetric Dirichlet problem outside a sphere, i.e. solve

subject to the boundary condition u(a, θ) = f(θ). How does the change in the range of r effect your choice of solution of the Euler equation for R(r)?

7. Show that the associated Legendre differential equation for a fixed value of m

on the interval [-1, 1] can be put in terms of a singular Sturm-Liouville problem. Specifically identify all of the parameters in this problem. Note that from the Rodrigues’ formula that P_n^m(x) is zero at x = ±1 if m ≠ 0.

MAPLE

In order to be able to access the Legendre polynomials in MAPLE you first need to use the command:

> with(orthopoly);

After this the Legendre polynomial P_n(x) is given by P(n,x).

MAPLE uses a different notation for the spherical coordinates. In order to have our notation, use the command:

> addcoords(my_spherical,[r,theta,phi],[r*sin(theta)*cos(phi),r*sin(theta)*sin(phi),r*cos(theta)]);

Then you can plot the function f(theta,phi) as

> plots3d(f,theta=0..Pi,phi=0..2*Pi,coords=my_spherical,axes=boxed);

In question 4, the simplest way to define the function to plot is to use the explicit formula in each case. If you are feeling adventurous, you could use the Rodrigues’ formula to generate P_n^m(x) from P_n(x). There are also the functions LegendreP in MAPLE but you have to be careful to define them correctly and they don’t plot properly near theta = 0 and Pi.