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 Pn(x).

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

 

3.  Associated Legendre functions

(a)  Use Rodrigues’ formula to show that

and Pnm(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 |Y2,0(θ,φ)| and |Y4,2(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 j0.  Note that j0 has the same zeros as J1/2.  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 Pnm(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 Pn(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 Pnm(x) from Pn(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.