**MATH3271**

**FW02**

**Assignment 5 Due
date: Monday, Dec. 2**

1. Legendre polynomials

(a) Use Rodrigues’ formula to find the Legendre
polynomials P_{0}, P_{1} and P_{2}.

(b) Using Bonnet’s recurrence relation and the
results from part (a) find the Legendre polynomials P_{3}, P_{4}
and P_{5}.

(c) Show that P_{2} and P_{4} are
orthogonal over the interval [-1, 1].

(d) Show that P_{3} satisfies the
normalization condition that

_{}

(e) Expand the generating function using the
expansion

_{}

with t = h^{2}
– 2xh. Gather up the coefficients of 1,
h and h^{2 } and show that these
are P_{0}, P_{1} and P_{2}.

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

2. (a)
Expand x^{4} 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 P_{3}^{2}(x) and P_{4}^{2}(x).

(c) Show that P_{2}^{2}(x) and P_{4}^{2}(x)
are orthogonal on the interval [-1, 1].

(d) Show that P_{3}^{2}(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 j_{0}
and j_{1} over the interval [0, 20] on the same graph. What can you say about the zeros of j_{1}
relative to the zeros of j_{0}.

(b) Find the first ten zeros of j_{0.} Note that j_{0} 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.