Due Monday, April 2


We worked in class a problem about Legendre polynomials. These appear in
solving partial differential equation on the sphere, or in representing
functions in R^3 using  spherical harmonics. This is a similar
problem about polynomials, which are called Hermite polynomials. 
As I recall, they appear in the solution of the quantum mechanical oscillator.


Cinsider the equation

         y" -2xy' + \lambda y = 0.


Find the first four terms in a solution in terms of a power series about 0.

Derive the recursion relationship among coefficients of an arbitrary solution.

Show that if \lambda = 2n, there is a polynomial solution to the equation.

The n-th Hermite polynomial is this polynomial, normalized so that the
coefficient of x^n is 2^n.  Find  the first six  Hermite polynomials, starting
with n = 0 up to n=5.