Homework 6

Due 12/17 (Tuesday of Finals week)

1. Prove that the Legendre polynomials normalized to have
   $P_n(1)=1$ satisfy the Rodridgues Formula
   $$p_n(x)=\frac{1}{2^{n}n!}\frac{d^n}{d{x}^n}(x^2-1{)}^n.$$
   Use this to show that they satisfy the Legendre Differential Equation
   $$[(x^2-1)u^{\prime }{]}^{\prime }=\lambda u,$$
   with
   $\lambda =n(n+1)$.
2. Suppose
   $p_n(x)$ are monic OPRL on a finite interval
   $(-a,a)$ with even weight function
   $w(x)=w(-x)$. Show that the Jacobi coefficients
   $a_n=0$ for all
   $n$.
3. (Exercise 8.7, Trefethen) The function
   $|x-i|$ is real analytic for
   $x\in [-1,1]$. This means it can be analytically continued to an analytic function
   $f(x)$ in a neighborhood of
   $[-1,1]$ in the complex
   $x-$plane. The formula
   $|x-i|$ itself does not define an analytic function in any complex neighborhood. Find another formula for
   $f$ that does, and use it to explain what singularities
   $f$ has in the complex plane.
4. (Exercise 8.10, Trefethen) Suppose we wish to approximate
   $f(x)=1/x$ on the interval
   $[m,M]$ with
   $0<m<M$. Show that for any
   $\kappa <M/m$, there exist polynomials
   $p_n\in P_n$ such that
   $\Vert f-p_n\Vert =O((1+2/\surd \kappa {)}^{-n})$ as
   $n\to \infty$, where
   $\Vert ·\Vert$ is the ∞-norm on
   $[m,M]$. This result is famous in numerical linear algebra as providing an upper bound for the convergence of the conjugate gradient iteration applied to a symmetric positive definite system of equations
   $Ax=b$ with condition number
   $\kappa$.
5. Exercise 4.7, Trefethen http://www.chebfun.org/ATAP/atap-first6chapters.pdf
6. Make a 2-3 page typed cheat sheet summarizing what you consider the important topics and ideas covered in this class, and the connections between them. Keep a copy for future reference.