# 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* $$p_n(x)=\frac{1}{2^nn!}\frac{d^n}{dx^n} (x^2-1)^n.$$ Use this to show that they satisfy the *Legendre Differential Equation* $$[(x^2-1)u']'=\lambda u,$$ 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 $x∈[−1,1]$. 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 $κ < M/m$, there exist polynomials $p_n∈P_n$ such that $‖f−p_n‖=O((1 + 2/√κ)^{−n})$ as $n→∞$, where $‖·‖$ 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 $κ$. 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.