**MATH4141/MATH6651/PHYS5070**

**FW02**

**Assignment
5 Due
date: Friday, Nov. 29**

1. Use the Legendre polynomials to find the
least squares polynomial approximation to exp(x) of degree 2 on [-1, 1] with
weight function w(x) = 1.

2. Find the Taylor series of degree 6 about x = 0 for cosh(x) and use Chebyshev economization to obtain a polynomial approximation of degree 4. Estimate the error of this approximation on [-1, 1].

3. Find the first four Hermite polynomials, i.e.
the orthogonal polynomials H_{0}, H_{1}, H_{2} and H_{3} on

[-¥, ¥] with weight function exp(-x^{2}).
Take H_{0} = 1. The usual normalization for the Hermite
polynomial H_{n}(x) is to take the coefficient
of x^{n} as 2^{n} which makes

_{}

4. Using the results of problem 3
find the least squares polynomial approximation of order 2 to the function e^{-x}
on [-¥, ¥] with weight function w(x) = exp(-x^{2}). Show (i.e.
plot or produce a table of) the **relative**
error of this approximation on [-2, 2].

5. (a) Find the [2/2] Padé
approximant to x cot(x). Compare the
error in this approximation with the Taylor series of degree 4 about x = 0 on
[0, 1]. Also compare with the error in
the approximation to x cot(x) obtained by writing it as x cos(x)/sin(x)
and
using the Maclaurin series of degree 2 for both sin(x) and cos(x).

(b) Find the location of the singularity in this
Padé approximant closest to zero. How
does this compare to the location of the first singularity in x cot(x)?

(c) Write
out this Padé approximant in continued fraction form.

6. Use the Fast Fourier Transform to find the
Discrete Fourier Transform of the set of data

(x_{j}, y_{j}), j = 0 ..3 where x_{i}
= -p + jp/2 and y_{j} = sin(x_{j}).

Integration
formula:

_{}

_{}