Homework 3

Due to the dropbox by 11:00pm on October 24, 2013

Do the following exercises from ATAP:

• Exercises 6.5, 7.1, 8.1, 8.2, 8.3, 10.4, 10.5, 10.6, 11.1
• For Exercise 11.1, add an additional part (d): Compute the integral (11.9) for the two choices of contours \(|x|=3\) and \(|x|=3/2\) and compare to the actual error \(f(2)-p(2)\). Confirm that in one case it gives the right result and in the other it does not (and make sure you understand why).

Notes:

For 6.5, note that the binomial coefficient formula for “\(k\) choose \(n\)” given by \(k(k-1)...(k-n+1)/n!\) can be used also when \(k=1/2\).

For 11.1, see http://www2.maths.ox.ac.uk/chebfun/guide/html/guide5.shtml for a discussion of how to compute Cauchy integrals. The function \(f(x)\) defined in the problem is an analytic function that takes the values we want to interpolate at the interpolation points. Any other analytic function taking the same values at these 3 points should result in the same interpolating polynomial. (But a different error at \(x=2\), of course.)