Math 136 Fall 2004

Review for 2nd Midterm

Prof Ward

Here's some advice on preparing for the 2nd midterm.

The test concentrates on Chapter 4 (Complex Integration) and Chapter 5 (Series), along with Sections 6.1 (Residue Theorem) and 6.3 (Improper Integrals of rational functions). Of course, you will also need many of the concepts from Chapters 1-3.

You may prepare one page of notes (double-sided if you wish) to use during the test. You may also use Saff and Snider. Otherwise the test is closed-book, closed-notes, closed-everything, and not to be discussed with anyone except me. The test will be a 3-hour takehome; the three hours should be in a continuous block.

In Chapters 4 and 5 there's a lot of theory. Make sure you know the theorems we studied, and can apply them. (Know the hypotheses and how to check them, and the conclusions.) A partial list follows. Also, we've learned plenty of computational techniques. This test will have a mixture of proofs and computations.

List of stuff to know (I think this covers just about everything):
  • formula for computing integrals by parametrization ('brute force')
  • the 'ABC' theorem
  • Cauchy's Theorem
  • Cauchy's Integral Formula
  • Cauchy Derivative Formulas
  • Morera's Theorem
  • the Cauchy Estimates
  • Liouville's Theorem
  • the Mean Value Property
  • the Maximum Modulus Principle
  • pointwise and uniform convergence of series of functions
  • theorems on Taylor series
  • theorems on Laurent series
  • theorems on zeroes and isolated singularities
  • examples of specific integrals
  • classification of singularities
  • examples of Taylor series and Laurent series
  • how to compute Taylor and Laurent series, in a given region
  • the point at infinity, limits involving infinity
  • the Residue Theorem
  • how to compute residues
  • how to do real integrals of rational functions (Sec 6.3)


Last modified October 2003, by ward at hmc.edu