### 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