Math 407-500 — Fall 2023

Test 2 Review

General Information

I will have office hours on Wednesday (11/15), Thursday (11/16), 12-1 and 2-3, and Friday (11/17) 12-1.

Calculators. You may use calculators to do arithmetic, although you will not need them. You may not use any calculator that has the capability of doing linear algebra or storing programs or other material.

Other devices. You may not use cell phones, computers, or any other device capable of storing, sending, or receiving information.

Structure and coverage. There will be 4 to 6 questions, some with multiple parts. In what follows, SO stands for Schaum's Outline and BC for Brown/Churchill. The test will cover SO, chapter 4 and chapter 5, 5.1-5.6; BC, chapter 5, sections 57-62 (pgs. 189-207).

The problems will be similar to ones done for homework, and as examples in class and in the texts. There will also be questions about definitions, and statements and proofs of theorems. These are indicated below.

Topics Covered

Line and Contour Integrals

• Line integrals. Know the definition of a line integral, real and complex (SO 4.2,4.3), how to parameterize a curve, definition of a simple closed curve, a simply connected region and a multiply connected region (SO 4.6-4.8), know and be able to use Green's Theorem.
• Be able to state Cauchy's Theorem. Be able to use Green's Theorem and the Cauchy-Riemann conditions to prove it for a simply connected region (SO 4.11). Know its statement for a multiply connected region (SO 4.15, Theorems 4.4 and 4.5). See the homework problems for Assignment 4.

Cauchy's Integral Formulas and Related Theorems

• Know the Cauchy's Integral formulas (SO 5.1) and be able to use them to compute various contour integrals (see Assignment 5).
• In SO 5,2, be able to state theorems 2-7, and be able to prove Cauchy's Inequality (theorems 2) and Liouville's Theorem (theorem 3).

Taylor and Laurent Series

• Be able to state Taylor's Theorem and be able to use the fact that the power series for a function is the Taylor series for a given analytic function $f(z)$ (see assignment 6). Be able to state the circle where the Taylor series converges to the given function. (See class notes for Nov. 8)
• Be able to state Laurent's Theorem and be able to use the fact that the Laurent series for a function $f(z)$ is unique, no matter how it is obtained. Know that a function may have different Laurent series in different annuli about a given point $z_0$. (See class notes for Nov. 13 and problems in Assignment 7.)

Updated 11/14/23.