Math 407-500 — Fall 2023
Final Exam Review
General Information
The final exam will be given on Monday, 12/11/2023, 3:30-5:30, in our
usual classroom. Please bring an 8½×11
bluebook.
I will have office hours on Wednesday (12/6) and Thursday (12/7),
12-1; on Friday (12/8), 11-12; and on Monday (12/11), 11-12:30.
-
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 8 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 5, 5.1-5.6. In
BC, it will cover chapter 5, sections 57-62 (pgs. 189-207); chapter 6,
sections 68-70 (pgs. 229-237), 72-76 (pgs. 240-257); chapter 7,
sections 72 (pgs. 264-269) and 85 (pgs. 288-290). All of these
sections were listed as reading assignments for HW 5 through HW 8.
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
Cauchy's Integral Formulas and Related Theorems
- Know Cauchy's Theorem, the Cauchy's Integral formulas (SO 5.1)
and be able to prove Cauchy's integral formula for the simplest case,
$n=0$. Be able to state and prove Cauchy's inequalities and
Liouville's Theorem.
- Be able to state the Maximum Modulus Theorem and to use it to
solve problems similar to ones one in class, on the homework and in
Test 2.
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 $f$ is the Taylor series for
$f$ about a point $z_0$ (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, problems in Assignment 7 and
a Laurent
series example.) Be able to find the principal part of a Laurent
series.
Isolated Singularities, Isolated Zeros and Residues
- Be able to classify isolated singularities of a function and
isolated zeros of a function.
- Be able to find residues at isolated singularities of a function.
The Residue Theorem and Its Applications
- Be able to state the Residue Theorem and to use it to evaluate
definite integrals. (See class notes for Dec. 1 and problems in
Assignment 8.).
Updated 12/3/23.