### Math 301

This page is located at https://www.math.ubc.ca/~rfroese/math301/ and will be updated regularly throughout the term.

#### Instructor information

• Instructor: Richard Froese
• Email: rfroese@math.ubc.ca
• Office: Math Annex 1106
• Hours: By appointment. I might set fixed hours later in the term.
• Office Phone: 822-3042

#### Course Overview

Topics. Timings are approximate.

1. Complex integration - 1.5 weeks
2. Multivalued functions, branch points and branch cuts - 1.5 weeks
3. Integrals involving multivalued functions - 1.5 weeks
4. Conformal mappings and applications - 2.5 weeks
5. Poles and zeros of complex functions - 1 week
6. Fourier analysis - 2 weeks
7. Laplace transform - 2 weeks

Text

• Fundamentals of Complex Analysis by Saff and Snider (Third Edition).

You may also consult

• Handwritten notes by Michael Ward (see below)
• Complex Variables, Introduction and Applications by Ablowitz and Fokas

We may cover some material not in the textbook.

#### Location and Time

MWF 11:00-12:00 in LSK 460

#### Homework and Tests

There will be weekly homework assignments. The assignments and due dates will be posted on this page. Late homework will not be accepted. Even if you miss the deadline, its a good idea to do the problems, since this is the best way to prepare for the tests and exam. You are welcome to discuss the homework problems with your friends, but are expected to hand in your own work.

There will be two midterm tests in class on Monday, February 2 and Friday March 16 as well as a final exam during the April exam period. You will not be permitted to bring calculators or formula sheets to the tests and exam.

 Homework (lowest two scores dropped): 10% Midterms: 2 x 20% Exam: 50% 40%

If you miss the test for a legitimate reason (e.g., illness with doctors note), the weight of the final exam will be increased.

#### Problem sets

Homework assignments are due in class on the due date.

 Homework 1 Section 5.6 p.285: 1 adeg, 5abcd, 12, 13, 14, 15 Due: Monday Jan 8 solutions1.1.pdf solutions1.2.pdf Homework 2 Section 6.2 p.317: 2, 9, 10 (evaluate for all $n\in\mathbb Z$), Section 6.3 p.325: 3, 10, 11, 15bc Due: Monday Jan 15 solutions2.pdf Homework 3 Section 6.4 p.336: 3, 6, Section 6.5 p.344: 2, 5, 10 Due: Monday Jan 22 -

#### Files

Here are a collection of handwritten notes by Michael Ward that you might find useful.

This file contains some basic examples of the residue calculus.

Here are some basic estimates that we use repeatedly.

Here are some notes on evaluating infinite sums using residues.

Here are some notes by Rosales on branch points and cuts.

#### Class notes

Wed Jan 3 5.6, 5.7 Introduction, Classification of singularities
Fri Jan 5 6.1, 6.2 Review of residue calculus: computing the residue. See here for some worked examples. Don't worry about the residue at infinity for now, we will discuss this on Monday. I have also posted some basic estimates as a reference. I might add to this file occasionally. Note the homework assignment due on Monday! Some of these problems might be challenging if it has been a while since you took Math 300.
Mon Jan 8 6.3 Residue at infinity, trig integrals (example). This should be review from math 300
Wed Jan 10 6.4 Examples where we add a contour to an integral we want to evaluate to use residues. Then we have to deal with the added contour by showing e.g. (i) the additional term goes away in a limit, (ii) the additional term is a multiple of the integral we are trying to compute, (iii) the additional term can be evaluated explicitly. Examples include integrals of the form $\int_{-\infty}^\infty e^{ikx} p(x)/q(x) dx$, (Jordan's lemma),$\int_0^\infty 1/(1+x^3) dx$ $\int_0^\infty \sin(x^2) dx$.
Fri Jan 12 - more examples, evaluating infinite sums of the form $\displaystyle{\sum_{k\in\mathbb Z}\frac{p(k)}{q(k)}}$. Here are some notes.
Mon Jan 15 6.5 Principal value integrals at a singularity, indented contours.