### 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 Friday, 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 solutions1.2 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 Homework 3 Section 6.4 p.336: 3, 6, Section 6.5 p.344: 2, 5, 10 Due: Monday Jan 22 solutions3 solutions3.2 Homework 4 hmk4.problems.pdf Due: Monday Jan 29 solutions4 Homework 5 Section 6.6: 1, 4, 10, 12 Not collected. Solutions posted to help prepare for midterm solutions5 Homework 6 hmk6.problems.pdf Due: Wednesday Feb 14 solutions6 Homework 7 7.3: 2, 3, 6, 9, 11, 7.4: 4, 8, 9 Due Monday Feb 26 solutions7 Homework 8 7.6: 1, 3, 4, 6, 8, 9 Due Monday Mar 5 solutions8 Homework 9 hmk9 Due Wednesday Mar 14 solutions9 Homework 10 hmk10 Due Wednesday Mar 28 solutions10 Homework 11 hmk11 Due Wednesday Mar 28 solutions11

#### 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.
Wed Jan 17 Rosales notes Multivalued functions, examples are inverse functions. Choosing a branch (i) by restricting the range or (ii) finding branch points, introducing branch cuts, picking a starting value and defining other values by continuity. Warm up with real functions,
Fri Jan 19 review: 3.3, 3.5 Branches of the argument, eg principal branch Arg$(z)$. The logarithm and some branches. (Graph of arg is the Riemann surface for the log. For this course Riemann surfaces are just a side note.) Definition of the complex power $z^\alpha$
Mon Jan 22 - Designer branches: finding branches of $\log(z^2-1)$ with specified cuts
Wed Jan 24 - Range of angles method for choosing a branch of $p(z)^\alpha$. Application: dogbone contour integral.
Fri Jan 26 6.6 Integrals involving multivalued functions. Old midterms posted above.
Mon Jan 29 - A few more examples. Here are some handwritten notes multiintroughnotes.pdf
Wed Jan 31 7.1 Behaviour of Laplace's equation under conformal mapping
Fri Feb 2 - Test 1
Mon Feb 5 7.2 Conformal maps, ctd
Wed Feb 7 - examples of solving Laplace's equation
Fri Feb 9 - Dirichlet problem on upper half plane with piecwewise constant boundary conditions on the real axis
Wed Feb 14 7.3 Fractional Linear Transformations (Moebius Transformations) basic properties
Fri Feb 16 7.4 FLT's mapping circles/lines to circles/line, mapping distinct points $z_1,z_2,z_3$ to $w_1,w_2,w_3$
Mon Feb 26 - cross ratio recap, symmetric points. Please fill out a course survey here
Wed Feb 28 7.6 Symmetric points ctd.
Fri Mar 2 - ideal fluid flow. Please have a look at Michael Ward's notes
Mon Mar 5 - fluid flow ctd. This material is discussed in Ablowitz and Fokas' book. Here are a couple of pages AF
Wed Mar 7 6.7 Argument principle and Rouche's theorem.
Fri Mar 9 - Open mapping theorem, Nyquist criterion. Test next Friday will cover conformal maps (solving Laplace's equation) with the maps we considered in class or in the homework. Also FLT's, their mapping properties, symmetric points. Fluid flow. Finally, Rouche's theorem and Nyquist. You should be able to do questions like 6.7: 6-12, 14.
Mon Mar 12 - Nyquist, ctd., definition of Fourier transform
Wed Mar 14 8.2 Fourier transform, inversion theorem, examples to illustrate this.
Fri Mar 16 - Test 2
Mon Mar 19 - Properties of the Fourier transform
Wed Mar 21 - Solving $-u''+\omega^2 u = f$ using the Fourier transform. The delta 'function' $\delta(x)$ and its Fourier transform. Fundamental solution and Green’s function. Fourier transform method only will find solutions whose Fourier transform exists. This does not include exponentially growing solutions. In this example, the solutions to the associated homogeneous equation are exponentially growing either to the right or to the left. The Fourier transform method computest the unique solution that is bounded at infinity.
Fri Mar 23 - Solving $-u''-\omega^2 u = f$. Non-uniqueness. An integral formula for the Airy function (solution to $u''(x) = x u(x)$). The heat equation.
Mon Mar 26 8.3 Laplace transforms. Definition, basic properties and inversion formula.
Wed Mar 28 - Computing inverse Laplace transforms using residue calculus. Old exams posted above. Last homework on Laplace transforms posted. This set will not be collected. I'll post the solutions next week.
Wed Apr 6 - Laplace examples: second order linear initial value problems, integral formula for the Bessel function $J_0$. Delay differential equations.
Fri Apr 8 - Examples, ctd.