### Math 301

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

Note: The focus has now shifted to the canvas page canvas page. Information about tests and exams on this page is out of date.

#### Announcements

The second midterm is at https://www.math.ubc.ca:~rfroese/math301/2020WT2midterm2.pdf

#### 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: review and more applications of residue calculus - 1.5 weeks
2. Multivalued functions, branch points and branch cuts - 1.5 weeks
3. Integrals involving multivalued functions: e.g. $\displaystyle{\int_0^\infty \dfrac{x^{\alpha-1}}{x+1} dx = \dfrac{\pi}{\sin(\alpha \pi)}}$ - 1.5 weeks
4. Conformal mappings and applications: Laplace's equation, ideal fluid flow in 2d - 2.5 weeks
5. Poles and zeros of complex functions: Rouche's theorem. - 1 week
6. Fourier analysis: Green's functions, delta function - 2 weeks
7. Laplace transform: ODE's, delay equations, stability. - 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
• A First Course in Complex Analysis (Free online textbook) by Beck, Marchesi, Pixton and Sabalka.

We may cover some material not in the textbook.

#### Location and Time

MWF 11:00-12:00 in MATX 1100

#### Homework and Tests

There will be weekly homework assignments. The assignments and due dates will be posted on this page and also on the canvas page for this course. 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 7 and Friday March 20 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.

#### Grades

The following weightings will be used in computing your final grade:

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

I will replace your lowest midterm grade with the final exam grade, if this improves your final grade. So if you are sick for one of the midterms, no doctor's note is needed.

#### Problem sets

Please upload your homework as a pdf file on the canvas page for this course.

 Homework 1 5.6: 1adeg, 5abcd, 12, 13, 14, 15; 6.1: 1bdf, 3ceg, 5, 7 Due: Monday Jan 13 hmk01.solutions.pdf Homework 2 6.2: 2, 9, 10 (in 10 evaluate for all $n\in {\mathbb Z}$); 6.3: 3, 10, 11 Due: Monday Jan 20 hmk02.solutions.pdf Homework 3 6.4: 8, 9, 10; 6.5: 6, 9, 12 Due: Monday Jan 27 hmk6.4.pdf, saffsnider6.5_69.pdf, saffsnider6.5_12.pdf Homework 4 6.6: 3, 5, 8, 10, 12 Due:Monday Feb 3 sec6.6.solns.pdf <- some more details added Homework 5 6.7 (p.~364) 6, 7, 8, 9, 10, 11. Due:Monday Feb 24 ss6.7.pdf Homework 6 hmk6.problems.pdf Due:Monday Mar 2 conformal.extras.solutions.5.pdf conformal.extras.solutions.1.scan.pdf conformal.extra.solutions.pdf ss7.2.pdf Homework 7 7.3: 2, 3, 6, 9, 11; 7.4: 4, 8, 9 Due: Wednesday March 11 ss7.3 ss7.4 Homework 8 7.6: 1, 3, 4, 6, 8, 9 Due: not due to help you prepare for the midterm ss7.6 Homework 9 . Due: . Homework 10 . Due: .

#### Class notes

Date Reading Topics
Mon Jan 6 5.6, 5.7 Introduction, Classification of singularities, determining the type of singularity from the local behaviour
Wed Jan 8 6.1, 6.2 Review of residue calculus, computing the residue
Fri Jan 10 . Computing the residue ctd. Review of Cauchy residue formula. residue at infinity, examples
Mon Jan 13 6.3, 6.4 Examples
Wed Jan 15 . SNOW
Fri Jan 17 6.5 Indented contours
Mon Jan 20 Rosales notes Multivalued functions, branch points, branch cuts
Wed Jan 22 . more branch cuts
Fri Jan 24 6.6 finding the branch points for eg $log(z^2-1)$
Mon Jan 27 . Range of angles method, cancellation of cuts.
Wed Jan 29 . Dogbone contour example (completed), integrals using pie contours.
Fri Jan 31 Some practice problems for the test: hmk4.problems.pdf You don't need to hand these in! Here are solutions hmk4.solutions.pdf .
Mon Feb 3 . argument principal and winding number
Wed Feb 5 . Rouche theorem, examples, fundamental theorem of algebra, (for the test: integralexample.pdf)
Fri Feb 7 . TEST 1
Mon Feb 10 . open mapping theorem, proof using Rouche and consequences
Wed Feb 12 6.7 Nyquist criterion for counting zeros un in half plane.
Fri Feb 14 . .
Mon Feb 24 . conformal maps: mapping properties of $\sin(z)$
Wed Feb 26 . conformal maps:upper half plane with piecwise constant boundary values
Fri Feb 28 . conformal maps: Joukowski map handwritten notes
Mon Mar 2 . conformal maps: examples using $\sin^{-1}(x)$
Wed Mar 4 . FLT
Fri Mar 6 . FLT
Mon Mar 9 . FLT
Wed Mar 11 . FLT:symmetric points tablet notes notes
Fri Mar 13 . Fluid Flow tablet notesnotes
Mon Mar 16 . .
Wed Mar 18 . .
Fri Mar 20 . TEST 2
Mon Mar 23 . .
Wed Mar 25 . .
Fri Mar 27 . .
Mon Mar 30 . .
Wed Apr 1 . .
Fri Apr 3 . .
Mon Apr 6 . .
Wed Apr 8 . .

#### Files

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

Scans of the first homework problems (oops, last page ) from text.

This file contains some basic examples of the residue calculus.

Here is the Math 300 exam from last term. It might help you review even if you were not in that class.

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.

Notes for the lecture on symmetric points.

This year’s midterm 1: 2019WT2midterm1.pdf