MATH20101 Real and Complex Analysis: Complex Analysis 2018-19

MATH20101 Real and Complex Analysis: Complex Analysis learning materials, 2018-19

Learning outcomes

At the end of the course you will be able to
• prove the Cauchy-Riemann Theorem and its converse and use them to decide whether a given function is holomorphic;
• use power series to define a holomorphic function and calculate its radius of convergence;
• define the complex integral and use a variety of methods (the Fundamental Theorem of Contour Integration, Cauchy's Theorem, the Generalised Cauchy Theorem and the Cauchy Residue Theorem) to calculate the complex integral of a given function;
• define and perform computations with elementary holomorphic functions such as sin, cos, sinh, cosh, exp, log, and functions defined by power series;
• use Taylor's Theorem and Laurent's Theorem to expand a holomorphic function in terms of power series on a disc and Laurent series on an annulus, respectively;
• identify the location and nature of a singularity of a function and, in the case of poles, calculate the order and the residue;
• apply techniques from complex analysis to deduce results in other areas of mathematics, including proving the Fundamental Theorem of Algebra and calculating infinite real integrals, trigonometric integrals, and the summation of series.

Lecture notes

The lecture notes for the course can be found here:

The lecture notes also contain the exercises and solutions to the exercises. I will trust you to have a serious attempt at the exercises before referring to the solutions.

There are separate files containing just the exercises here and just the solutions here

Note to people from outside Manchester: I understand that various people outside Manchester have recommended my notes for part of their Complex Analysis course. If any of you spot any typos/mistakes/etc in the notes then please let me know (email: charles dot walkden at manchester dot ac dot uk) - thanks!

Hand-written slides from the lectures

Below will be the visualiser slides from the lectures. My advice would be to read the printed lecture notes above rather than to read the slides; however they may be useful if you want to double-check what I wrote in a lecture.

Week 1, Week 2, Week 3, Week 4, Week 5, Week 7, Week 8, Week 9, Week 10, Week 11, Week 12.

Coursework

The coursework for this course will take the form of a 40 minute closed-book test held during Week 6 (Reading Week). All questions on th e test are compulsory and it will be in the format of an exam question. Thus, looking at past exam papers will provide excellent preparation for the test. You will need to know sections 2,3,4 from the course for the test (this is the material that we will cover in weeks 1--5).

A mock coursework test is available here.

Videos

There are a series of short videos explaining either key points from the course or commonly-asked questions that are slightly off-topic to address in lectures. The appropriate time for watching each video is given in the lecture notes. (Note that these videos are only available to Manchester students.)

Feedback

You will get feedback on your understanding in this course from (i) the Week 6 coursework test, (ii) the regular Kahoot! quizzes, (iii) during the support classes from the PhD demonstrators, (iv) during my office hour, and (v) via the generic feedback (and exam script viewing) on the final exam. This feedback will allow you to gauge your understanding of the material during the course, understand the mark that you got in the assessment, understand how you could do better in the future.

Past exams

Below are the exam papers, together with feedback, for the complex analysis part of the course for the last 3 years.

Tibees, an Australian Youtuber, has uploaded a video talking about a past exam for this course. You can watch the video here