Real and Complex Analysis
|Unit level:||Level 2|
|Teaching period(s):||Semester 1|
|Offered by||School of Mathematics|
|Available as a free choice unit?:||N|
The course unit unit aims to introduce the basic ideas of real analysis (continuity, differentiability and Riemann integration) and their rigorous treatment, and then to introduce the basic elements of complex analysis, with particular emphasis on Cauchy's Theorem and the calculus of residues.
The first half of the course describes how the basic ideas of the calculus of real functions of a real variable (continuity, differentiation and integration) can be made precise and how the basic properties can be developed from the definitions. It builds on the treatment of sequences and series in MATH10242. Important results are the Mean Value Theorem, leading to the representation of some functions as power series (the Taylor series), and the Fundamental Theorem of Calculus which establishes the relationship between differentiation and integration.
The second half of the course extends these ideas to complex functions of a complex variable. It turns out that complex differentiability is a very strong condition and differentiable functions behave very well. Integration is along paths in the complex plane. The central result of this spectacularly beautiful part of mathematics is Cauchy's Theorem guaranteeing that certain integrals along closed paths are zero. This striking result leads to useful techniques for evaluating real integrals based on the 'calculus of residues'.
On completion of this unit successful students will be able to:
- understand the concept of limit for real functions and be able to calculate limits of standard functions and construct simple proofs involving this concept;
- understand the concept of continuity and be familiar with the statements and proofs of the standard results about continuous real functions;
- understand the concept of the differentiability of a real valued function and be familiar with the statements and proofs of the standard results about differentiable real functions;
- appreciate the definition of the Riemann integral, and be familiar with the statements and proofs of the standard results about the Riemann integral including the Fundamental Theorem of Calculus;
- understand the significance of differentiability for complex functions and be familiar with the Cauchy-Riemann equations;
- evaluate integrals along a path in the complex plane and understand the statement of Cauchy's Theorem;
- compute the Taylor and Laurent expansions of simple functions, determining the nature of the singularities and calculating residues;
- use the Cauchy Residue Theorem to evaluate integrals and sum series.
- Other - 20%
- Written exam - 80%
Assessment Further Information
- Coursework; An in-class test in reading week for Real Analysis, an on-line test for Complex Analysis, each counting 10%.
- 3 hours end of semester examination; Weighting within unit 80%.
- Limits. Limits of real-valued functions, sums, products and quotients of limits. [5 lectures]
- Continuity. Continuity of real-valued functions, sums, products and quotients of continuous functions, the composition of continuous functions. Boundedness of continuous functions on a closed interval. The Intermediate Value Theorem. The Inverse Function Theorem. 
- Differentiability. Differentiability of real-valued functions, sums, products and quotients of continuous functions, Rolle's Theorem, the Mean Value Theorem, Taylor's Theorem. 
- Integration. Definition of the Riemann integral, integrability of monotonic and continuous function, the Fundamental Theorem of Calculus. 
- The complex plane. The topology of the complex plane, open sets, complex sequences and series, power series, and continuous functions. 
- Differentiation. Differentiable complex functions and the Cauchy-Riemann equations. 
- Integration. Integration along paths, the Fundamental Theorem of Calculus, the Estimation Lemma, Cauchy's Theorem, Argument and Logarithm. 
- Taylor and Laurent Series. Cauchy's Integral Formula and Taylor series, Louiville's Theorem and the Fundamental Theorem of Algebra, zeros and poles, Laurent series. 
- Residues. Cauchy's Residue Theorem, the evaluation of definite integrals and summation of series. 
- Mary Hart, Guide to Analysis, Macmillan Mathematical Guides, Palgrave Macmillan; second edition 2001.
- Rod Haggerty, Fundamentals of Mathematical Analysis, Addison-Wesley, second edition 1993.
- Ian Stewart and David Tall, Complex Analysis, Cambridge University Press, 1983.
Feedback tutorials will provide an opportunity for students' work to be discussed and provide feedback on their understanding. Coursework or in-class tests (where applicable) also provide an opportunity for students to receive feedback. Students can also get feedback on their understanding directly from the lecturer, for example during the lecturer's office hour.
- Lectures - 44 hours
- Tutorials - 22 hours
- Independent study hours - 134 hours
Teaching staffMark Coleman - Unit coordinatorCharles Walkden - Unit coordinator
MATH20101 Complex Analysis Lecture Notes
Lecture notesThe 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 Leicester (and others): I understand that Jeremy Levesley has recommended my notes for part of the Complex Analysis course at Leicester. 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!
CourseworkThe 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.
VideosThere 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.
- Argument and modulus in the complex plane
- Why can't we draw the graph of a complex function?
- The Cauchy-Riemann Theorem and its converse
- The complex logarithm and complex powers
- Paths and contours
- Complex integration
- Poles and residues
Hand-written slides from the lecturesBelow 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 (part 1), Week 2 (part 2), Week 3, Week 4, Week 5, Week 7 (part 1), Week 7 (part 2), Week 8 (part 1), Week 8 (part 2), Week 9, Week 10, Week 11, Week 12 (revision lecture),