## Week 12: Monday March 28th to Friday April 1st

Although there is no new exercise sheet, there will be exercise classes as usual, to hand back your marked Sheet 9 scripts and to discuss any difficulties on the course.

In Thursday's lectures I hope to talk about the two (non-examinable) topics in Section VI, and in Friday's lecture I will answer any questions you may have about the course.

If at any time between now and the exam, any of you want to try the 2010 paper and give your script to me to mark, I would be happy to do so.

## Module Information

Level:6. Semester:6.

Lecturer: Prof S Bullett (Room 252, Maths Building)
e-mail: s.r.bullett [at] qmul.ac.uk tel: 020 7882 5474

Timetable: Lectures: Thurs 11-1 (MLT) and Fri 12-1 (Maths 103). Exercise classes: Mon 1-2 (ENG 324) and Thurs 10-11 (MLT). Course began in the week beginning 10th January 2011 and finishes in the week ending 1st April 2011.

Office Hours: Wednesdays and Fridays 1.30-2.30.

Prerequisites: MTH5103 Complex Variables and MTH5104 Convergence and Continuity. (MTH5105 Differential and Integral Analysis is not a formal prerequisite but could be helpful too.)

Assessment: 90% final examination (May 2011) and 10% based on weekly exercise sheets.

## OVERVIEW

As you will have seen in MTH5103 "Complex Variables", when we do calculus with complex numbers in place of real numbers we find a whole new world of mathematics. Some results become simpler: for example Taylor's Theorem, which for real functions is about approximating differentiable functions by power series, now tells us that complex differentiable functions are equal to power series. But the complex world brings completely new results, for example Cauchy's Theorem, and the Calculus of Residues, which are not just beautiful mathematics, but have consequences and applications to many different problems which at first sight have nothing to do with complex numbers.

In this course we are going to take the subject much further than did "Complex Variables". This will involve us first covering some of the same ground again (for example Cauchy's Theorem), but with more precise statements and detailed proofs. This part of the course will apply the mathematical rigour of "Convergence and Continuity" and "Differential and Integral Analysis" to the main results of "Complex Variables". We shall then explore a number of applications - some will be developments of themes already introduced in "Complex Variables" (for example the geometry of conformal maps, and harmonic maps, and the application of the calculus of residues to the summation of power series), and others will be new to most students (for example analytic continuation and Riemann surfaces). We shall encounter many famous results (Picard's Theorem, the Riemann Mapping Theorem, the Prime Number Theorem,...) most of which we shall only state, but some of which we may prove, or at least sketch prove. And I hope that we shall have time in the final couple of weeks to look at how complex analysis is involved in two very different areas of very active current research:

(i) Iteration of complex functions (the theory behind the beautiful pictures of Julia sets and the Mandelbrot set..)

(ii) The Riemann Hypothesis (the most important unresolved question in mathematics today...)

A detailed syllabus will appear bit by bit as the course progresses. The official course description on http://www.maths.qmul.ac.uk/undergraduate/modules is quite flexible as regards the contents of the second half of the course, and I would like to take your interests into account in deciding exactly what we cover.

## Learning Outcomes

A rigorous understanding of classical complex analysis.
(A more detailed version of this objective is available on the School of Mathematical Sciences learning outcomes webpage.)

## Lecture Notes

These will appear here chapter by chapter in downloadable pdf form, one to two weeks after the relevant lecture. The list of headings below is from the 2009-2010 course, and may change a little in 2010-11.

Apologies that the downloadable pdf notes below have no examples or pictures, and for any misprints (please bring these to my attention).

I. Holomorphic Functions:
1. The complex numbers. 2. Open sets and closed sets. 3. Limits. continuity and compactness. 4. Differentiation. 5. Power series.
Section I lecture notes.

II. Integration and Cauchy's Theorem:
6. Paths and integration. 7.The fundamental theorem of calculus. 8. Lengths and the estimation lemma. 9. Cauchy's Theorem.
Section II lecture notes

III. Consequences of Cauchy's Theorem:
10. Cauchy's formulae. 11.Taylor's theorem. 12. Laurent series and the Residue Theorem. 13.Evaluating real integrals and sums of series using the Residue Theorem. 14. Types of singularity and behaviour near singularities. 15. Differentiable maps on the Riemann sphere
Section III lecture notes

IV. Conformal maps:
16. Geometric interpretation of differentiability. 17. Automorphisms of the Riemann sphere (Moebius transformations). 18. Conformal maps between subsets of the plane (Riemann Mapping Theorem). 19. Harmonic maps. 20. Holomorphic maps of the unit disc (Schwarz's Lemma).
Section IV lecture notes

V. Analytic Continuation and Riemann Surfaces:
21. Continuation along paths. 22. Riemann surfaces. 23. The Schwarz reflection principle. 24. Elliptic functions. 25. Picard's Theorem.
Section V lecture notes
Handwritten notes of lecture on 25th March (including illustrations)
Analysis of course questionnaires

At the end of week 11 we had completed Section V.

VI. Further Topics (not for exam):
26. The Zeta function: Riemann's Hypothesis. 27. Complex iteration: Julia sets and the Mandelbrot set.
Section VI lecture notes

## Summary of Main Results

Summary of Main Results and Big Ideas in Proofs (now completed, covering all examinable sections of the course, but you might like to assemble your own version to help you with your exam revision)

## Exercise Sheets

Best 7 of 9 sheets count towards the final mark.

Exercise sheet 1 to be handed in by 12 noon Fri 21 Jan 2011.

Exercise sheet 2 to be handed in by 12 noon Fri 28 Jan 2011.

Exercise sheet 3 to be handed in by 12 noon Fri 4 Feb 2011.

Exercise sheet 4 to be handed in by 12 noon Fri 11 Feb 2011.

Exercise sheet 5 exceptionally to be handed in by 12 noon on Mon 28 Feb (extra week because of READING WEEK 21-25 Feb).

Exercise sheet 6 to be handed in by 12 noon on Fri 4 March 2011.

Exercise sheet 7 to be handed in by 12 noon on Fri 11 March 2011.

Exercise sheet 8 to be handed in by 12 noon on Fri 18 March 2011.

Exercise sheet 9 to be handed in by 12 noon on Fri 25 March 2011. This is a corrected version. Apologies that the version that originally appeared here and was handed out in class had several lines in question 1 repeated due to a "cut and paste" error.

## EXAMINATION

The examination will last 2 hours. The paper will contain SIX questions and the rubric will state:

"You may attempt as many questions as you wish and all questions carry equal marks. Except for the award of a bare pass, only the best FOUR questions answered will be counted."

Here is the 2010 Examination Paper. If anyone who tries the 2010 paper would like me to mark their solutions, I would be happy to do so.

You are reminded that examinations are designed to test "Learning Outcomes", and that the Learning Outcome of MTH6111 is:

"A rigorous understanding of classical complex analyis". (A more detailed version of this objective is available on the School of Mathematical Sciences learning outcomes webpage.)

## BOOKS

The following books are good references introducing 'Complex Analysis' from a pure mathematical point of view and covering much of the course except for some advanced topics. There are advantages and disadvantages to each textbook, and I have not chosen any particular textbook to follow as regards notation and terminology this year:

H.A.Priestley, 'Introduction to Complex Analysis' (Oxford University Press 1990)

John M.Howie, 'Complex Analysis' (Springer Undergraduate Mathematics Series 2003)

I.N Stewart and D.OTall, 'Complex Analysis' (Cambridge University Press 1983)

For a geometric point of view, see:

Tristan Needham, 'Visual Complex Analysis' (Oxford University Press 1999)

There are also many more advanced texts on complex function theory you might find it interesting to dip into, for example Lars Ahlfors, 'Complex Analysis' (McGraw-Hill, 3rd edition 1979), or Serge Lang, 'Complex Analysis' (Springer Graduate Texts in Mathematics No. 103, 4th edition 2003). And there are specialist texts on individual topics which also cover much more than we shall: some of these will be mentioned in the course as we go along.

I shall also try to sort out some interesting web references, for example relevant articles in Plus Magazine (such as Marcus du Sautoy's article and Bob Devaney's article). For biographies of some of the celebrated mathematicians responsible for the development of the subject, Cardano, Bombelli, Wessel, Argand, Gauss, Cauchy, Riemann, Liouville, Weierstrass, Picard, Jordan, Poincare, ... see the University of St Andrews mathematical biographies website.

I recommend a video made a couple of years ago by Etienne Ghys, of the Ecole Normale Superieure de Lyon. Chapters 5 and 6 have material on complex numbers and complex iteration, but you should find the other Chapters interesting too. The video can be watched on the web, downloaded for free, or bought as a DVD. The website address is www.dimensions-math.org

Watch this space, and let me know of anything interesting you find too.

Finally, for bed-time reading on the Riemann Hypothesis (and for much more about the properties and uses of prime numbers):

Marcus du Sautoy, 'The Music of the Primes' (Fourth Estate 2003)

Shaun Bullett (last up-dated 1/4/2011).

http://www.maths.qmul.ac.uk/~sb