Unit Information Sheet for MATH2962: Real and Complex Analysis (Advanced)

Time and Place

Lectures: Mon/Tue/Wed 9-10am Carslaw Room 275.
Practice Class: Thu 9-10am Carslaw Room 275 (the purpose of this class is to show examples, let students do simple problems/proofs under close guidance, or review lecture material).

Lecturer

Florica Cîrstea, Carslaw Room 719, phone 9351 2965
email: florica.cirstea@sydney.edu.au
Consultation: TBA

Web Page

http://www.maths.usyd.edu.au/u/UG/IM/MATH2962/

Lecture Notes

Real and Complex Analysis by D Daners available from Kopystop. Also make use of the reference books (you do not need to buy any of these).

Tutorials

You should attend at the time and place given on your timetable. See the timetable for the tutorial times. Please always take your lecture notes to the tutorial, so you can look up what you need to solve the problems.

Tutorial sheets are available online as PDF files from the handouts page on the Friday of the week before the tutorial takes place. No hard copies will be distributed.

Tutorials are an integral part of the course. You can only learn mathematics (or any other subject) by doing problems yourself, so attending tutorials is absolutely essential for performing well in the course.

Tutors: Florica Cîrstea Daniel Daners and James Parkinson

Assessment

There will be one assignment worth 10%, two 40 min quizzes worth 15% each. The final exam counts 60% of the total assessment.

Quiz Dates:

Quiz 1: Thursday 5th April (during Practice Class)
Quiz 2: Thursday 24th May (during Practice Class)

Assignment Collection Date:

Friday 11th May

Group assignments are not permitted. You are encouraged to collaborate with others in solving the problems, but the work submitted must be your own!

Late assignments are not accepted without prior arrangement well before the deadline!

Final Exam:

There will be a two-hour final exam. Only material covered in lectures and tutorials will be tested. The exam will also contain questions on the theory and proofs, and not just problems to solve.

The Course

For the official description see the entry for MATH2962 in the Units of Study Handbook.

In previous calculus courses many facts were introduced in an informal way using examples or diagrams as evidence. This course will give a rigorous treatment of many concepts known from calculus. Starting off with an axiomatic description of the real number system, this first course in analysis concentrates on the limiting behaviour of infinite sequences and series on the real line and the complex plane. These concepts are then applied to sequences and series of functions, looking at point-wise and uniform convergence. Particular attention is given to power series leading into the theory of analytic functions and complex analysis. Topics in complex analysis include elementary functions on the complex plane, the Cauchy integral theorem, Cauchy integral formula, residues and related topics with applications to real integrals.

Assumed Knowledge: A working knowledge of the calculus and complex numbers components of Mathematics 1.

Outcomes: Students who successfully complete this course should

• know the basic completeness properties of the real numbers in various forms;
• be familiar with the basic topological properties of the real and complex numbers;
• have solid foundations in the more formal aspects of analysis, including the ability to do simple proofs;
• be able to work with inequalities;
• know and be able to apply convergence tests for sequences and series;
• have a good knowledge of (real and complex) power series;
• know the basic facts on analytic functions;
• know how to apply the basic theorems of complex analysis such as the Cauchy integral theorem and formula, Laurent expansions, the residue theorem with application to real integrals.

Reference Books

• H. Amann, J. Escher, Analysis I, Birkhäuser, 2005. (Library 515 82)
• T.M. Apostol, Mathematical Analysis, Addison-Wesley, 1974. (Library 517 23A)
• R.G. Bartle, The Elements of Real Analysis, Oxford University Press, 1986. (Library 517 175)
• R. Courant and F. John, Introduction to Calculus and Analysis , Interscience Publishers, New York (Library 517 28)
• H.S. Gaskill, P.P. Narayanaswami, Elements of Real Analysis, Prentice Hall, 1998. (Library 515.822 1)
• M.H. Protter and C.B. Morrey, A First Course in Real Analysis, Springer, 1991. (Library 517.52 158)
• S. Lang, Complex Analysis, Springer (Library 517.8 208 or 515.9 2)
• S. Lang, Undergraduate Analysis, Springer, New York 1983. (Library 517 107A)
• H.A. Priestley, Introduction to Complex Analysis, Oxford University Press (Library copies)
• W. Rudin, Principles of Mathematical Analysis, McGraw-Hill, 1976. (Library 517 203)
• M Spivak, Calculus, Cambridge University Press, 2006 (Library 517 139 B)