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Undergraduate Study

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).

Florica Cîrstea
email: MATH2962@sydney.edu.au (please include your SID with all correspondence)
Consultation: 1-2pm, Thursday (Carslaw 719)

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 resources 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.

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 13 April (during Practice Class)
Quiz 2: Thursday 1st June (during Practice Class)

Assignment Collection Date:

Assignment: Friday 19 May

All assignments must be submitted electronically. Late assignments will not be marked.

Electronic copies must be submitted through the LMS, where they will be passed through the text matching software Turnitin (scanned copies of handwritten assignments are fine).

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

For all assessments, the rules for special consideration/arrangement apply. The maximum possible extension is 7 days.

The final mark is determined by the following criteria

  • High Distinction (HD), 85-100: representing complete or close to complete mastery of the material;
  • Distinction (D), 75–84: representing excellence, but substantially less than complete mastery;
  • Credit (CR), 65–74: representing a creditable performance that goes beyond routine knowledge and understanding, but less than excellence;
  • Pass (P), 50–64: representing at least routine knowledge and understanding over a spectrum of topics and important ideas and concepts in the course;
  • Fail (F), 0–49: representing rather limited understanding on a significant range of topics and concepts.

Course Description

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 from first year advanced mathematics.

Outcomes

Students who successfully complete this course should

  • know the basic completeness properties of the real numbers in various forms, including supremum and infimum;
  • be familiar with the basic topological properties of the real and complex euclidean space;
  • have solid foundations in the more formal aspects of analysis, including a knowledge of precise definitions, how to apply them and the ability to write simple proofs;
  • be able to work with inequalities, limits and limit inferior/superior;
  • know and be able to apply convergence tests for sequences and series;
  • have a working knowledge of (real and complex) power series and their properties;
  • know the basic facts on analytic functions including the uniqueness theorem;
  • be able to distinguish uniform and pointwise convergence;
  • work with sequences of uniformly convergent sequences of functions with applications to integration and differentiation;
  • 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)