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Undergraduate Study

Unit Information Sheet for MATH3068: Analysis

Lecturer

  • Daniel Hauer
  • Consultation hour : Wednesdays, 1-2pm, Carslaw Building, Room 635.

To contact the lecturer please email math3068@sydney.edu.au including your name and SID.

Please use the Ed - Discussion forum if you have questions to the content of the lecture, tutorial, assignment or quizzes.

Resource page

All resources of this course such as tutorial sheets, solutions to tutorials, assignment problem sheet and further handouts can be found on the Resource page.

Ed - Discussion forum

If you have questions to the content of the lecture, tutorial, assignment or quizzes, then please write your question in the Ed - Discussion forum .

What's New

All important announcements will be posted here.

  • Reading for week 1: Chapter 1. To get ready for our first lecture on Monday, 25 July 2016: Read Section 1.7 Summary of definitions and properties .
  • Reading for week 2: Rest of Chapter 1.

Lecture Notes

This lecture follows the lecture notes written by Donald I Cartwright, "Analysis" (available from Kopystop).

  • Here is a list of typos in the lecture notes. Updated September 2, 2013.
The lecture will be recorded and can be found online in Echocenter/Blackboard. In addition, we will provide the handwritten lecture notes on the Resources page.

Tutorials

The tutorials are on

  • Mondays, 4pm, Carslaw Building, Room 361
  • Wednesdays, 4pm, Abercrombie Business School, Collaborative Learning Studio 2130
  • Thursdays, 4pm, Carslaw Building, Room 451.

Tutorials begin in week 2. and will be held in the form of board tutorials. This means that all students will work on white boards together with the tutor on a problem sheet, which is related to the lecture. After the tutorial the students should be able to do the homework sheet independently. The tutorial sheets are not available online, but the homework sheets can be downloaded as PDF files from the Resources page at the end of each week.

Tutorials are an integral part of the course and rolls will be kept. Mathematics is best learned by doing problems yourself, so attending tutorials is essential for performing well in the course.

Assessment

There will be one assignment worth 5%, two 40 min Quizzes counting 15% each, and one final exam worth 65% of the total assessment.

Quizzes
  • Quiz 1: Monday, 28 August 2017 at 10am in Room 173.
  • Quiz 2: Monday, 23 October 2017 at 10am in Room 173.
Assignment

The assignment will be due on Sunday, 24 September 2017 at 11:59pm. Please note that the maximum possible extension for the assignment is 7 days (including the weekend). The handwritten assignment must be scanned/imaged and submitted in PDF format online via LMS with Turnitin. Please ensure your submitted pdf is legible and keep your original handwritten version.

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

Final Exam

There will be a two-hour final exam at a date to be announced, during the official two-week examination period. The following PDF-file contains more information about the final exam: Exam Information

The Course

Analysis grew out of calculus, which leads to the study of limits of functions, sequences and series. The aim of the unit is to present enduring beautiful and practical results that continue to justify and inspire the study of analysis. This course will be useful not just to students of mathematics but also to engineers and scientists, and to future school mathematics teachers, because we shall explain why common practices in the use of calculus are correct, and understanding this is important for correct applications and explanations. The unit has three parts: the foundations of calculus including the study of sequences and series (including power series), the theory of Fourier series, and complex analysis.

Assumed Knowledge: The prerequisite for MATH3068 is 12cp intermediate mathematics. You will need a good working knowledge of the first year differential and integral calculus units, including the basics of complex numbers.

Objectives

The overall objectives of this unit of study are to improve your ability to think logically, analytically and abstractly, and to enhance your problem-solving skills. These attributes belong to the 'Research and Enquiry' category of the University's desired generic skills for its graduates.The objectives will be achieved through the study of specific topics in real and complex analysis. This unit will extend your knowledge of complex numbers, functions and calculus through the study of infinite series of numbers and infinite series of functions.These topics, which are placed in a more general context than seen in the earlier part of your degree, lead to the study of power series, Fourier series and contour integration. Along the way, you should improve your proof-writing skills and your ability to take active notice of details of mathematical statements.

Learning Outcomes

Students who successfully complete this unit of study will have demonstrated satisfactory performance in most of the following tasks:

  • determine the convergence or divergence of a variety of sequences and series of real or complex numbers;
  • pointwise and uniform convergence of sequences and series of functions;
  • find the radius of convergence of power series with real or complex coefficients;
  • find the sums of some particular convergent series;
  • determine where some particular functions of a complex variable are differentiable;
  • understand the definitions of the complex exponential and logarithm functions, also what is meant by taking a complex power of a complex number;
  • be able to calculate the Fourier coefficients of a number of different functions and understand both pointwise and Cesaro convergence of the Fourier series;
  • understand and use Parseval's identity to evaluate certain series;
  • understand the definition of the Bernoulli numbers and Bernoulli polynomials and be able to use them in the Euler Maclaurin and Stirling formulas;
  • understand the Cauchy Integral Formula and the Residue Theorem and be able to evaluate contour integrals.

Reference Books

  • Kosmala, W.A.J. A Friendly Introduction to Analysis, Pearson Prentice Hall International Edition.
  • Ruel V. Churchill, James Ward Brown Fourier series and boundary value problems, McGraw-Hill, New York
  • Ruel V. Churchill, James Ward Brown Complex variables and applications, McGraw-Hill, New York