37234 Complex Analysis
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Subject handbook information prior to 2025 is available in the Archives.
Credit points: 6 cp
Result type: Grade and marks
Requisite(s): ((35102 Introduction to Analysis and Multivariable Calculus OR 37132 Introduction to Mathematical Analysis and Modelling)) OR ((33230 Mathematics 2 OR 33290 Statistics and Mathematics for Science) AND 35007 Real Analysis)
These requisites may not apply to students in certain courses. See access conditions.
Anti-requisite(s): 35232 Advanced Calculus AND 68038 Advanced Mathematics and Physics
Description
Transform methods such as the Laplace transform are useful in solving differential equations that arise in many areas of applications including signal analysis, mathematical finance and various queuing models in quantitative management. This subject highlights the areas of advanced calculus needed to justify the use of complex integration to invert the Laplace Transform when solving such problems. Topics include line integrals; Green's theorem; functions of a complex variable; analytic functions; Cauchy-Riemann equations; complex integrals; Cauchy's integral theorem; residues and poles; contour integration; and inversion of Laplace Transform.
Subject learning objectives (SLOs)
Upon successful completion of this subject students should be able to:
| 1. | Correctly perform routine calculations, such as the evaluation of integrals using complex variable theory. |
|---|---|
| 2. | Understand and apply the various theoretical results which justify these methods of calculation |
| 3. | Understand the relationship of advanced calculus and complex variables to other areas of mathematics and its applications and be able to apply this understanding to other disciplines such as aeronautical engineering, finance and physics. |
| 4. | Correctly apply the subject matter covered in lectures, tutorials and assignments to previously unseen problems |
Course intended learning outcomes (CILOs)
This subject also contributes specifically to the development of following course intended learning outcomes:
- Demonstrate theoretical and technical knowledge of mathematical sciences including calculus, discrete mathematics, linear algebra, probability, statistics and quantitative management. (1.1)
- Evaluate mathematical and statistical approaches to problem solving, analysis, application, and critical thinking to make mathematical arguments, and conduct experiments based on analytical, numerical, statistical, algorithms to solve new problems. (2.1)
- Design creative solutions to contemporary mathematical sciences-related issues by incorporating innovative methods, reflective practices and self-directed learning. (4.1)
- Use succinct and accurate presentation of reasoning and conclusions to communicate mathematical solutions, and their implications, to a variety of audiences, using a variety of approaches. (5.1)
Contribution to the development of graduate attributes
1. Disciplinary knowledge and its appropriate applications. Complex variable theory is one of the powerful tools used by mathematicians, physicists, engineers. A sound knowledge of its methods and principles is fundamental to most areas of mathematics and its applications. In this subject these methods are presented and assessed in every assessment task.
2. An Inquiry-oriented approach. Throughout the course mathematics is presented as a tool, which students are challenged to use in the solution of theoretical and real-world problems, such as the solution of differential equations and the evaluation of integrals that arise in areas such as physics, finance and engineering. In this subject these methods are presented and assessed in every assessment task.
Students learn the techniques of complex variable theory, which can be used to solve many different theoretical and applied problems. For example, they learn the techniques of Laplace transform inversion which are used in the pricing of certain financial products. This is assessed in the final exam and in the Tests and tutorials.
4. Reflection, Innovation, Creativity
5. Communication
Teaching and learning strategies
Lectures: Approximately two-three hours/week, with lecture recordings given online
Tutorials: one hour/week on campus.
Class tests: These will be timed tests given remotely, but at a particular time specified in advance
A complete set of lecture notes and recording will be posted, as well as tutorials, class tests, and complete solutions to all problems. In addition past final exams with solutions will be posted - students should be aware that the form of the final exam will be different to past years because the exam will be held remotely.
Students are expected to prepare for each class each week by viewing the lectures, then by doing the Tutorial problem sets and reading material from the notes.
In tutorials students are able to collaborate remotely with their peers on the solution of the weekly problem sets. The solutions to the problem sets will be recorded and placed online.
Students will be given regular written and verbal feedback in the tutorial sessions. The most common errors being made by the students will be addressed in lectures with explanations and advice on how to avoid them. Feedback is given on Class Tests when they are returned to the students. The tutors also provide feedback and assistance in the tutorial classes as the students work through problems. Significant feedback will also be given through official student email.
The assessment tasks are based upon the application of the theory learned in lectures and through other resources, such as the online notes, which provide greater depth than are covered in the lectures.
Content (topics)
Topics include:
- Paths in the complex plane
- Functions of a complex variable
- Exponential and logarithmic functions
- Complex differentiation
- Analytic functions
- The Cauchy-Riemann equations
- Integration in the complex plane
- Cauchy's integral theorem and integral formula
- Taylor and Laurent series
- Residues and poles
- Contour integration
- The Laplace Transform and Laplace transform integrals.
Assessment
Assessment task 1: Tutorials
| Intent: | This assessment task contributes to the development of Graduate attributes : 1. Disciplinary knowledge and its appropriate applications. 5. Communication |
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| Objective(s): | This assessment task addresses subject learning objective(s): 1, 2 and 4 This assessment task contributes to the development of course intended learning outcome(s): 1.1 and 5.1 |
| Weight: | 10% |
| Criteria: | Engagement with the material during the meetings. Helping other students solve problems, presenting and explaining correct solutions to problems. If attendance at the remote meeting is not possible, then a submission of an attempt of the tutorial questions will be accepted. |
Assessment task 2: Class Tests
| Intent: | This assessment task contributes to the development of Graduate attributes : 1. Disciplinary knowledge and its appropriate applications. 2. An Inquiry-oriented approach.
|
|---|---|
| Objective(s): | This assessment task addresses subject learning objective(s): 1 and 2 This assessment task contributes to the development of course intended learning outcome(s): 1.1 and 2.1 |
| Weight: | 40% |
| Criteria: | accuracy of proofs and calculations, Clarity of answers, Correctness of results |
Assessment task 3: Final Exam
| Intent: | This assessment task contributes to the development of Graduate attributes : 1. Disciplinary knowledge and its appropriate applications. 2. An Inquiry-oriented approach. 4. Reflection, Innovation, Creativity |
|---|---|
| Objective(s): | This assessment task addresses subject learning objective(s): 1, 2 and 4 This assessment task contributes to the development of course intended learning outcome(s): 1.1, 2.1 and 4.1 |
| Weight: | 50% |
| Length: | The examination will be two hours and ten minutes long. |
| Criteria: | accuracy of proofs and calculations, Clarity of answers, Correctness of results |
Minimum requirements
A mark of 50 or higher is required to pass the subject.
Recommended texts
D G Zill & M R Cullen, Advanced Engineering Mathematics,
2nd edition, Jones & Bartlett, 2000,
References
Needham, T. Visual Complex Analysis, Oxford University Press, 1999.
Priestley, H. A. Introduction to Complex Analysis, Oxford University Press, 2003.
Saff, E. B. & Snider, A. D. Fundamentals of Complex Analysis, 3rd edition, Prentice Hall, 2002.
Shaw, W. Complex Analysis with Mathematica, Cambridge University Press, 2006.
Other resources
Lecture notes covering the subject will be provided on UTSOnline