PURE MTH 7054 - Complex Analysis

North Terrace Campus - Semester 1 - 2017

When the real numbers are replaced by the complex numbers in the definition of the derivative of a function, the resulting (complex-)differentiable functions turn out to have many remarkable properties not enjoyed by their real analogues. These functions, usually known as holomorphic functions, have numerous applications in areas such as engineering, physics, differential equations and number theory, to name just a few. The focus of this course is on the study of holomorphic functions and their most important basic properties. Topics covered are: Complex numbers and functions; complex limits and differentiability; elementary examples; analytic functions; complex line integrals; Cauchy's theorem and the Cauchy integral formula; Taylor's theorem; zeros of holomorphic functions; Rouch?'s Theorem; the Open Mapping theorem and Inverse Function theorem; Schwarz' Lemma; automorphisms of the ball, the plane and the Riemann sphere; isolated singularities and their classification; Laurent series; the Residue Theorem; calculation of definite integrals and evaluation of infinite series using residues; Montel's Theorem and the Riemann Mapping Theorem.

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Course Details

Course Code	PURE MTH 7054
Course	Complex Analysis
Coordinating Unit	Mathematical Sciences
Term	Semester 1
Level	Postgraduate Coursework
Location/s	North Terrace Campus
Units	3
Contact	Up to 3 hours per week
Available for Study Abroad and Exchange	Y
Prerequisites	MATHS 2100 or MATHS 2101 or MATHS 2202
Assumed Knowledge	MATHS 2101 or MATHS 2202
Assessment	ongoing assessment 30%, exam 70%

Course Staff

Course Coordinator: Associate Professor Nicholas Buchdahl

Course Timetable

The full timetable of all activities for this course can be accessed from Course Planner.

Course Learning Outcomes

1. Demonstrate understanding of the basic concepts underlying complex analyis.

2. Demonstrate familiarity with a range of examples of these concepts.

3. Prove basic results in complex analysis.

4. Apply the methods of complex analysis to evaluate definite integrals and infinite series.

5. Demonstrate understanding and appreciation of deeper aspects of complex analysis such as the Riemann Mapping theorem.

6. Demonstrate skills in communicating mathematics orally and in writing.

University Graduate Attributes

This course will provide students with an opportunity to develop the Graduate Attribute(s) specified below:

University Graduate Attribute	Course Learning Outcome(s)
Deep discipline knowledge informed and infused by cutting edge research, scaffolded throughout their program of studies acquired from personal interaction with research active educators, from year 1 accredited or validated against national or international standards (for relevant programs)	1,2,3,4,5
Critical thinking and problem solving steeped in research methods and rigor based on empirical evidence and the scientific approach to knowledge development demonstrated through appropriate and relevant assessment	1,2,3,4
Teamwork and communication skills developed from, with, and via the SGDE honed through assessment and practice throughout the program of studies encouraged and valued in all aspects of learning	6
Career and leadership readiness technology savvy professional and, where relevant, fully accredited forward thinking and well informed tested and validated by work based experiences	5,6
Self-awareness and emotional intelligence a capacity for self-reflection and a willingness to engage in self-appraisal open to objective and constructive feedback from supervisors and peers able to negotiate difficult social situations, defuse conflict and engage positively in purposeful debate	6

Learning Resources

Required Resources

None.

Recommended Resources

In increasing order of difficulty, the following books are available in the BSL. The closest to the level of this course is 2.

1. Churchhill & Brown: Complex Variables and Applications; 517.53 C563
2. Marsden & Hoffman: Basic Complex Analysis; 517.54 M363b
3. Conway: Functions of One Complex Variable; 517.53 C767f
4. Ahlfors: An Introduction to the Theory of Analytic Functions
of One Complex Variable; 517.53 A28

Online Learning

This course uses MyUni exclusively for providing electronic resources, such as lecture notes, assignment papers, sample solutions, discussion boards, etc. It is recommended that students make appropriate use of these resources.

Learning & Teaching Modes

Over the course of 30 lectures, the lecturer presents the material to the students and guides them through it. During this time students are expected to engage with the material being presented in lectures, identifying any difficulties that may arise in their understanding of it, and interacting with the lecturer to overcome these difficulties. It is expected that students will attend all lectures, but lectures will be recorded (when facilities allow for this) to help with incidental absences and for revision purposes. In fortnightly tutorials students present their solutions to assigned exercises and discuss them with the lecturer and their peers. Fortnightly homework assignments help students strengthen their understanding of the theory and their skills in applying it, allowing them to gauge their progress.

Workload

The information below is provided as a guide to assist students in engaging appropriately with the course requirements.

Activity	Quantity	Workload Hours
Lectures	30	90
Tutorials	6	18
Assignments	5	50
Total		158

Learning Activities Summary

	Lecture Schedule
Week 1	Complex numbers, functions and differentiation.
Week 2	Cauchy-Riemann equations. Elementary functions.
Week 3	Further examples, harmonic functions, complex series.
Week 4	Analytic functions. Complex antiderivatives.
Week 5	Integration of complex functions.
Week 6	Cauchy-Goursat theorem. The Cauchy integral formula.
Week 7	Consequences of the Cauchy integral formula.
Week 8	Taylor's theorem. Zeros of holomorphic functions.
Week 9	The open mapping and inverse function theorems. Isolated singularities of holomorphic functions.
Week 10	Meromorphic functions, Laurent series; residues.
Week 11	Applications of residues. Simply connected domains.
Week 12	The Riemann Mapping theorem.

A tutorial at the end of each even week will cover the material of the preceding 5 lectures.

The University's policy on Assessment for Coursework Programs is based on the following four principles:

Assessment must encourage and reinforce learning.
Assessment must enable robust and fair judgements about student performance.
Assessment practices must be fair and equitable to students and give them the opportunity to demonstrate what they have learned.
Assessment must maintain academic standards.

Assessment Summary

Assessment Task	Task Type	Due	Weighting	Learning Outcomes
Exam	Summative	Examination Period	70%	All
Tutorials	Formative and summative	Even weeks	5%	All
Assignments	Formative and summative	Weeks 3,5,7,9,11	25%	All

Assessment Related Requirements

An aggregate score of 50% is required to pass the course.

Assessment Detail

Assessment	Set	Due	Weighting
Tutorial 1	Week 1	Week 2	0%
Assignment 1	Week 2	Week 3	5%
Tutorial 2	Week 3	Week 4	1%
Assignment 2	Week 4	Week 5	5%
Tutorial 3	Week 5	Week 6	1%
Assignment 3	Week 6	Week 7	5%
Tutorial 4	Week 7	Week 8	1%
Assignment 4	Week 8	Week 9	5%
Tutorial 5	Week 9	Week 10	1%
Assignment 5	Week 10	Week 11	5%
Tutorial 6	Week 11	Week 12	1%

Submission

Assignments will have a maximum two-week turn-around time for feedback to students. Late assignments will not be accepted.

Course Grading

Grades for your performance in this course will be awarded in accordance with the following scheme:

M10 (Coursework Mark Scheme)
Grade	Mark	Description
FNS		Fail No Submission
F	1-49	Fail
P	50-64	Pass
C	65-74	Credit
D	75-84	Distinction
HD	85-100	High Distinction
CN		Continuing
NFE		No Formal Examination
RP		Result Pending

Further details of the grades/results can be obtained from Examinations.

Grade Descriptors are available which provide a general guide to the standard of work that is expected at each grade level. More information at Assessment for Coursework Programs.

Final results for this course will be made available through Access Adelaide.

Student Feedback

The University places a high priority on approaches to learning and teaching that enhance the student experience. Feedback is sought from students in a variety of ways including on-going engagement with staff, the use of online discussion boards and the use of Student Experience of Learning and Teaching (SELT) surveys as well as GOS surveys and Program reviews.

SELTs are an important source of information to inform individual teaching practice, decisions about teaching duties, and course and program curriculum design. They enable the University to assess how effectively its learning environments and teaching practices facilitate student engagement and learning outcomes. Under the current SELT Policy (http://www.adelaide.edu.au/policies/101/) course SELTs are mandated and must be conducted at the conclusion of each term/semester/trimester for every course offering. Feedback on issues raised through course SELT surveys is made available to enrolled students through various resources (e.g. MyUni). In addition aggregated course SELT data is available.
Student Support
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Authorised by:	Deputy Vice-Chancellor and Vice-President (Academic)
Maintained by:	Course Outlines Administrator