PURE MTH 7054 - Complex Analysis

North Terrace Campus - Semester 1 - 2014

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	Pure Mathematics
Term	Semester 1
Level	Postgraduate Coursework
Location/s	North Terrace Campus
Units	3
Contact	Up to 3 hours per week
Prerequisites	MATHS 1012
Assumed Knowledge	MATHS 2101 or MATHS 2202
Assessment	ongoing assessment 30%, exam 70%

Course Staff

Course Coordinator: Dr Melissa Tacy

Course Timetable

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

Course Learning Outcomes

Demonstrate an understanding of the fundamental concepts of complex analysis.
Demonstrate an understanding of the application of the theory both to other mathematical areas and to physics and engineering.
Prove the basic results relating to holomorphic functions.
Apply the theory learnt in the course to solve a variety of problems at an appropriate level of difficulty.
Demonstrate skills in communicating mathematics orally and in writing.

最新糖心Vlog Graduate Attribute	Course Learning Outcome(s)
Knowledge and understanding of the content and techniques of a chosen discipline at advanced levels that are internationally recognised.	1,2,3,4,5
The ability to locate, analyse, evaluate and synthesise information from a wide variety of sources in a planned and timely manner.	2,3,4
An ability to apply effective, creative and innovative solutions, both independently and cooperatively, to current and future problems.	4
Skills of a high order in interpersonal understanding, teamwork and communication.	5
A commitment to continuous learning and the capacity to maintain intellectual curiosity throughout life.	1,2,3,4,5
An awareness of ethical, social and cultural issues within a global context and their importance in the exercise of professional skills and responsibilities.	4,5

Learning & Teaching Modes

The lecturer guides the students through the course material in 30 lectures. Students are expected to engage with the material in the lectures. Interaction with the lecturer and discussion of any difficulties that arise during the lecture is encouraged and attendance at the lecturer's consultation hour is particularly encouraged. Students are expected to attend all lectures, but lectures will be recorded to help with occasional absences and for revision purposes. In fortnightly tutorials students will work through exercises designed to practise their skills. Fortnightly homework assignments help students strengthen their understanding of the theory and their skills in applying it, and allow 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	100
Tutorials	4	16
Assignments	5	40
Total		156

Learning Activities Summary

	Lecture Schedule
Week 1	The complex numbers and the complex plane. Continuity and complex differentiation.
Week 2	Holomorphic functions and power series.
Week 3	Integration along curves, Goursat's theorem.
Week 4	Cauchy's integral formula and consequences, Liouville's theorem.
Week 5	Zeros and poles of holomorphic functions.
Week 6	Residues and the residue formula.
Week 7	Applications of the residue formula, Morera's theorem.
Week 8	Classfication of singularities and Laurent series.
Week 9	Rouche's theorem, the maximum principle and applications.
Week 10	The Riemann sphere. Generalisations to simply connected regions.
Week 11	The complex logarithm. Links to harmonic functions and consequences.
Week 12	Conformal maps and the Riemann mapping theorem.

Tutorials in Weeks 3, 5, 9, and 11 cover the material of the previous two weeks.

The 最新糖心Vlog'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
Test	Summative	Week 7	10%	All
Assignments	Formative and summative	Weeks 4, 6, 8, 10, and 12	20%	All

Assessment Related Requirements

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

Assessment Detail

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

Submission

Homework assignments must be submitted on time with a signed assessment cover sheet. Late assignments will not be accepted. Assignments will be returned within two weeks. Students may be excused from an assignment for medical or compassionate reasons. Documentation is required and the lecturer must be notified as soon as possible.

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 .

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Maintained by:	Course Outlines Administrator

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