PURE MTH 4119 - Complex Analysis - Honours

North Terrace Campus - Semester 1 - 2016

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; Rouche'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 4119
Course	Complex Analysis - Honours
Coordinating Unit	Mathematical Sciences
Term	Semester 1
Level	Undergraduate
Location/s	North Terrace Campus
Units	3
Contact	Up to 3 hours per week
Available for Study Abroad and Exchange
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: 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)
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
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	4,5
Career and leadership readiness technology savvy professional and, where relevant, fully accredited forward thinking and well informed tested and validated by work based experiences	4,5
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	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. Integration along curves.
Week 3	Goursat's theorem. Cauchy's theorem.
Week 4	Cauchy's integral formula and consequences, Liouville's theorem.
Week 5	Zeros and poles of holomorphic functions, residues.
Week 6	The residue formula and applications, Morera's theorem.
Week 7	Classfication of singularities and Laurent series.
Week 8	Rouche's theorem, the maximum principle and applications.
Week 9	Transformations of the complex plane.
Week 10	The Riemann sphere. Generalisations to simply connected regions.
Week 11	The complex logarithm, Riemann mapping theorem.
Week 12	Connections to harmonic functions and PDE.

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:

M11 (Honours Mark Scheme)
Grade	Grade reflects following criteria for allocation of grade	Reported on Official Transcript
Fail	A mark between 1-49	F
Third Class	A mark between 50-59	3
Second Class Div B	A mark between 60-69	2B
Second Class Div A	A mark between 70-79	2A
First Class	A mark between 80-100	1
Result Pending	An interim result	RP
Continuing	Continuing	CN

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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