PURE MTH 7054 - Complex Analysis
North Terrace Campus - Semester 1 - 2018
General Course Information
Course Code PURE MTH 7054 Course Complex Analysis Coordinating Unit School of 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 Course Description 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.
Course Coordinator: Associate Professor Nicholas Buchdahl
The full timetable of all activities for this course can be accessed from Course Planner.
Course Learning Outcomes1. 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
Recommended ResourcesIn 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 LearningThis 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 Activities
Learning & Teaching ModesOver 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.
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 5 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.
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 Task Task Type Due Weighting Learning Outcomes Exam Summative Examination Period 55% All Mid-semester test Summative Week 6 20% 1,2,3,6 Tutorials Formative and summative Even weeks 5% All Assignments Formative and summative Weeks 3,5,7,9,11 20% All
Assessment Related RequirementsAn aggregate score of 50% is required to pass the course.
Assessment Set Due Weighting Tutorial 1 Week 1 Week 2 1% Assignment 1 Week 2 Week 3 4% Tutorial 2 Week 3 Week 4 1% Assignment 2 Week 4 Week 5 4% Mid-semester test Week 6 Week 6 20% Assignment 3 Week 6 Week 7 4% Tutorial 3 Week 7 Week 8 1% Assignment 4 Week 8 Week 9 4% Tutorial 4 Week 9 Week 10 1% Assignment 5 Week 10 Week 11 4% Tutorial 5 Week 11 Week 12 1%
SubmissionAssignments will have a maximum two-week turn-around time for feedback to students.
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.
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.
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