PURE MTH 3019 - Complex Analysis III

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; 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.

• General Course Information
Course Details
Course Code PURE MTH 3019 Complex Analysis III Pure Mathematics Semester 1 Undergraduate North Terrace Campus 3 Up to 3 hours per week MATHS 1012 (Note: from 2015 the prerequisites for this course will be MATHS 2100 or MATHS 2101 or MATHS 2202 . Please plan your 2014 enrolment accordingly). MATHS 2101 or MATHS 2202 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.
Course Staff

Course Coordinator: Dr Melissa Tacy

Course Timetable

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

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

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

University 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 Resources
None.
Recommended Resources
The course will loosely follow E. M. Stein and R. Shakarchi, Complex Analysis (available to us as an e-book).

Other resources you may wish to use are:

J. Bak and D. J. Newman, Complex Analysis (available as an e-book).
T. W. Gamelin, Complex Analysis.
R. E. Greene and S. G. Krantz, Function theory of one complex variable.
Online Learning
Assignments, tutorial exercises, handouts, and course announcements will be posted on MyUni.
• Learning & Teaching Activities
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.

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.
• Assessment

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

1. Assessment must encourage and reinforce learning.
2. Assessment must enable robust and fair judgements about student performance.
3. Assessment practices must be fair and equitable to students and give them the opportunity to demonstrate what they have learned.
4. 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.

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

M10 (Coursework Mark Scheme)
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
• Policies & Guidelines
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