## PURE MTH 4119 - Complex Analysis - Honours

### 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; 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 4119 Complex Analysis - Honours School of Mathematical Sciences Semester 1 Undergraduate North Terrace Campus 3 Up to 3 hours per week MATHS 2100 or MATHS 2101 or MATHS 2202 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: Associate Professor Nicholas Buchdahl

##### Course Timetable

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

• Learning Outcomes
##### 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
• 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
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 Activities
##### Learning & Teaching Modes
Overthe 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 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.
• 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 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 5 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
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:

M11 (Honours Mark Scheme)
GradeGrade reflects following criteria for allocation of gradeReported 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 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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