## PURE MTH 7054 - Complex Analysis

### North Terrace Campus - Semester 1 - 2022

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.

• General Course Information
##### Course Details
Course Code PURE MTH 7054 Complex Analysis Mathematical Sciences Semester 1 Postgraduate Coursework North Terrace Campus 3 Up to 3 hours per week Y MATHS 2100 or MATHS 2101 or MATHS 2202 or ELEC ENG 2106 Ongoing assessment, exam
##### Course Staff

Course Coordinator: Dr David Baraglia

##### 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.

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

University Graduate Attribute Course Learning Outcome(s)

Attribute 1: Deep discipline knowledge and intellectual breadth

Graduates have comprehensive knowledge and understanding of their subject area, the ability to engage with different traditions of thought, and the ability to apply their knowledge in practice including in multi-disciplinary or multi-professional contexts.

1,2,3,4,5

Attribute 2: Creative and critical thinking, and problem solving

Graduates are effective problems-solvers, able to apply critical, creative and evidence-based thinking to conceive innovative responses to future challenges.

1,2,3,4

Attribute 3: Teamwork and communication skills

Graduates convey ideas and information effectively to a range of audiences for a variety of purposes and contribute in a positive and collaborative manner to achieving common goals.

6

Graduates engage in professional behaviour and have the potential to be entrepreneurial and take leadership roles in their chosen occupations or careers and communities.

5,6
• 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
The course material will be presented in the form of course notes which students are expected to read and supplementary videos. In the videos, the lecturer guides the students throught the course material by presenting key concepts and examples from the lecture notes. Students are expected to engage with the material being presented, identifying any difficulties that may arise in their understanding of it, and interacting with the lecturer to overcome these difficulties. In fortnightly tutorials students will form small groups and work through
tutorial problems 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 Lecture Notes 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.
• 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 50% All Mid-semester test Summative Week 6 20% 1,2,3,6 Tutorial Participation 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 1% Assignment 1 Week 2 Week 3 5% Tutorial 2 Week 3 Week 4 1% Assignment 2 Week 4 Week 5 5% Mid-semester test Week 6 Week 6 20% Assignment 3 Week 6 Week 7 5% Tutorial 3 Week 7 Week 8 1% Assignment 4 Week 8 Week 9 5% Tutorial 4 Week 9 Week 10 1% Assignment 5 Week 10 Week 11 5% Tutorial 5 Week 11 Week 12 1%
##### Submission
Assignments 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)
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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The University of Adelaide is committed to regular reviews of the courses and programs it offers to students. The University of Adelaide therefore reserves the right to discontinue or vary programs and courses without notice. Please read the important information contained in the disclaimer.

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