## PURE MTH 3019 - Complex Analysis III

### North Terrace Campus - Semester 1 - 2020

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 School of Mathematical Sciences Semester 1 Undergraduate North Terrace Campus 3 Up to 3 hours per week Y MATHS 2100 or MATHS 2101 or MATHS 2202 or ELEC ENG 2106 MATHS 2101 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 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.
##### 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
• 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
Over 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.
• 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 Mid-semester test Summative Week 6 10% 1,2,3,6 Tutorials Formative and summative Weeks 2,4,8,10,12 5% All Assignments Formative and summative Weeks  3,5,7,9,11 15% All

Due to the current COVID-19 situation modified arrangements have been made to assessments to facilitate remote learning and teaching. Assessment details provided here reflect recent updates.

To support the changes to teaching, the following revisions to assessment have been made:-

The University will not hold exams as usual this semester. As such it has become necessary to change the assessment
of this course.

The new assessment is as follows:

The usual exam will be replaced by an online exam administered through MyUni. It will be worth 50% of the marks.

There will still be 5 assignments but now they will be worth 30% of you overall grade (6% per assignment).

Tutorial participation 10%. As we have already been doing, each student is assigned a question in one of the tutorials. You receive the mark for presenting a solution to this question to me.

Lastly, there will be an online quiz administered through MyUni which will be worth 10%.

Further details concerning the exam and online quiz will be posted in due course.
##### 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 3% Tutorial 2 Week 3 Week 4 1% Assignment 2 Week 4 Week 5 3% Midsemester test Week 6 Week 6 10% Assignment 3 Week 6 Week 7 3% Tutorial 3 Week 7 Week 8 1% Assignment 4 Week 8 Week 9 3% Tutorial 4 Week 9 Week 10 1% Assignment 5 Week 10 Week 11 3% 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
• Fraud Awareness

Students are reminded that in order to maintain the academic integrity of all programs and courses, the university has a zero-tolerance approach to students offering money or significant value goods or services to any staff member who is involved in their teaching or assessment. Students offering lecturers or tutors or professional staff anything more than a small token of appreciation is totally unacceptable, in any circumstances. Staff members are obliged to report all such incidents to their supervisor/manager, who will refer them for action under the university's studentâ€™s disciplinary procedures.

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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