# MATHS 740 : Complex Analysis

## Science

### Course Prescription

An introduction to functions of one complex variable, including Cauchy's integral formula, the index formula, Laurent series and the residue theorem. Many applications are given including a three line proof of the fundamental theorem of algebra. Complex analysis is used extensively in engineering, physics and mathematics. Strongly recommended: MATHS 333.

### Course Overview

The first half of the course studies the basics of complex analysis: Cauchy-Riemann equations, holomorphic functions, power series, and the Cauchy integral formula.  More advanced properties of holomorphic and harmonic functions, as well as their applications, are covered in the second half of the course.

### Course Requirements

Prerequisite: MATHS 332 Restriction: MATHS 341

### Capabilities Developed in this Course

 Capability 1: Disciplinary Knowledge and Practice Capability 2: Critical Thinking Capability 3: Solution Seeking Capability 4: Communication and Engagement Capability 5: Independence and Integrity
Graduate Profile: Master of Science

### Learning Outcomes

By the end of this course, students will be able to:
1. Present precise definitions of a range of important concepts in complex analysis. (Capability 1, 2, 3, 4 and 5)
2. Present proofs of important complex analysis theorems. (Capability 1, 2, 3, 4 and 5)
3. Identify the truth or otherwise of statements involving the properties referred to above. (Capability 1, 2, 3, 4 and 5)
4. Construct examples illustrating that certain statements involving certain complex analysis properties are false. (Capability 1, 2, 3, 4 and 5)
5. Apply complex analysis ideas to the applications of Cauchy’s theorem, Cauchy integral theorem, Schwarz’s lemma, Harnack’s inequality, etc., at a depth suitable for a graduate level course. (Capability 1, 2, 3, 4 and 5)

### Assessments

Assessment Type Percentage Classification
Assignments 30% Individual Coursework
Final Exam 70% Individual Examination
Assessment Type Learning Outcome Addressed
1 2 3 4 5
Assignments
Final Exam
The final grade will be based on Assignments (30%)  and Final Exam (70%), or on the Final Exam (100%), whichever is higher.

### Tuākana

Tuākana Science is a multi-faceted programme for Māori and Pacific students providing topic specific tutorials, one-on-one sessions, test and exam preparation and more. Explore your options at
https://www.auckland.ac.nz/en/science/study-with-us/pacific-in-our-faculty.html
https://www.auckland.ac.nz/en/science/study-with-us/maori-in-our-faculty.html

For further information on the Tuākana Maths programme, please visit
https://www.auckland.ac.nz/en/science/study-with-us/pacific-in-our-faculty.html
https://www.auckland.ac.nz/en/science/study-with-us/maori-in-our-faculty.html

This course is a standard 15-point course and students are expected to spend 10 hours per week involved in each 15-point course that they are enrolled in. For each week of this course, you can expect 3 hours of lectures, 3 hours of reading and thinking about the content and 4 hours of work on assignments.

### Delivery Mode

#### Campus Experience

• Lectures will be available as recordings where possible.
• The course will not include live online events.
• The activities for the course are scheduled as a standard weekly timetable.

### Learning Resources

Course materials are made available in a learning and collaboration tool called Canvas which also includes reading lists and lecture recordings (where available).

Please remember that the recording of any class on a personal device requires the permission of the instructor.

Lecture notes will be distributed in class and posted on canvas.

In the library there are many books on complex analysis with number 515.9.

### Student Feedback

During the course Class Representatives in each class can take feedback to the staff responsible for the course and staff-student consultative committees.

At the end of the course students will be invited to give feedback on the course and teaching through a tool called SET or Qualtrics. The lecturers and course co-ordinators will consider all feedback.

Your feedback helps to improve the course and its delivery for all students.

Students were generally satisfied with the course structure in 2021, and it will remain unchanged in 2023.

The University of Auckland will not tolerate cheating, or assisting others to cheat, and views cheating in coursework as a serious academic offence. The work that a student submits for grading must be the student's own work, reflecting their learning. Where work from other sources is used, it must be properly acknowledged and referenced. This requirement also applies to sources on the internet. A student's assessed work may be reviewed against online source material using computerised detection mechanisms.

### Class Representatives

Class representatives are students tasked with representing student issues to departments, faculties, and the wider university. If you have a complaint about this course, please contact your class rep who will know how to raise it in the right channels. See your departmental noticeboard for contact details for your class reps.

The content and delivery of content in this course are protected by copyright. Material belonging to others may have been used in this course and copied by and solely for the educational purposes of the University under license.

You may copy the course content for the purposes of private study or research, but you may not upload onto any third party site, make a further copy or sell, alter or further reproduce or distribute any part of the course content to another person.

### Inclusive Learning

All students are asked to discuss any impairment related requirements privately, face to face and/or in written form with the course coordinator, lecturer or tutor.

Student Disability Services also provides support for students with a wide range of impairments, both visible and invisible, to succeed and excel at the University. For more information and contact details, please visit the Student Disability Services’ website

### Special Circumstances

If your ability to complete assessed coursework is affected by illness or other personal circumstances outside of your control, contact a member of teaching staff as soon as possible before the assessment is due.

If your personal circumstances significantly affect your performance, or preparation, for an exam or eligible written test, refer to the University’s aegrotat or compassionate consideration page .

This should be done as soon as possible and no later than seven days after the affected test or exam date.

### Learning Continuity

In the event of an unexpected disruption, we undertake to maintain the continuity and standard of teaching and learning in all your courses throughout the year. If there are unexpected disruptions the University has contingency plans to ensure that access to your course continues and course assessment continues to meet the principles of the University’s assessment policy. Some adjustments may need to be made in emergencies. You will be kept fully informed by your course co-ordinator/director, and if disruption occurs you should refer to the university website for information about how to proceed.

The delivery mode may change depending on COVID restrictions. Any changes will be communicated through Canvas.

### Student Charter and Responsibilities

The Student Charter assumes and acknowledges that students are active participants in the learning process and that they have responsibilities to the institution and the international community of scholars. The University expects that students will act at all times in a way that demonstrates respect for the rights of other students and staff so that the learning environment is both safe and productive. For further information visit Student Charter .

### Disclaimer

Elements of this outline may be subject to change. The latest information about the course will be available for enrolled students in Canvas.

In this course students may be asked to submit coursework assessments digitally. The University reserves the right to conduct scheduled tests and examinations for this course online or through the use of computers or other electronic devices. Where tests or examinations are conducted online remote invigilation arrangements may be used. In exceptional circumstances changes to elements of this course may be necessary at short notice. Students enrolled in this course will be informed of any such changes and the reasons for them, as soon as possible, through Canvas.

Published on 28/10/2022 11:28 a.m.