Overview
Develops the differential and integral calculus of functions of a complex variable, and its applications.
This paper introduces the field of complex analysis, which is concerned with the properties of functions depending upon a complex variable. We study notions you are already familiar with – limits, differentiation, integration, convergence of sequences and series – for functions defined on the complex plane. Complex differentiation is much stronger than real differentiation, and we will be able to prove many nice theorems about complex-differentiable functions. As one example, we will prove that the integral of a complex differentiable function along any closed path in the complex plane is always zero.
Due to the two-dimensional nature of the complex plane, we can move around singularities (places where a function is not differentiable, such as a division-by-zero) rather than needing to pass through them (as is the case on the real line), and this allows for a number of useful results which utilize the singularities of a function. One example is the residue theorem, which enables us to calculate a wide variety of difficult integrals in terms of the singularities of their integrands.
One key feature of complex-differentiable functions is that they are equal to a Taylor series on any point at which they are differentiable. (This is not always the case for functions on the real line.) Therefore, the complex plane is the natural setting for studying infinite series of functions (power series are but one such example), and we will spend a good deal of time understanding the properties of infinite series, such as their convergence, differentiability, and integrability in the complex plane. It is also possible to develop a series expansion about a singularity, and to this end we introduce the notion of a Laurent series expansion for functions of a complex variable.
Along the way we will strengthen your knowledge of elementary analysis and point-set topology, while helping to improve your proof-writing and problem-solving abilities.
About this paper
| Paper title | Complex Analysis |
|---|---|
| Subject | Mathematics |
| EFTS | 0.15 |
| Points | 18 points |
| Teaching period | Semester 2 (On campus) |
| Domestic Tuition Fees ( NZD ) | $1,103.10 |
| International Tuition Fees | Tuition Fees for international students are elsewhere on this website. |
- Prerequisite
- MATH 201
- Schedule C
- Arts and Music, Science
- Eligibility
- This paper is particularly relevant to Mathematics and Physics majors.
- Contact
- Teaching staff
- Paper Structure
MATH302 focuses on both theory and methods of mathematical analysis on the complex plane. The key material will be partitioned into five chapters, comprising:
Chapter 1: Topology
- Point-set topology (open and closed sets, compactness, connectedness, homotopy)
- The notion of distance and construction of metric spaces
- Abstract notions of continuity and uniform continuity
Chapter 2: Integration
- Measures and simple functions
- Construction of the Riemann–Stieltjes and Lebesgue integrals
- Line integrals and formalization of integral theorems on the complex plane
- Monotone Convergence Theorem, Fatou's lemma, Dominated Convergence Theorem
Chapter 3: Key Results in Complex Analysis
- The notion of a holomorphic function
- Representation of holomorphic functions via power series
- Cauchy’s theorem (in multiple forms)
- Open mapping theorem
- Corollaries to Cauchy’s theorem
Chapter 4: Sequences, Series, and Singularities on the Complex Plane
- Uniform convergence of sequences and series
- Classification of singularities and Laurent series
- Residue theorem
- Calculation of definite integrals and infinite series
Chapter 5: Topics in Complex Analysis
- Multi-valued functions and Riemann surfaces
- Conformal maps
- Fourier series and PDE
- Analytic continuation
- The Riemann hypothesis
- Teaching Arrangements
Three lectures per week.
One tutorial per week.
- Textbooks
No text books required. A list of optional readings will be provided.
- Graduate Attributes Emphasised
- Lifelong learning, Scholarship, Communication, Critical thinking, Information literacy, Research, Self-motivation.
View more information about Otago's graduate attributes. - Learning Outcomes
On completion of the study of this paper, students are expected to:
1. Understand basic facts about the topological structure of the complex plane.
2. Understand key theorems concerning differentiation and integration of complex functions.
3. Have a working understanding of infinite series of complex functions.
4. Be able to use key facts and results to construct proofs in mathematical analysis.
5. Develop the ability to properly write-up solutions to mathematical problems and then interpret what such a solution means.- Assessment details
Final Exam: 60% of the final grade
Internal Assessment: 40% of the final grade
Internal assessment comprises tutorial activities and periodic in-class tests.
Overview
Develops the differential and integral calculus of functions of a complex variable, and its applications.
This paper introduces the field of complex analysis, which is concerned with the properties of functions depending upon a complex variable. We study notions you are already familiar with–limits, differentiation, integration, convergence of sequences and series–for functions defined on the complex plane. Complex differentiation is much stronger than real differentiation and we will be able to prove many nice theorems about complex-differentiable functions. As one example, we will prove that the integral of a complex differentiable function along any closed path in the complex plane is always zero.
Due to the two-dimensional nature of the complex plane, we can move around singularities (places where a function is not differentiable, such as a division-by-zero) rather than needing to pass through them (as is the case on the real line), and this allows for a number of useful results which utilize the singularities of a function. One example is the residue theorem, which enables us to calculate a wide variety of difficult integrals in terms of the singularities of their integrands.
One key feature of complex-differentiable functions is that they are equal to a Taylor series on any point at which they are differentiable; this is not always the case for functions on the real line. Therefore, the complex plane is the natural setting for studying infinite series of functions (power series are but one such example), and we will spend a good deal of time understanding the properties of infinite series, such as their convergence, differentiability and integrability in the complex plane. It is also possible to develop a series expansion about a singularity, and to this end we introduce the notion of a Laurent series expansion for functions of a complex variable.
Along the way we will strengthen your knowledge of elementary analysis and point-set topology, while helping to improve your proof-writing and problem-solving abilities.
About this paper
| Paper title | Complex Analysis |
|---|---|
| Subject | Mathematics |
| EFTS | 0.1500 |
| Points | 18 points |
| Teaching period | Semester 2 (On campus) |
| Domestic Tuition Fees | Tuition Fees for 2027 have not yet been set |
| International Tuition Fees | Tuition Fees for international students are elsewhere on this website. |
- Prerequisite
- MATH 201
- Schedule C
- Arts and Music, Science
- Eligibility
This paper is particularly relevant to Mathematics and Physics majors.
- Contact
- Teaching staff
- Paper Structure
MATH 302 focuses on both theory and methods of mathematical analysis on the complex plane. The key material will be partitioned into five chapters, comprising:
Chapter 1: Topology
- Point-set topology (open and closed sets, compactness, connectedness, homotopy)
- The notion of distance and construction of metric spaces
- Abstract notions of continuity and uniform continuity
Chapter 2: Integration
- Measures and simple functions
- Construction of the Riemann–Stieltjes and Lebesgue integrals
- Line integrals and formalization of integral theorems on the complex plane
- Monotone Convergence Theorem, Fatou's lemma, Dominated Convergence Theorem
Chapter 3: Key Results in Complex Analysis
- The notion of a holomorphic function
- Representation of holomorphic functions via power series
- Cauchy’s theorem (in multiple forms)
- Open mapping theorem
- Corollaries to Cauchy’s theorem
Chapter 4: Sequences, Series, and Singularities on the Complex Plane
- Uniform convergence of sequences and series
- Classification of singularities and Laurent series
- Residue theorem
- Calculation of definite integrals and infinite series
Chapter 5: Topics in Complex Analysis
- Multi-valued functions and Riemann surfaces
- Conformal maps
- Fourier series and PDE
- Analytic continuation
- The Riemann hypothesis
- Teaching Arrangements
Three lectures and one tutorial per week.
- Textbooks
No textbooks required. A list of optional readings will be provided.
- Graduate Attributes Emphasised
- Lifelong learning, Scholarship, Communication, Critical thinking, Information literacy, Research, Self-motivation.
View more information about Otago's graduate attributes. - Learning Outcomes
On completion of the study of this paper, students are expected to:
- Understand basic facts about the topological structure of the complex plane.
- Understand key theorems concerning differentiation and integration of complex functions.
- Have a working understanding of infinite series of complex functions.
- Be able to use key facts and results to construct proofs in mathematical analysis.
- Develop the ability to properly write-up solutions to mathematical problems and then interpret what such a solution means.
- Assessment details
- Final exam (60%)
- Internal assessment (40%)
- Internal assessment comprises tutorial activities and periodic in-class tests.