# MATH302 Complex Analysis

Develops the differential and integral calculus of functions of a complex variable, and its applications.

Complex differentiability has much stronger consequences than real differentiability and gives many new insights into the theory of functions of a real variable. A function of a complex variable is called analytic at a point z if it is differentiable in a neighbourhood of z. Because the real and imaginary parts of an analytic function must satisfy Laplace's equation, complex analysis is widely applicable to two-dimensional problems in physics. An important tool in complex analysis is the line integral, and one theme of this paper is to explore the classical integral theorems. For example, Cauchy's theorem says that the integral around a closed path of a function that is differentiable everywhere inside the area bounded by the path is always zero.

Paper title Complex Analysis MATH302 Mathematics 0.1500 18 points Second Semester \$851.85 \$3,585.00
Prerequisite
MATH 201
Schedule C
Arts and Music, Science
Eligibility
This paper is particularly relevant to Mathematics and Physics majors.
Contact
david.bryant@otago.ac.nz
Teaching staff
Associate Professor Boris Baeumer
Teaching Arrangements
Five lectures each fortnight
One tutorial per week
Textbooks
We will follow the book Complex Analysis, 3rd edition, by J. Bak and D.J. Newman, Springer (2010), XII, 328pp, available online from the Resources page.

Lecture notes are also available online free of charge.
Course outline
View course outline for MATH 302
Paper Structure
Main topics;
• Complex numbers (modulus, argument, etc; inequalities, powers, roots, geometry and topology of the complex plane)
• Analytic functions (Cauchy-Riemann equations, harmonic functions, polynomials, power series, exponential, trigonometric and logarithmic functions)
• Complex integration (curves, rectifiability, curve integrals, domains, starlikeness, homotopy, simple-connectedness, Cauchy's theorem via Goursat's lemma, Cauchy's integral formulae, Cauchy's inequalities, Liouville's theorem, mean value theorem (for harmonic functions), fundamental theorem of algebra, maximum modulus principle, Morera's theorem, isolated singularities, Weierstrass's theorem, residue theorem, real integrals, Rouche's theorem, Schwarz's lemma)
Critical thinking.
Learning Outcomes
Demonstrate in-depth knowledge of basic concepts of complex analysis and mathematical proof.

## Timetable

### Second Semester

Location
Dunedin
Teaching method
This paper is taught On Campus
Learning management system
Other

#### Lecture

Stream Days Times Weeks
Attend
L1 Monday 12:00-12:50 28-34, 36-41
Wednesday 12:00-12:50 28-34, 36-41
Friday 12:00-12:50 28, 30, 32, 34, 36, 38, 40

#### Tutorial

Stream Days Times Weeks
Attend
T1 Thursday 14:00-14:50 28-34, 36-41

Develops the differential and integral calculus of functions of a complex variable, and its applications.

This paper provides an introduction to the mathematics and analyis of complex numbers, which are a central topic of pure and applied mathematics.

Paper title Complex Analysis MATH302 Mathematics 0.1500 18 points Second Semester Tuition Fees for 2018 have not yet been set 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
david.bryant@otago.ac.nz
Teaching staff
Dr Melissa Tacy
Paper Structure
Main topics;
• Complex numbers (modulus, argument, etc; inequalities, powers, roots, geometry and topology of the complex plane)
• Analytic functions (Cauchy-Riemann equations, harmonic functions, polynomials, power series, exponential, trigonometric and logarithmic functions)
• Complex integration (curves, rectifiability, curve integrals, domains, starlikeness, homotopy, simple-connectedness, Cauchy's theorem via Goursat's lemma, Cauchy's integral formulae, Cauchy's inequalities, Liouville's theorem, mean value theorem (for harmonic functions), fundamental theorem of algebra, maximum modulus principle, Morera's theorem, isolated singularities, Weierstrass's theorem, residue theorem, real integrals, Rouche's theorem, Schwarz's lemma)
Teaching Arrangements
Five lectures each fortnight
One tutorial per week
Textbooks
We will follow the book Complex Analysis, 3rd edition, by J. Bak and D.J. Newman, Springer (2010), XII, 328pp, available online from the Resources page.

Lecture notes are also available online free of charge.
Course outline
View course outline for MATH 302
Critical thinking.
Learning Outcomes
Demonstrate in-depth knowledge of basic concepts of complex analysis and mathematical proof.

## Timetable

### Second Semester

Location
Dunedin
Teaching method
This paper is taught On Campus
Learning management system
Other

#### Lecture

Stream Days Times Weeks
Attend
L1 Monday 12:00-12:50 28-34, 36-41
Wednesday 12:00-12:50 28-34, 36-41
Friday 12:00-12:50 28, 30, 32, 34, 36, 38, 40

#### Tutorial

Stream Days Times Weeks
Attend
T1 Thursday 14:00-14:50 29-34, 36-41