## MATH302 Complex Analysis

Second Semester |

### Paper details

This paper 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 holomorphic at a point z if it is differentiable in a neighborhood of z. The requirement that a function be complex differentiable has far reaching consequences. One very important consequence is that the real and imaginary parts of an holomorphic function must satisfy Laplace's equation. This means that 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.

### Potential students

This paper is particularly relevant to Mathematics and Physics majors.

### Prerequisites

MATH 201 (Real Analysis) and solid knowledge of what constitutes a mathematical proof.

### Textbook

We have our own course notes which in parts follow the book Complex Analysis, 3rd edition, by J. Bak and D.J. Newman, Springer (2010), XII, 328pp.

### Course Outline

The paper will cover most of the following topics:

- Complex numbers (modulus, argument etc., inequalities, powers, roots, geometry and topology of the complex plane)
- Holomorphic functions (Cauchy-Riemann equations, harmonic functions, polynomials, power series, exponential, trigonometric and logarithmic functions)
- Complex integration on star shaped domains (line integrals, Cauchy's theorem via Goursat's lemma, Cauchy's integral formulas, 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)
- Extension to simply connected domains
- Mappings of the complex plane (time permitting)

### Lecturer

Dr Melissa Tacy, room 220

### Lectures

3 per week: Mon, Wed and every other Fri at 12 noon (St Dav 2)

### Tutorials

1 per week: Thursdays 2pm (MA241)

### Internal Assessment

There will be ten homework sheets (5 of them will be marked) and one midterm test based on unmarked homework problems.

### Final mark

Your final mark F in the paper will be calculated according to this formula:

**F = max((3E + A + T)/5, E)**

where:

- E is the Exam mark
- A is the Assignments mark
- T is the Tests mark

and all quantities are expressed as percentages.

### Students must abide by the University’s Academic Integrity Policy

**Academic endeavours at the University of Otago are built upon an essential commitment to academic integrity.**

The two most common forms of academic misconduct are *plagiarism* and *unauthorised collaboration*.

#### Academic misconduct: **Plagiarism**

Plagiarism is defined as:

- Copying or paraphrasing another person’s work and presenting it as your own.
- Being party to someone else’s plagiarism by letting them copy your work or helping them to copy the work of someone else without acknowledgement.
- Using your own work in another situation, such as for the assessment of a different paper or program, without indicating the source.
- Plagiarism can be unintentional or intentional. Even if it is unintentional, it is still considered to be plagiarism.

All students have a responsibility to be aware of acceptable academic practice in relation to the use of material prepared by others and are expected to take all steps reasonably necessary to ensure no breach of acceptable academic practice occurs. You should also be aware that plagiarism is easy to detect and the University has policies in place to deal with it.

#### Academic misconduct: **Unauthorised Collaboration**

Unauthorised Collaboration occurs when you work with, or share work with, others on an assessment which is designed as a task for individuals and in which individual answers are required. This form does not include assessment tasks where students are required or permitted to present their results as collaborative work. Nor does it preclude collaborative effort in research or study for assignments, tests or examinations; but unless it is explicitly stated otherwise, each student’s answers should be in their own words. If you are not sure if collaboration is allowed, check with your lecturer.

### Augustin Louis Cauchy

The French mathematician Augustin Louis Cauchy (1789-1857) proved a special case of the Cauchy-Schwarz Inequality, namely that given any 2n real numbers a_{1}, a

_{2}, ..., a

_{n}, b

_{1}, b

_{2}, ..., b

_{n}, then we have the inequality

(a

_{1}b

_{1}+a

_{2}b

_{2}+ ... + a

_{n}b

_{n})

^{2}≤

(a

_{1}

^{2}+ a

_{2}

^{2}+ ... + a

_{n}

^{2}) x

(b

_{1}

^{2}+ b

_{2}

^{2}+ ... + b

_{n}

^{2}).

In MATH302 this inequality is a special case of one established for vectors in a general inner product space. Apart from the inequality, there is a lunar crater named after Cauchy as well as a street in Paris (Rue Cauchy) and he is one of 72 prominent French scientists whose names are recorded on plaques on the Eiffel Tower.

### Vector field

Line integrals provide a central tool in complex analysis. Here the integral is taken along the curve which traverses a vector field.