## 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) is required, MATH 202 (Linear Algebra) and MATH 203 (Multivariate Calculus) are strongly recommended. Mathematical maturity and the ability to construct a mathematical proof are the keys to success in this paper.

### Textbook

No textbook is required. The course will be self-contained, and anything I want you to know will be discussed in the lectures.

### 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 R. A. Van Gorder (Science 3, room 2.12)

### Lectures

3 per week: Mon, Wed, Fri at 2PM (Science 3, room MA241)

Material from the first ten weeks of lectures will be fair game for the final exam. Lectures after this point will cover optional advanced material useful for those pursuing postgraduate study.

### Tutorials

1 per week: Thursdays at 2PM (Science 3, room MA241)

### Internal Assessment

There will be frequent homework problems worth 20% of your final grade in total. These are designed to keep you engaged with the material as we progress.

There will be one "midterm" test, worth 20% of your final grade. If you do better on the final exam, then your final exam grade will replace the midterm grade.

### Final Exam

There will be a closed-book final exam worth 60% of your final grade.

### Final mark

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

**F = max((6*E + 2*A + 2*T)/10, (6*E + 2*A)/8)**

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 integrity** means being honest in your studying and assessments. It is the basis for ethical decision-making and behaviour in an academic context. Academic integrity is informed by the values of honesty, trust, responsibility, fairness, respect and courage.

**Academic misconduct** is seeking to gain for yourself, or assisting another person to gain, an academic advantage by deception or other unfair means. The most common form of academic misconduct is plagiarism.

Academic misconduct in relation to work submitted for assessment (including all course work, tests and examinations) is taken very seriously at the University of Otago.

All students have a responsibility to understand the requirements that apply to particular assessments and also to be aware of acceptable academic practice regarding the use of material prepared by others. Therefore it is important to be familiar with the rules surrounding academic misconduct at the University of Otago; they may be different from the rules in your previous place of study.

Any student involved in academic misconduct, whether intentional or arising through failure to take reasonable care, will be subject to the University’s Student Academic Misconduct Procedures which contain a range of penalties.

If you are ever in doubt concerning what may be acceptable academic practice in relation to assessment, you should clarify the situation with your lecturer before submitting the work or taking the test or examination involved.

Types of academic misconduct are as follows:

**Plagiarism**

The University makes a distinction between unintentional plagiarism (Level One) and intentional plagiarism (Level Two).

- Although not intended,
*unintentional plagiarism*is covered by the Student Academic Misconduct Procedures. It is usually due to lack of care, naivety, and/or to a lack to understanding of acceptable academic behaviour. This kind of plagiarism can be easily avoided. *Intentional plagiarism*is gaining academic advantage by copying or paraphrasing someone elses work and presenting it as your own, or helping someone else copy your work and present it as their own. It also includes self-plagiarism which is when you use your own work in a different paper or programme without indicating the source. Intentional plagiarism is treated very seriously by the University.

**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 students answers should be in their own words. If you are not sure if collaboration is allowed, check with your lecturer..

**Impersonation**

Impersonation is getting someone else to participate in any assessment on your behalf, including having someone else sit any test or examination on your behalf.

**Falsiﬁcation**

Falsiﬁcation is to falsify the results of your research; presenting as true or accurate material that you know to be false or inaccurate.

**Use of Unauthorised Materials**

Unless expressly permitted, notes, books, calculators, computers or any other material and equipment are not permitted into a test or examination. Make sure you read the examination rules carefully. If you are still not sure what you are allowed to take in, check with your lecturer.

**Assisting Others to Commit Academic Misconduct**

This includes impersonating another student in a test or examination; writing an assignment for another student; giving answers to another student in a test or examination by any direct or indirect means; and allowing another student to copy answers in a test, examination or any other assessment.

### 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.