MATH302 Complex Analysis
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.
This paper is particularly relevant to Mathematics and Physics majors.
MATH 201 (Real Analysis) and solid knowledge of what constitutes a mathematical proof.
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.
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)
Dr Melissa Tacy, room 220
3 per week: Mon, Wed and every other Fri at 12 noon (St Dav 2)
1 per week: Thursdays 2pm (MA241)
There will be ten homework sheets (5 of them will be marked) and one midterm test based on unmarked homework problems.
Your final mark F in the paper will be calculated according to this formula:
F = max((3E + A + T)/5, E)
- E is the Exam mark
- A is the Assignments mark
- T is the Tests mark
and all quantities are expressed as percentages.
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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.
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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.
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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..
Impersonationis getting someone else to participate in any assessment on your behalf, including having someone else sit any test or examination on your behalf.
Falsiﬁcationis 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 CauchyThe French mathematician Augustin Louis Cauchy (1789-1857) proved a special case of the Cauchy-Schwarz Inequality, namely that given any 2n real numbers a1, a2, ..., an, b1, b2, ..., bn, then we have the inequality
(a1b1+a2b2 + ... + anbn)2 ≤
(a12+ a22+ ... + an2) x
(b12+ b22+ ... + bn2).
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.
Line integrals provide a central tool in complex analysis. Here the integral is taken along the curve which traverses a vector field.