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)
Professor Boris Baeumer, rm 213
3 per week: Mon, Wed and every other Fri at 12 noon
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 = (6*E + 2*A + 2*T)/10
- 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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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.