| MATH 323 (COMPLEX ANALYSIS)
2009
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Lec: Mon 11:25-12:10, Wed
8:40-9:25, Thur 13:15-14:00 Tut:
Wed
15:05 - 16:30 Venue: F24 |
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Topics: Complex numbers and complex plane,
elementary complex functions, complex differentiation, analyticity, Cauchy-Riemann
equations, branches of a function, line integrals, Cauchy's
Theorem, Cauchy Integral Formulas, Liouville's Theorem, Fundamental
Theorem of Algebra, Taylor series, Laurent series, singularities, Residue Calculus |
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Lectures: 29 hours Tutorials:
20 hours |
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Text: Complex Analysis by George Cain, 1999/2005
http://www.freetechbooks.com/about491.html |
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Assessment: 33% tests (2
best marks from 3 one-hour tests), 67% exam (3 hours)
DP: 30% class marks |
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| Lec 1 |
(20 July) |
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| Lec
2 |
(22 July) |
1.1 arithmetic of complex
numbers, relation to real numbers, complex plane |
| Lec
3 |
(23 July) |
1.2 modulus, real part,
imaginary part, conjugate , geometry of the complex plane |
| Lec
4 |
(27 July) |
1.3 polar form of a
complex number, argument, principal argument, rotation of a complex
number |
| Lec
5 |
(29 July) |
1.3 arithmetic using
polar form, rotation about the origin, complex number as 2x2
matrices |
| Lec
6 |
(30 July) |
1.3 some examples about the polar form, casual remarks about
FTA and the quaternions |
| Lec
7 |
(03 Aug) |
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2.1
parametric curves in complex plane,
2.2
complex functions |
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| Lec
8 |
(05 Aug) |
2.2 limit, continuity of a
complex function, examples |
| Lec
9 |
(06 Aug) |
2.2 differentiability, analyticity,
2.3
Cauchy-Riemann equations |
| Lec
10 |
(12 Aug) |
2.3
sufficient condition for differentiability |
| Lec
11 |
(13 Aug) |
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3.1
& 3.2 the exponential function |
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| Lec
12 |
(17 Aug) |
3.3 trigonometric functions |
| Lec
13 |
(19 Aug) |
3.4 logarithm and complex
exponentiation |
| Lec
14 |
(20 Aug) |
revision |
| Lec
15 |
(24 Aug) |
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4.1
line integral of a complex function |
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| Lec
16 |
(26 Aug) |
4.1
& 4.2 more on integration |
| Lec
17 |
(27 Aug) |
Test 1 solutions |
| Lec
18 |
(31 Aug) |
4.3
antiderivative and path-independent |
| Lec
19 |
(02 Sept) |
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5.1 piecewise smooth curves, simple closed curves,
orientation, region, homotopy |
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| Lec
20 |
(03 Sept) |
5.2 Cauchy's Theorem |
| Lec
21 |
(07 Sept) |
5.2 two equivalent versions of Cauchy's Theorem |
| Lec
22 |
(09 Sept) |
5.2 applications of the two versions of Cauchy's Theorem |
| Lec
23 |
(10 Sept) |
5.2 Cauchy's Theorem, anti-derivative, path-independence |
| Lec
24 |
(14 Sept) |
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6.1 Cauchy
integral formula |
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| Lec
25 |
(16 Sept) |
6.2 general Cauchy
integral formula and Morera's Theorem |
| Lec
26 |
(17 Sept) |
6.2 proof of the general
Cauchy
integral formula |
| Lec
27 |
(28 Sept) |
6.3 Liouville's Theorem,
the Fundamantal Theorem of Algebra |
| Lec
28 |
(30 Sept) |
6.4
Maximum Modulus Theorem, etc |
| Lec
29 |
(01 Oct) |
Test 2 solutions |
| Lec
30 |
(05 Oct) |
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8.1
complex sequence, power series |
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| Lec
31 |
(07 Oct) |
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| Lec
32 |
(08 Oct) |
9.2 Laurent series |
| Lec
33 |
(12 Oct) |
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10.1
singularities, residues, Laurent series |
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| Lec
34 |
(14 Oct) |
10.1
the Residue Theorem, examples |
| Lec
35 |
(15 Oct) |
10.2 poles and
calculation of residues |
| Lec
36 |
(19 Oct) |
10.2 poles and
calculation of residues |
| Lec
37 |
(21 Oct) |
revision |
| Lec
38 |
(22 Oct) |
Test 2 solutions |
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