Honours Complex Analysisatombe2/classnotes...Honours Complex Analysis MATH-366 Lecture notes by...
Transcript of Honours Complex Analysisatombe2/classnotes...Honours Complex Analysis MATH-366 Lecture notes by...
Honours Complex AnalysisMATH-366
Lecture notes by Alexandre Tomberg
Prof. Robert VermesMcGill University
Fall 2008
Lecture Notes Page 1
Complex numbers, polar representation, conjugate numbers, C as
a complete metric, etc. (*)
1.
Complex functions, continuity, uniform continuity, etc. (*)2.Complex differentiable functions, Cauchy-Riemann differential equations, analytic functions. (*)
3.
Cauchy-Hadamard formula; (*)a.Abel-Stolz theorem, Cauchy product. (*)b.
Power series (as example of analytic functions on a disc):4.
Elementary functions: zn, z+1/z, ez, log z, za, sin z, cos z, arcsin z, arccos z, linear fractional transformations, conformal mapping. (*)
5.
Cauchy theorem and its consequences; (*)a.Cauchy Integral formula; (*)b.Morera's theorem; (*)c.Uniform convergence of analytic functions on compact subsets. (*)
d.
Complex integration:6.
Taylor expansion, power series representation of analytic functions, Cauchy's estimate. (*)
a.
Entire functions, Liouville theorem, fundamental theorem of algebra. (*)
b.
Power series and analyticity:7.
Uniqueness theorem, zeros of analytic functions. (*)8.Maximum, minimum principle, Schwartz's lemma. (*)9.Laurent expansion, classification of isolated singularities, Casorati-Wierstrass theorem, the residue theorem. (*)
10.
Rouché's theorem, argument principle and Hurwitz theorem. (*)11.Evaluation of definite integrals. (*)12.Harmonic functions, Poisson formula, boundary values of harmonic functions defined by Poisson's formula. (*)
13.
14. Concept of analytic continuation, singular point, natural boundary, Pringsheim theorem. (*)
Table of ContentsDecember-03-08
12:16 PM
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(*)Complex PlaneSeptember-03-08
11:38 AM
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Complex sequences & seriesSeptember-08-08
11:41 AM
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(*)Complex calculusSeptember-10-08
11:41 AM
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(*)Cauchy-Riemann EquationsSeptember-12-08
11:51 AM
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Analytic functionsSeptember-15-08
11:33 AM
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(*)Examples of analytic functions.September-17-08
11:42 AM
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Trigonometric functionsSeptember-19-0811:34 AM
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(*)Complex Power SeriesSeptember-22-08
11:35 AM
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(*)Complex Power SeriesSeptember-22-08
12:04 PM
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Differentiation of Power SeriesSeptember-24-08
11:41 AM
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Abel continuity theoremSeptember-24-08
12:14 PM
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Complex curvesSeptember-29-0811:35 AM
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IntegrationSeptember-29-08
11:57 AM
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(*)
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(*)Cauchy's theoremOctober-08-0811:39 AM
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(*)Morera's TheoremOctober-10-08
11:48 AM
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(*)Liouville's TheoremOctober-10-08
11:52 AM
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Fundamental Theorem of AlgebraOctober-10-08
12:01 PM
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(*)Sequences of Analytic functionsOctober-10-08
12:16 PM
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(*)
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Removable singularitiesOctober-17-08
11:41 AM
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(*)More Properties of Analytic FunctionsOctober-17-08
12:03 PM
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(*)
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Lemma of SchwartzOctober-22-08
11:44 AM
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(*)Laurent expansionOctober-24-08
11:41 AM
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Meromorphic functionsOctober-29-08
11:58 AM
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Famous Meromorphic functionsOctober-31-08
11:37 AM
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Counting the poles of a functionOctober-31-08
11:46 AM
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(*)Rouche 's TheoremOctober-31-08
12:01 PM
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Argument PrincipleNovember-05-08
12:16 PM
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One-point Compactification of Complex PlaneNovember-07-08
12:03 PM
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(*)Computing Real IntegralsNovember-10-08
11:55 AM
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12:01 PM
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(*)Harmonic functionsNovember-19-08
11:40 AM
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Linear Fractional TransformationsNovember-21-08
12:07 PM
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(*)
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