Honours Complex AnalysisMATH-366

Lecture notes by Alexandre Tomberg

Prof. Robert VermesMcGill University

Fall 2008

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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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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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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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November-17-08

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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