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Here you will find what we have covered in the lectures Notes_10Jan25, with the reference to the pages in the Lectures Notes and the books.
| Date | Content | Reference | |
|---|---|---|---|
| 22 august | - Complex numbers - Conjugate and Modulus. Representation of circles and lines - Polar coordinate representation. The argument | notes pp. 5–10 also, Sarason Sections I.1–I.5 and I.9 | |
| 23 august | - The Exponential Form, and De Moivre's Formula - Roots of Complex Numbers - The Extended Complex plane \(\mathbb C_\infty \) - The Stereographic Projection | notes pp. 10–15 also, Sarason Sections I.9–I.12 | |
| 29 august | - The Spherical metric - Topology and convergence in \(\mathbb C \) - Continuity of complex functions - We covered pp. 21–29 from the notes, mostly without proofs. I suggest to skim through this part. | notes p. 16 and pp. 21–29 also, Sarason Sections I.8, I.13, I.14. | |
| 30 august | - Continuity of Principal Argument and Root - Take a quick look at uniform continuity - Connectedness and Path-Connectedness (proofs in the notes) - Complex Differentiability - Operations with differentiable functions (proofs in the notes) - The Cauchy–Riemann Equations | notes pp. 30–37 (right before Corollary 2.34) also, Sarason Sections II.1–II.6 | |
| 5 september | - Consequences of the Cauchy-Riemann Equations - Inverse function Theorem for Holomorphic maps - Differentiation of curves in \(\mathbb C \) - Conformal mappings | notes pp. 38–41 also, Sarason Sections II.7, II.8, II.11, II.12 | |
| 6 september | - End of the proof of Theorem 2.42 - Harmonic maps - Harmonic conjugates - The Complex Exponential - Theorem 2.48 (proved until part (vi)) | notes pp. 41–44 also, Sarason Sections IV.1–IV.5 and II.12–II.16 | |
| 12 september | - End of the proof of Theorem 2.48 - Complex trigonometric/hyperbolic functions - Branches of the argument - Holomorphic Roots | notes pp. 44–48 also, Sarason Sections IV.6–IV.13 | |
| 13 september | - Holomorphic Logarithms - Complex Powers - Series of complex numbers - Convergence and Absolute convergence - The Cauchy Product of two series | notes pp. 48–50, 55–58 also, Sarason Sections V.1–V.7 | |
| 19 september | - Pointwise, absolute, and uniform convergence - Weierstrass M-test - Power Series. The Radius of Convergence. Abel's Lemma - The Cauchy-Hadamard Theorem/Formula - Worked Examples | notes pp. 58–64 also, Sarason Chapter V | |
| 20 september | - Worked Examples - Convergence in the Boundary. - Abel's Summation by Parts. Picard's Criterion - Holomorphicity of power series (entirely rigorous proof in Theorem 3.20 of the notes, not in the lectures) - \(\mathbb C^\infty \) regularity and Taylor expansions | notes pp. 64–69 also, Sarason Chapter V | |
| 26 september | - Analytic Functions. Analyticity of Power Series - Worked Examples - Operations with Power Series and Analytic Functions - Identity Principles for Analytic Functions | notes pp. 69–76 | |
| 27 september | - Path Integral and Arc-Length Integral - Elementary properties of the integrals - The Fundamental Theorem of Calculus - Examples of Path-Integrals | notes 79–86 also, Sarason Chapter VI | |
| 03 oktober | - Differentiation under integral sign - Computing a fundamental integral - The Cauchy-Goursat Theorem - Cauchy Theorem/Primitives in Convex Domains - Thw Winding Numbers | notes pp. 86–93 also, Sarason Chapter VII | |
| 04 oktober | - Cauchy Integral Formula in Convex Domains - Local Cauchy Theorem - Holomorphic Extension to a point - Mean Value Property - Cauchy Formulae and Estimates for the Derivatives - Morera Theorem - Weierstrass Convergence Theorem | notes pp. 93–97 also, Sarason Chapter VII | |
| 10 oktober | - Analiticity of holomorphic functions - Order of a zero of a function - Identity Principles - Liouville's Theorem - The Fundamental Theorem of Algebra | notes pp. 98–101 also, Sarason Chapter VII | |
| 11 oktober | - The Fundamental Theorem of Algebra (Proof) - The Maximum Modulus Principles - Schwarz's Lemma | notes pp. 101–104 also, Sarason Chapter VII | |
| 17 oktober | - Proof of Schwarz's Lemma - Laurent Series - Cauchy Theorem in Annuli - Laurent Series Expansion of Functions - Isolated Singularities | notes pp. 104, pp. 111–117 also, Sarason Chapter VIII | |
| 18 oktober | - Types of Singularities - Removable Singularities: Riemann Theorem - Characterizations for Poles - Essential Singularities: The Casorati-Weierstrass Theorem - Definition of Residue | notes pp. 117–121 also, Sarason Chapter VIII | |
| 24 oktober | - Useful results to compute residues: See Propositions 5.19, 5.20, 5.22, 5.23, 5.25 and Examples 5.21, 5.24. - Cauchy Global/Homological Theorem - The Cauchy Residues Theorem - Evaluation of a complex integral via Residues Theorem | notes pp. 121–130 also, Sarason Chapter VIII | |
| 25 oktober | - Trigonometric Integrals (of bounded functions) in \(\mathbb [0,2\pi] \) - Integrals in \( \mathbb R \) of Functions without real singularities - Mixed Trigonometric-Rational Integrals in \( \mathbb R \) - Integrals in \( \mathbb R \) of Functions with simple real poles | notes pp. 131–137 also, Sarason Chapter X | |
| 31 oktober | - Example Integral with simple real poles - Trigonometric Polynomials - Fourier Coefficients, Sums, Series - Examples - Bessel's Inequality | notes pp. 137-138, pp. 143–146 also, Krantz Chapter XIII | |
| 01 november | - Convergence of Fourier Series of Lipschitz functions - Fourier Coefficients of a Derivative - The Dirichlet and the Féjer Kernel - Approximation by Trigonometric Polynomials (Quick look at Subsection 6.1.4) - The Fourier Transform | notes pp. 147–152 also, Krantz Chapter XIII | |
| 07 november | - Fourier Transform of Gaussians - Fourier Transform of the Derivative - The Heat Equation - Decay of the Fourier Transform at infinity - The Poisson Kernel - The Dirichlet Problem in the Disk | notes pp. 152–155 also, Krantz Chapter XIII | |
| 08 november | - Solution to the Dirichlet Problem in the Disk - Fourier Series of Sines and Cosines - The Heat Equation in an Interval: Separation of Variables and Superposition - The Heat Equation in \( \mathbb R \). The Heat Kernel - The Wave Equation. Separation of Variables | notes pp. 156–165 also, Krantz Chapter XIII | |
| 14 november | - Conclusions for the Wave Equation - Repetition of Holomorphicity and Harmonicity - Repetition of Elementary Functions - Exercises 1.3, 1.5, 1.12, 1.20 in Collection_Exercises_MA2106 | notes pp. 165–166 | |
| 15 november | - Exam Preparation - Exercises 1.14, 1.19, 1.20 in Collection_Exercises_MA2106 - Exercises 1–5 in Prøveeksamen_11Nov24 | ||
| 21 november | - Exam Preparation - Repetition Identity Principles and Cauchy Estimates - Exercises 6–7 in Prøveeksamen_11Nov24 - Exercise 5 in Prøveeksamen_II_19Nov24 | ||
| 22 november | - Exam Preparation - Repetition Evaluation of Integrals. Fourier Coefficients. - Exercises 4,6,7 in Prøveeksamen_II_19Nov24 |