Advanced Complex Analysis I
Video Lectures
Displaying all 43 video lectures.
I. Theorems of Rouche and Hurwitz | |
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Lecture 1 Play Video |
Fundamental Theorems Connected with Zeros of Analytic Functions Keywords: Zeros of analytic functions are isolated, domain in the complex plane, isolated singularity, removable singularity, pole, essential singularity Taylor expansion, Laurent expansion, residue at singular point, Residue Theorem, uniform convergence allows termwise integration and differentiation, Argument (Counting) Principle, multiplicity or order of the pole or zero, Rouche's theorem, small perturbation of an analytic function, normal convergence (uniform convergence on compact subsets), Hurwitz's theorem Open Mapping theorem, Inverse Function theorem |
Lecture 2 Play Video |
The Argument (Counting) Principle, Rouche's Theorem and The Fundamental Theorem Keywords: Fundamental Theorem of Algebra, simple closed contour, self-intersection, piecewise-smooth, parametrization, meromorphic function, compact set, euclidean space, limit point, non-isolated singularity, simple zeros and poles, logarithmic derivative, Cauchy's Theorem, simply connected, analytic branch of logarithm, zeros of analytic functions are isolated, domain in the complex plane, isolated singularity, removable singularity, pole, essential singularity, Laurent expansion, residue at singular point, Residue Theorem, Argument (Counting) Principle, multiplicity or order of the pole or zero, Rouche's theorem |
Lecture 3 Play Video |
Morera's Theorem and Normal Limits of Analytic Functions Keywords: Analytic perturbation of analytic function, counting zeros and poles with multiplicity inside a simple closed contour, zero of the limit of a sequence of analytic functions, Hurwitz's theorem, pointwise convergence, uniform convergence, normal convergence (or uniform convergence on compact subsets), Morera's theorem, Cauchy-Riemann equations, interior and exterior of a contour, orientation or sense of a contour, multiply connected domain, piecewise continuous, Fundamental theorem of Integral Calculus, simple closed contour, self-intersection, piecewise-smooth, parametrization, meromorphic function, non-isolated singularity, logarithmic derivative, Cauchy's theorem, simply connected, zeros of analytic functions are isolated, domain in the complex plane, isolated singularity, removable singularity, pole, essential singularity, residue at singular point, Residue Theorem, Argument (Counting) Principle, multiplicity or order of the pole or zero, Rouche's theorem |
Lecture 4 Play Video |
Hurwitz's Theorem and Normal Limits of Univalent Functions |
II. Open Mapping Theorem | |
Lecture 5 Play Video |
Local Constancy of Multiplicities of Assumed Values |
Lecture 6 Play Video |
The Open Mapping Theorem |
III. Inverse Function Theorem | |
Lecture 7 Play Video |
Introduction to the Inverse Function Theorem |
Lecture 8 Play Video |
Completion of the Proof of the Inverse Function Theorem: The Integral Inversion |
Lecture 9 Play Video |
Univalent Analytic Functions have never-zero Derivatives and are Analytic Isomorphisms |
IV. Implicit Function Theorem | |
Lecture 10 Play Video |
Introduction to the Implicit Function Theorem |
Lecture 11 Play Video |
Proof of the Implicit Function Theorem: Topological Preliminaries |
Lecture 12 Play Video |
Proof of the Implicit Function Theorem: The Integral Formula for & Analyticity |
V. Riemann Surfaces for Multi-Valued Functions | |
Lecture 13 Play Video |
Doing Complex Analysis on a Real Surface: The Idea of a Riemann Surface |
Lecture 14 Play Video |
F(z,w)=0 is naturally a Riemann Surface |
Lecture 15 Play Video |
Constructing the Riemann Surface for the Complex Logarithm |
Lecture 16 Play Video |
Constructing the Riemann Surface for the m-th root function |
Lecture 17 Play Video |
The Riemann Surface for the functional inverse of an analytic |
Lecture 18 Play Video |
The Algebraic nature of the functional inverses of an analytic |
VI. Analytic Continuation | |
Lecture 19 Play Video |
The Idea of a Direct Analytic Continuation or an Analytic Extension |
Lecture 20 Play Video |
General or Indirect Analytic Continuation and the Lipschitz Nature of the Radius |
Lecture 21 Play Video |
Analytic Continuation Along Paths via Power Series Part A |
Lecture 22 Play Video |
Analytic Continuation Along Paths via Power Series Part B |
Lecture 23 Play Video |
Continuity of Coefficients occurring in Families of Power Series defining Analytic |
VII. Monodromy | |
Lecture 24 Play Video |
Analytic Continuability along Paths: Dependence on the Initial Function |
Lecture 25 Play Video |
Maximal Domains of Direct and Indirect Analytic Continuation: Second |
Lecture 26 Play Video |
Deducing the Second (Simply Connected) Version of the Monodromy Theorem |
Lecture 27 Play Video |
Existence and Uniqueness of Analytic Continuations on Nearby Paths |
Lecture 28 Play Video |
Proof of the First (Homotopy) Version of the Monodromy Theorem |
Lecture 29 Play Video |
Proof of the Algebraic Nature of Analytic Branches of the Functional Inverse |
VIII. Harmonic Functions, Maximum Principles, Schwarz's Lemma and Uniqueness of Riemann Mappings | |
Lecture 30 Play Video |
The Mean-Value Property, Harmonic Functions and the Maximum Principle |
Lecture 31 Play Video |
Proofs of Maximum Principles and Introduction to Schwarz's Lemma |
Lecture 32 Play Video |
Proof of Schwarz's Lemma and Uniqueness of Riemann Mappings |
Lecture 33 Play Video |
Reducing Existence of Riemann Mappings to Hyperbolic Geometry of Sub-domains |
IX. Pick's Lemma and Hyperbolic Geometry on the Unit Disc | |
Lecture 34 Play Video |
Part A: Differential or Infinitesimal Schwarz's Lemma, Pick's Lemma, Hyperbolic |
Lecture 35 Play Video |
Part B: Differential or Infinitesimal Schwarz's Lemma, Pick's Lemma, Hyperbolic |
Lecture 36 Play Video |
Hyperbolic Geodesics for the Hyperbolic Metric on the Unit Disc |
Lecture 37 Play Video |
Schwarz-Pick Lemma for the Hyperbolic Metric on the Unit Disc |
X. Theorems of Arzela-Ascoli and Montel | |
Lecture 38 Play Video |
Arzela-Ascoli Theorem: Under Uniform Boundedness, Equicontinuity and Uniform |
Lecture 39 Play Video |
Completion of the Proof of the Arzela-Ascoli Theorem and Introduction |
Lecture 40 Play Video |
The Proof of Montel's Theorem |
XI. Existence of a Riemann Mapping | |
Lecture 41 Play Video |
The Candidate for a Riemann Mapping |
Lecture 42 Play Video |
Completion of Proof of The Riemann Mapping Theorem - Part A |
Lecture 43 Play Video |
Completion of Proof of The Riemann Mapping Theorem - Part B |