Advanced Complex Analysis I

Video Lectures

Displaying all 43 video lectures.
I. Theorems of Rouche and Hurwitz
Lecture 1
Fundamental Theorems Connected with Zeros of Analytic Functions
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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
The Argument (Counting) Principle, Rouche's Theorem and The Fundamental Theorem
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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
Morera's Theorem and Normal Limits of Analytic Functions
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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
Hurwitz's Theorem and Normal Limits of Univalent Functions
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Hurwitz's Theorem and Normal Limits of Univalent Functions
II. Open Mapping Theorem
Lecture 5
Local Constancy of Multiplicities of Assumed Values
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Local Constancy of Multiplicities of Assumed Values
Lecture 6
The Open Mapping Theorem
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The Open Mapping Theorem
III. Inverse Function Theorem
Lecture 7
Introduction to the Inverse Function Theorem
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Introduction to the Inverse Function Theorem
Lecture 8
Completion of the Proof of the Inverse Function Theorem: The Integral Inversion
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Completion of the Proof of the Inverse Function Theorem: The Integral Inversion
Lecture 9
Univalent Analytic Functions have never-zero Derivatives and are Analytic Isomorphisms
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Univalent Analytic Functions have never-zero Derivatives and are Analytic Isomorphisms
IV. Implicit Function Theorem
Lecture 10
Introduction to the Implicit Function Theorem
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Introduction to the Implicit Function Theorem
Lecture 11
Proof of the Implicit Function Theorem: Topological Preliminaries
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Proof of the Implicit Function Theorem: Topological Preliminaries
Lecture 12
Proof of the Implicit Function Theorem: The Integral Formula for & Analyticity
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Proof of the Implicit Function Theorem: The Integral Formula for & Analyticity
V. Riemann Surfaces for Multi-Valued Functions
Lecture 13
Doing Complex Analysis on a Real Surface: The Idea of a Riemann Surface
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Doing Complex Analysis on a Real Surface: The Idea of a Riemann Surface
Lecture 14
F(z,w)=0 is naturally a Riemann Surface
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F(z,w)=0 is naturally a Riemann Surface
Lecture 15
Constructing the Riemann Surface for the Complex Logarithm
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Constructing the Riemann Surface for the Complex Logarithm
Lecture 16
Constructing the Riemann Surface for the m-th root function
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Constructing the Riemann Surface for the m-th root function
Lecture 17
The Riemann Surface for the functional inverse of an analytic
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The Riemann Surface for the functional inverse of an analytic
Lecture 18
The Algebraic nature of the functional inverses of an analytic
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The Algebraic nature of the functional inverses of an analytic
VI. Analytic Continuation
Lecture 19
The Idea of a Direct Analytic Continuation or an Analytic Extension
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The Idea of a Direct Analytic Continuation or an Analytic Extension
Lecture 20
General or Indirect Analytic Continuation and the Lipschitz Nature of the Radius
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General or Indirect Analytic Continuation and the Lipschitz Nature of the Radius
Lecture 21
Analytic Continuation Along Paths via Power Series Part A
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Analytic Continuation Along Paths via Power Series Part A
Lecture 22
Analytic Continuation Along Paths via Power Series Part B
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Analytic Continuation Along Paths via Power Series Part B
Lecture 23
Continuity of Coefficients occurring in Families of Power Series defining Analytic
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Continuity of Coefficients occurring in Families of Power Series defining Analytic
VII. Monodromy
Lecture 24
Analytic Continuability along Paths: Dependence on the Initial Function
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Analytic Continuability along Paths: Dependence on the Initial Function
Lecture 25
Maximal Domains of Direct and Indirect Analytic Continuation: Second
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Maximal Domains of Direct and Indirect Analytic Continuation: Second
Lecture 26
Deducing the Second (Simply Connected) Version of the Monodromy Theorem
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Deducing the Second (Simply Connected) Version of the Monodromy Theorem
Lecture 27
Existence and Uniqueness of Analytic Continuations on Nearby Paths
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Existence and Uniqueness of Analytic Continuations on Nearby Paths
Lecture 28
Proof of the First (Homotopy) Version of the Monodromy Theorem
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Proof of the First (Homotopy) Version of the Monodromy Theorem
Lecture 29
Proof of the Algebraic Nature of Analytic Branches of the Functional Inverse
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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
The Mean-Value Property, Harmonic Functions and the Maximum Principle
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The Mean-Value Property, Harmonic Functions and the Maximum Principle
Lecture 31
Proofs of Maximum Principles and Introduction to Schwarz's Lemma
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Proofs of Maximum Principles and Introduction to Schwarz's Lemma
Lecture 32
Proof of Schwarz's Lemma and Uniqueness of Riemann Mappings
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Proof of Schwarz's Lemma and Uniqueness of Riemann Mappings
Lecture 33
Reducing Existence of Riemann Mappings to Hyperbolic Geometry of Sub-domains
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Reducing Existence of Riemann Mappings to Hyperbolic Geometry of Sub-domains
IX. Pick's Lemma and Hyperbolic Geometry on the Unit Disc
Lecture 34
Part A: Differential or Infinitesimal Schwarz's Lemma, Pick's Lemma, Hyperbolic
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Part A: Differential or Infinitesimal Schwarz's Lemma, Pick's Lemma, Hyperbolic
Lecture 35
Part B: Differential or Infinitesimal Schwarz's Lemma, Pick's Lemma, Hyperbolic
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Part B: Differential or Infinitesimal Schwarz's Lemma, Pick's Lemma, Hyperbolic
Lecture 36
Hyperbolic Geodesics for the Hyperbolic Metric on the Unit Disc
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Hyperbolic Geodesics for the Hyperbolic Metric on the Unit Disc
Lecture 37
Schwarz-Pick Lemma for the Hyperbolic Metric on the Unit Disc
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Schwarz-Pick Lemma for the Hyperbolic Metric on the Unit Disc
X. Theorems of Arzela-Ascoli and Montel
Lecture 38
Arzela-Ascoli Theorem: Under Uniform Boundedness, Equicontinuity and Uniform
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Arzela-Ascoli Theorem: Under Uniform Boundedness, Equicontinuity and Uniform
Lecture 39
Completion of the Proof of the Arzela-Ascoli Theorem and Introduction
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Completion of the Proof of the Arzela-Ascoli Theorem and Introduction
Lecture 40
The Proof of Montel's Theorem
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The Proof of Montel's Theorem
XI. Existence of a Riemann Mapping
Lecture 41
The Candidate for a Riemann Mapping
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The Candidate for a Riemann Mapping
Lecture 42
Completion of Proof of The Riemann Mapping Theorem - Part A
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Completion of Proof of The Riemann Mapping Theorem - Part A
Lecture 43
Completion of Proof of The Riemann Mapping Theorem - Part B
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Completion of Proof of The Riemann Mapping Theorem - Part B