This is the first part of a series of lectures on advanced topics in Complex Analysis. By advanced, we mean topics that are not (or just barely) touched upon in a first course on Complex Analysis. The theme of the course is to study zeros of analytic (or holomorphic) functions and related theorems. These include the theorems of Hurwitz and Rouche, the Open Mapping theorem, the Inverse and Implicit Function theorems, applications of those theorems, behaviour at a critical point, analytic branches, constructing Riemann surfaces for functional inverses, Analytic continuation and Monodromy, Hyperbolic geometry and the Riemann Mapping theorem. For more details, please look at the titles, goals and keywords for each lecture given below.

Unit Number / Title

Lecture Number / Title

UNIT 1: Theorems of Rouche and Hurwitz

Lecture 1:
Fundamental Theorems Connected with
Zeros of Analytic Functions

Lecture 2:
The Argument (Counting) Principle, Rouche's Theorem
and The Fundamental Theorem of Algebra

Lecture 3:
Morera's Theorem and Normal Limits of Analytic Functions

Lecture 4:
Hurwitz's Theorem and Normal Limits of Univalent Functions

UNIT 2: Open Mapping Theorem

Lecture 5:
Local Constancy of Multiplicities of Assumed Values

Lecture 6:
The Open Mapping Theorem

UNIT 3: Inverse Function Theorem

Lecture 7:
Introduction to the Inverse Function Theorem

Lecture 8:
Completion of the Proof of the Inverse Function Theorem: The Integral Inversion Formula for the Inverse Function

Lecture 9:
Univalent Analytic Functions have never-zero Derivatives and are Analytic Isomorphisms

UNIT 4: Implicit Function Theorem

Lecture 10:
Introduction to the Implicit Function Theorem

Lecture 11:
Proof of the Implicit Function Theorem: Topological Preliminaries

Lecture 12:
Proof of the Implicit Function Theorem: The Integral Formula for & Analyticity of the Explicit Function

UNIT 5: Riemann Surfaces for Multi-Valued Functions

Lecture 13:
Doing Complex Analysis on a Real Surface: The Idea of a Riemann Surface

Lecture 14:
F(z,w)=0 is naturally a Riemann Surface

Lecture 15
Constructing the Riemann Surface for the Complex Logarithm

Lecture 16
Constructing the Riemann Surface for the m-th root function

Lecture 17
The Riemann Surface for the functional inverse of an analytic mapping at a critical point

Lecture 18
The Algebraic nature of the functional inverses
of an analytic mapping at a critical point

UNIT 6: Analytic Continuation

Lecture 19
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 of Convergence

Lecture 21A
Analytic Continuation Along Paths via Power Series Part A

Lecture 21B
Analytic Continuation Along Paths via Power Series Part B

Lecture 22
Continuity of Coefficients occurring in Families of Power Series defining Analytic Continuations along Paths

UNIT 7: Monodromy

Lecture 23:
Analytic Continuability along Paths: Dependence on the Initial Function and on the Path - First Version of the Monodromy Theorem

Lecture 24:
Maximal Domains of Direct and Indirect Analytic Continuation: Second Version of the Monodromy Theorem

Lecture 25:
Deducing the Second (Simply Connected)
Version of the Monodromy Theorem from the First (Homotopy) Version

Lecture 27:
Existence and Uniqueness of Analytic Continuations on Nearby Paths

Lecture 28:
Proof of the First (Homotopy) Version of the Monodromy Theorem

Lecture 30:
Proof of the Algebraic Nature of Analytic Branches of the Functional Inverse of an Analytic Function at a Critical Point

UNIT 8: Harmonic Functions, Maximum Principles,
Schwarz's Lemma and Uniqueness of Riemann Mappings

Lecture 31:
The Mean-Value Property, Harmonic Functions and the Maximum Principle

Lecture 32:
Proofs of Maximum Principles and Introduction to Schwarz's Lemma

Lecture 33:
Proof of Schwarz's Lemma and Uniqueness of Riemann Mappings

Lecture 34:
Reducing Existence of Riemann Mappings to
Hyperbolic Geometry of Sub-domains of the Unit Disc

UNIT 9: Pick's Lemma and Hyperbolic Geometry on the Unit Disc

Lecture 35A: Differential or Infinitesimal Schwarz's Lemma,Pick's Lemma, Hyperbolic Arclengths, Metric and Geodesics on the Unit Disc

Lecture 35B: Differential or Infinitesimal Schwarz's Lemma,Pick's Lemma, Hyperbolic Arclengths, Metric and Geodesics on the Unit Disc

Lecture 36: Hyperbolic Geodesics for the Hyperbolic Metric on the Unit Disc

Lecture 37: Schwarz-Pick Lemma for the Hyperbolic Metric on the Unit Disc

UNIT 10: Theorems of Arzela-Ascoli and Montel

Lecture 38: Arzela-Ascoli Theorem: Under Uniform Boundedness, Equicontinuity and Uniform Sequential Compactness are Equivalent

Lecture 39: Completion of the Proof of the Arzela-Ascoli Theorem and
Introduction to Montel's Theorem

Lecture 40: The Proof of Montel's Theorem

UNIT 11: Existence of a Riemann Mapping

Lecture 41: The Candidate for a Riemann Mapping

Lecture 42A: Completion of Proof of The Riemann Mapping Theorem

Lecture 42B: Completion of Proof of The Riemann Mapping Theorem

A first course in Topology covering the euclidean spaces (real line and real plane), and a first course in Complex Analysis covering Cauchy's Integration theory, Taylor series, Laurent series and the Residue theorem.

Complex Variables with Applications, by Saminathan Ponnusamy & Herb Silverman, 2006, 524 pp, Birkhaeuser, Boston.

Complex Analysis (UTM) by Theodore Gamelin, Springer, 2003.

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