"Complex Analysis Core Class Notes" Webpage

Complex Analysis Core Class Notes
Functions of One Complex Variable, Second Edition, John Conway

Copies of the classnotes are on the internet in PDF format as given below. The "Proofs of Theorems" files were prepared in Beamer. The "Printout of Proofs" are printable PDF files of the Beamer slides without the pauses.

Introduction.
Chapter I. The Complex Number System.
Chapter II. Metric Spaces and the Topology of ℂ.
Chapter III. Elementary Properties and Examples of Analytic Functions.
Chapter IV. Complex Integration.
Chapter V. Singularities.
Chapter VI. The Maximum Modulus Theorem.
Mobius Transformations, Transformational Geometry, and Hyperbolic Geometry.
Additional Notes on Chapters VII-XII

Introduction.

Supplement. Introduction to Math Philosophy and Meaning Intro to Math Philosophy notes.
Study Guide 0.

Chapter I. The Complex Number System.

I.1. The Real Number System. Section I.1 notes
I.2. The Field of Complex Numbers. Section I.2 notes
Supplement. Ordering the Complex Numbers. Ordering the Complex Numbers notes
I.3. The Complex Plane. Section I.3 notes

Supplement. Proofs of Theorems in Section I.3. Beamer file of Section I.3 proofs (prepared in Beamer).
Supplement. Printout of the Proofs of Theorems in Section I.3. Print file of Section I.3 proofs

Supplement. Location of Zeros of Polynomials. Zeros of Polynomials notes

Supplement. Proofs of Theorems in Location of Zeros of Polynomials. Beamer file of Zeros of Polynomials proofs (prepared in Beamer).
Supplement. Printout of the Proofs of Theorems in Location of Zeros of Polynomials. Print file of Zeros of Polynomials proofs

I.4. Polar Representation and Roots of Complex Numbers. Section I.4 notes
I.5. Lines and Half Planes in the Complex Plane. Section I.5 notes
Supplement. The Ilief-Sendov Conjecture. Ilief-Sendov Conjecture notes
Supplement. Dr. Bob's Favorite Results on Complex Polynomials. Results on Complex Polynomials notes
I.6. The Extended Plane and its Spherical Representation. Section I.6 notes
Study Guide 1.

Chapter II. Metric Spaces and the Topology of ℂ.

II.1. Definition and Examples of Metric Spaces. Section II.1 notes

Supplement. Proofs of Theorems in Section II.1. Beamer file of Section II.1 proofs (prepared in Beamer).
Supplement. Printout of the Proofs of Theorems in Section II.1. Print file of Section II.1 proofs

II.2. Connectedness. Section II.2 notes

Supplement. Proofs of Theorems in Section II.2. Beamer file of Section II.2 proofs (prepared in Beamer).
Supplement. Printout of the Proofs of Theorems in Section II.2. Print file of Section II.2 proofs

II.3. Sequences and Completeness. Section II.3 notes

Supplement. Proofs of Theorems in Section II.3. Beamer file of Section II.3 proofs (prepared in Beamer).
Supplement. Printout of the Proofs of Theorems in Section II.3. Print file of Section II.3 proofs

II.4. Compactness. Section II.4 notes (This section includes an example of a closed and bounded set which is not compact and a discussion of Cauchy sequences in a topological space.)

Supplement. Proofs of Theorems in Section II.4. Beamer file of Section II.4 proofs (prepared in Beamer).
Supplement. Printout of the Proofs of Theorems in Section II.4. Print file of Section II.4 proofs

Supplement. The Extended Complex Plane. Extended Complex Plane notes

Supplement. Proofs of Theorems in Supplement, The Extended Complex Plane. Beamer file of Extended Complex Plane proofs (prepared in Beamer).
Supplement. Printout of the Proofs of Theorems in Supplement, The Extended Complex Plane. Print file of Extended Complex Plane proofs

II.5. Continuity. Section II.5 notes

Supplement. Proofs of Theorems in Section II.5. Beamer file of Section II.5 proofs (prepared in Beamer).
Supplement. Printout of the Proofs of Theorems in Section II.5. Print file of Section II.5 proofs

II.6. Uniform Convergence. Section II.6 notes

Supplement. Proofs of Theorems in Section II.6. Beamer file of Section II.6 proofs (prepared in Beamer).
Supplement. Printout of the Proofs of Theorems in Section II.6. Print file of Section II.6 proofs

Study Guide 2.

Chapter III. Elementary Properties and Examples of Analytic Functions.

Supplement. A Primer on Lipschitz Functions. Lipschitz Functions notes
III.1. Power Series. PDF.

Supplement. Proofs of Theorems in Section III.1. Beamer file of Section III.1 proofs (prepared in Beamer).
Supplement. Printout of the Proofs of Theorems in Section III.1. Print file of Section III.1 proofs

III.2. Analytic Functions. Section III.2 notes

Supplement. Proofs of Theorems in Section III.2. Beamer file of Section III.2 proofs (prepared in Beamer).
Supplement. Printout of the Proofs of Theorems in Section III.2. Print file of Section III.2 proofs

III.3. Analytic Functions as Mapping, Mobius Transformations. Section III.3 notes

Supplement. Proofs of Theorems in Section III.3. Beamer file of Section III.3 proofs (prepared in Beamer).
Supplement. Printout of the Proofs of Theorems in Section III.3. Print file of Section III.3 proofs

Study Guide 3.

Chapter IV. Complex Integration.

IV.1. Riemann-Stieltjes Integrals. Section IV.1 notes

Supplement. Proofs of Theorems in Section IV.1. Beamer file of Section IV.1 proofs (prepared in Beamer).
Supplement. Printout of the Proofs of Theorems in Section IV.1. Print file of Section IV.1 proofs

IV.2. Power Series Representation of Analytic Functions. Section IV.2 notes

Supplement. Proofs of Theorems in Section IV.2. Beamer file of Section IV.2 proofs (prepared in Beamer).
Supplement. Printout of the Proofs of Theorems in Section IV.2. Print file of Section IV.2 proofs

IV.3. Zeros of Analytic Functions. Section IV.3 notes (This section includes a proof of the Fundamental Theorem of Algebra and a version of the Maximum Modulus Theorem.)

Supplement. Proofs of Theorems in Section IV.3. Beamer file of Section IV.3 proofs (prepared in Beamer).
Supplement. Printout of the Proofs of Theorems in Section IV.3. Print file of Section IV.3 proofs

IV.4. The Index of a Closed Curve. Section IV.4 notes

Supplement. Proofs of Theorems in Section IV.4. Beamer file of Section IV.4 proofs (prepared in Beamer).
Supplement. Printout of the Proofs of Theorems in Section IV.4. Print file of Section IV.4 proofs

IV.5. Cauchy's Theorem and Integral Formula. Section IV.5 notes

Supplement. Proofs of Theorems in Section IV.5. Beamer file of Section IV.5 proofs (prepared in Beamer).
Supplement. Printout of the Proofs of Theorems in Section IV.5. Print file of Section IV.5 proofs

IV.6. The Homotopic Version of Cauchy's Theorem and Simple Connectivity. Section IV.6 notes

Supplement. Proofs of Theorems in Section IV.6. Beamer file of Section IV.6 proofs (prepared in Beamer).
Supplement. Printout of the Proofs of Theorems in Section IV.6. Print file of Section IV.6 proofs

IV.7. Counting Zeros; The Open Mapping Theorem. Section IV.7 notes

Supplement. Proofs of Theorems in Section IV.7. Beamer file of Section IV.7 proofs (prepared in Beamer).
Supplement. Printout of the Proofs of Theorems in Section IV.7. Print file of Section IV.7 proofs

Supplement. The Stability Theorem for Orders of Zeros of Equations (Theorem IV.7.4). Stability Theorem notes
IV.8. Goursat's Theorem. Section IV.8 notes

Supplement. Proofs of Theorems in Section IV.8. Beamer file of Section IV.8 proofs (prepared in Beamer).
Supplement. Printout of the Proofs of Theorems in Section IV.8. Print file of Section IV.8 proofs

Study Guide 4.

Chapter V. Singularities.

V.1. Classification of Singularities. Section V.1 notes

Supplement. Proofs of Theorems in Section V.1. Beamer file of Section V.1 proofs (prepared in Beamer).
Supplement. Printout of the Proofs of Theorems in Section V.1. Print file of Section V.1 proofs

Supplement. Laurent Series Development (Theorem V.1.11). Laurent Series Development notes
V.2. Residues. Section V.2 notes

Supplement. Proofs of Theorems in Section V.2. Beamer file of Section V.2 proofs (prepared in Beamer).
Supplement. Printout of the Proofs of Theorems in Section V.2. Print file of Section V.2 proofs

V.3. The Argument Principle. Section V.3 notes (This section includes another proof of the Fundamental Theorem of Algebra.)

Supplement. Proofs of Theorems in Section V.3. Beamer file of Section V.3 proofs (prepared in Beamer).
Supplement. Printout of the Proofs of Theorems in Section V.3. Print file of Section V.3 proofs

Study Guide 5.

Chapter VI. The Maximum Modulus Theorem.

VI.1. The Maximum Principle. Section VI.1 notes

Supplement. Proofs of Theorems in Section VI.1. Beamer file of Section VI.1 proofs (prepared in Beamer).
Supplement. Printout of the Proofs of Theorems in Section VI.1. Print file of Section VI.1 proofs

Supplement. Applications of the Maximum Modulus Theorem to Polynomials. Max Modulus Theorem Applications notes

Supplement. Proofs of Theorems in Supplement to VI.1. Beamer file of Max Modulus Theorem Applications proofs (prepared in Beamer).
Supplement. Printout of the Proofs of Theorems in Supplement to VI.1. Print file of Max Modulus Theorem proofs

VI.2. Schwarz's Lemma. Section VI.2 notes

Supplement. Proofs of Theorems in Section VI.2. Beamer file of Section VI.2 proofs (prepared in Beamer).
Supplement. Printout of the Proofs of Theorems in Section VI.2. Print file of Section VI.2 proofs

VI.3. Convex Functions and Hadamard's Three Circles Theorem. Section VI.3 notes

Supplement. Proofs of Theorems in Section VI.3. Beamer file of Section VI.3 proofs (prepared in Beamer).
Supplement. Printout of the Proofs of Theorems in Section VI.3. Print file of Section VI.3 proofs

VI.4. Phragmen-Lindelof Theorem. Section VI.4 notes

Supplement. Proofs of Theorems in Section VI.4. Beamer file of Section VI.4 proofs (prepared in Beamer).
Supplement. Printout of the Proofs of Theorems in Section VI.4. Print file of Section VI.4 proofs

Study Guide 6.

Supplemental Material on Mobius Transformations, Transformational Geometry, and Hyperbolic Geometry

Supplemental Notes on III.3. Analytic Functions as Mappings: Mobius Transformations - with Supplemental Material from Hitchman's Geometry with an Introduction to Cosmic Topology. Analytic Functions as Mappings notes
Supplement. Transformations of ℂ and ℂ_∞ - An Approach to Geometry. Transformations notes
Supplement. Hyperbolic Geometry and the Poincare Disk Hyperbolic Geometry notes
Study Guide.

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