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Complex Analysis - IITM

Prof. Pranav Haridas · Kerala School of Mathematics

Start with lecture 1 At NPTEL

About

This is a first course in Complex Analysis focussing on holomorphic functions and its basic properties like Cauchy’s theorem and residue theorems, the classification of singularities, and the maximum principle. We shall study the singularities of holomorphic functions. If time permits, we shall also study Branches of the complex logarithm through covering spaces and attempt proving Picard’s theorem.

Intended audience: Third year Undergraduate or first year Master’s students in various universities.

Prerequisites: Real Analysis, Linear Algebra

Lectures

Week 1

  1. Field of Complex Numbers YouTube
  2. Conjugation and Absolute value YouTube
  3. Topology on Complex plane YouTube
  4. Topology on Complex Plane (Contd) YouTube

Week 2

  1. Problem Session YouTube
  2. Isometries on the Complex Plane YouTube
  3. Functions on the Complex Plane YouTube
  4. Complex differentiability YouTube

Week 3

  1. Power Series YouTube
  2. Differentiation of power series YouTube
  3. Problem Session YouTube
  4. Cauchy-Riemann equations YouTube

Week 4

  1. Harmonic functions YouTube
  2. Möbius transformations YouTube
  3. Problem session YouTube
  4. Curves in the complex plane YouTube

Week 5

  1. Complex Integration over curves YouTube
  2. First Fundamental theorem of Calculus YouTube
  3. Second Fundamental theorem of Calculus YouTube
  4. Problem session YouTube

Week 6

  1. Homotopy of curves YouTube
  2. Cauchy-Goursat theorem YouTube
  3. Cauchy's theorem YouTube
  4. Problem Session YouTube

Week 7

  1. Cauchy Integral Formula YouTube
  2. Principle of analytic continuation and Cauchy estimates YouTube
  3. Further consequences of Cauchy Integral Formula YouTube
  4. Problem session YouTube

Week 8

  1. Winding number YouTube
  2. Open mapping theorem YouTube
  3. Schwarz reflection principle YouTube
  4. Problem session YouTube

Week 9

  1. Singularities of a holomorphic function YouTube
  2. Pole of a function YouTube
  3. Laurent Series YouTube
  4. Casorati Weierstrass theorem YouTube

Week 10

  1. Problem Session YouTube
  2. Residue theorem YouTube
  3. Argument principle YouTube
  4. Problem Session YouTube

Week 11

  1. Branch of the Complex logarithm YouTube
  2. Automorphisms of the Unit disk YouTube
  3. Phragmen Lindelof method YouTube
  4. Problem Session YouTube

Week 12

  1. Lifting of maps YouTube
  2. Covering spaces YouTube
  3. Bloch's theorem YouTube
  4. Little Picard's theorem YouTube

Syllabus and materials

At NPTEL, as the course publishes them.

Credits and licence

Institution
Kerala School of Mathematics
Department
Mathematics
Teacher
Prof. Pranav Haridas
Level
Undergraduate
Language
English
Subjects
Mathematics
Source
NPTEL
Licence
CC BY-NC-SA, as NPTEL states it. Non-commercial use only. NPTEL's terms
Attribution
NPTEL states: "Distributed under Creative Commons Attribution-ShareAlike / CC BY - NC - SA", beside a BY-NC-SA badge. Coordinated by IIT Madras.
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