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Course11110300273 lectures

Complex Analysis - I

Prof. Arup Chattopadhyay · IIT Guwahati

Start with lecture 1 At NPTEL

About

Complex analysis is one of the fundamental mathematical subjects that is introduced to undergraduate and postgraduate students of Mathematical Science, Engineering, and Technology programmes. Real analysis (or Single-Variable Calculus) is introduced in some form at the Higher Secondary Level, whereas Complex analysis is introduced in the next level. Complex analysis contain some of the most powerful and beautiful tools in Mathematics, which are required to solve many mathematical problems. For example, some indefinite integrals (that cannot be handled through real analysis techniques) can be evaluated using contour integration and residue theorems. Apart from higher mathematics, complex analysis has wide applications in Engineering and Physics. My plan for this course is to go through the relevant concepts rigorously so that students who are interested in higher mathematics should benefit, as well as students who are from engineering background should also be able to apply those concepts in engineering and technology fields.

Intended audience: Second year UG level students who completed basic Real Analysis and basic Linear Algebra courses are eligible to take this course. First year PG students are eligible to take this course.

Prerequisites: Basic Real Analysis and Basic Linear Algebra courses are required for this proposed course. Second year UG level students who completed basic Real Analysis and basic Linear Algebra courses are eligible to take this course. First year PG students are eligible to take this course.

Lectures

Week 01: Review of Complex Numbers, Topology of Sets, Complex Functions: Limit, Continuity

  1. Lecture 01: Review of Complex Numbers YouTube
  2. Lecture 02: Properties of Complex Numbers YouTube
  3. Lecture 03: More properties, polar form and Trignometric form YouTube
  4. Lecture 04: Principle argument and powers of complex numbers YouTube
  5. Lecture 05: Topology of sets in the complex plane YouTube
  6. Lecture 06: More on topology and complex functions YouTube
  7. Lecture 07: Limit of complex functions YouTube
  8. Lecture 08: Stereographic projection and continuity of complex functions YouTube

Week 02: Differentiability, Analytic functions, and C-R Equation, Harmonic functions and finding the Harmonic conjugate

  1. Lecture 01: Notion of differentiability and CR equation YouTube
  2. Lecture 02: Proof of CR equations and Examples YouTube
  3. Lecture 03: Necessary and Sufficient conditions for differentiabilty YouTube
  4. Lecture 04: Concept of analytic function and few results of analytic functions YouTube
  5. Lecture 05: Results of analytic functions, harmonic functions and harmonic conjugate YouTube
  6. Lecture 06: How to calculate harmonic conjugate YouTube
  7. Lecture 07: More of harmonic functions and singular points YouTube

Week 03: Elementary Analytic Functions, Multivalued functions and concept of branches, Riemann-Stieltjes integral,…

  1. Lecture 01: Exponential Function YouTube
  2. Lecture 02: Other trigonometric functions and log function YouTube
  3. Lecture 03: Branch of log function, branch cut and branch point, complex exponents YouTube
  4. Lecture 04: Integration of complex valued function of real variable, concept of curve YouTube
  5. Lecture 05: Curves, Differentiable curves, Smooth curves, Contour, Opposite curves YouTube
  6. Lecture 06: Simple closed curves, Orientation, Function of bounded variation YouTube
  7. Lecture 07: Riemann Stieltjes integral, Definition of contour integral/line integral, Properties of contour integral/line integral YouTube

Week 04: Path independency, Cauchy-Goursat Theorem, Antiderivatives, Cauchy’s theorem and integral formula, Cauchy’s…

  1. Lecture 01: Path independency and proof of Cauchy-Goursat theorem YouTube
  2. Lecture 02: Concept of antiderivatives or primitives and related theorems and examples YouTube
  3. Lecture 03: Consequence of Cauchy-Goursat Theorem and existence of antiderivatives YouTube
  4. Lecture 04: Deformation of contours and Cauchy-Goursat theorem for multiply connected domain YouTube
  5. Lecture 05: Cauchy's integral formula and its consequences YouTube
  6. Lecture 06: Cauchy's integral formula for n-th derivatives and its consequences YouTube
  7. Lecture 07: Cauchy’s Estimate and Liouville’s Theorem YouTube

Week 05: Application of Liouville’s Theorem, Fundamental Theorem of Algebra, Maximum Modulus Principle and its…

  1. Lecture 01: Application of Liouville’s Theorem YouTube
  2. Lecture 02: Fundamental Theorem of Algebra and Gauss Mean value Theorem YouTube
  3. Lecture 03: Maximum Modulus Theorem and its proof YouTube
  4. Lecture 04: Maximum Modulus Principle and its consequences YouTube
  5. Lecture 05: Schwarz’s Lemma and Morera’s Theorem YouTube

Week 06: Power Series, Power series representation of an analytic function, Taylor series

  1. Lecture 01: Sequence and Series of Functions YouTube
  2. Lecture 02: Notion of power series and radius of convergence YouTube
  3. Lecture 03: Power series with gaps, Analyticity of the Sum Function of a Power Series YouTube
  4. Lecture 04: Proof of the Analyticity of the Sum Function of a Power Series YouTube
  5. Lecture 05: Statement and Proof of Taylor's Theorem, Examples YouTube
  6. Lecture 06: Taylor series, justification of Euler's formula, comparison between real and complex functions YouTube

Week 07: Laurent series, Zeros of Analytic Functions, Singularities: Classification of singularities

  1. Lecture 01: Analyticity at the point at infinity and definition of singular points YouTube
  2. Lecture 02: Laurent's theorem and an example of finding Laurent series YouTube
  3. Lecture 03: Zeros of an analytic function, zeros of an analytic function are isolated, factorization YouTube
  4. Lecture 04: Zeros of an analytic function, Identity and Uniqueness Theorem YouTube
  5. Lecture 05: Application of Identity and Uniqueness Theorem, definition of isolated and non-isolated singularities YouTube
  6. Lecture 06: Classification of isolated singularities: removable singularity, pole, and essential singularity YouTube

Week 08: More on singularities, Meromorphic functions, Residues, Argument principle

  1. Lecture 01: Properties of removable singularity, pole, and essential singularity, singularity at Infinity YouTube
  2. Lecture 02: Entire functions with isolated singularities and a meromorphic function YouTube
  3. Lecture 03: Concept of residues, residue at removable singularity, residue formula for a pole of order m YouTube
  4. Lecture 04: Residue Formula for Quotient of Analytic Functions, Residue at an Essential Singularity YouTube
  5. Lecture 05: Residue at an isolated singularity infinity, Cauchy's residue theorem YouTube
  6. Lecture 06: Cauchy's residue theorem and a few examples YouTube
  7. Lecture 07: Argument principle and examples YouTube

Week 9: Rouche’s theorem, Open Mapping Theorem, Casorati-Weierstrass theorem, Evaluating definite integrals of…

  1. Lecture 01: Statement and proof of Rouche's theorem, and a few examples YouTube
  2. Lecture 02: Statement and proof of Open Mapping Theorem YouTube
  3. Lecture 03: Statement and proof of the Casorati-Weierstrass theorem YouTube
  4. Lecture 04: Evaluation of definite integrals of trigonometric functions (Type-I) YouTube
  5. Lecture 05: Definition of improper integrals of rational functions and the Cauchy principal value YouTube
  6. Lecture 06: Evaluation of improper integrals of rational functions (Type-II) YouTube

Week 10: Evaluating improper integrals from Fourier Analysis, Indented Contour Integration: Indention around a Branch…

  1. Lecture 01: Jordan's Lemma and its proof (concerning Type-III integrals) YouTube
  2. Lecture 02: Evaluating improper integrals from Fourier Analysis (Type-III) YouTube
  3. Lecture 03: Evaluating Indented Contour Integration (Type-IV(a)) YouTube
  4. Lecture 04: Evaluating Indented Contour Integration (Type-IV(b)) YouTube
  5. Lecture 05: Evaluating Indented Contour Integration (Type-IV(c)) YouTube

Week 11: Integration along a Branch Cut (Type-V), Finding the Inverse Laplace Transform, Conformal mapping, Mobius…

  1. Lecture 01: Evaluating Keyhole Contour Integration (Type-V) YouTube
  2. Lecture 02: Finding the Inverse Laplace Transform YouTube
  3. Lecture 03: Conformal mapping and its properties YouTube
  4. Lecture 04: Mobius transformations and their properties YouTube
  5. Lecture 05: Finding Mobius transformations fixing points YouTube

Week 12: More on properties of Mobius transformations, Riemann Mapping Theorem, Weierstrass Factorization Theorem

  1. Lecture 01: Finding Mobius transformations fixing points and the orientation principle YouTube
  2. Lecture 02: Cross Ratio and Symmetry Principle YouTube
  3. Lecture 03: Mobius transformation that map upperhalf plane onto the open unit disc YouTube
  4. Lecture 04: Riemann Mapping Theorem and Weierstrass Factorization Theorem YouTube

Syllabus and materials

At NPTEL, as the course publishes them.

Credits and licence

Institution
IIT Guwahati
Department
Mathematics
Teacher
Prof. Arup Chattopadhyay
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.
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