Complex Analysis - I EJJAEYSW3M

URI	`m2://res/EJJAEYSW3M`
Kind	course
Creators	Arup Chattopadhyay (instructor)
Date	2026-07-10
Scope	in
Ids	course 0022nd079:111103002

Expressions

e1: edition

Language	en
Date	2026-07-10

m1: page, IIT Guwahati (record). nptel 111103002
m2: page, 2026-07-10 (record). nptel 111103002

Contents

part 1: Week 01: Review of Complex Numbers, Topology of Sets, Complex Functions: Limit, Continuity
session 1: Lecture 01: Review of Complex Numbers
session 2: Lecture 02: Properties of Complex Numbers
session 3: Lecture 03: More properties, polar form and Trignometric form
session 4: Lecture 04: Principle argument and powers of complex numbers
session 5: Lecture 05: Topology of sets in the complex plane
session 6: Lecture 06: More on topology and complex functions
session 7: Lecture 07: Limit of complex functions
session 8: Lecture 08: Stereographic projection and continuity of complex functions
part 2: Week 02: Differentiability, Analytic functions, and C-R Equation, Harmonic functions and finding the Harmonic conjugate
session 9: Lecture 01: Notion of differentiability and CR equation
session 10: Lecture 02: Proof of CR equations and Examples
session 11: Lecture 03: Necessary and Sufficient conditions for differentiabilty
session 12: Lecture 04: Concept of analytic function and few results of analytic functions
session 13: Lecture 05: Results of analytic functions, harmonic functions and harmonic conjugate
session 14: Lecture 06: How to calculate harmonic conjugate
session 15: Lecture 07: More of harmonic functions and singular points
part 3: Week 03: Elementary Analytic Functions, Multivalued functions and concept of branches, Riemann-Stieltjes integral, Contour integral
session 16: Lecture 01: Exponential Function
session 17: Lecture 02: Other trigonometric functions and log function
session 18: Lecture 03: Branch of log function, branch cut and branch point, complex exponents
session 19: Lecture 04: Integration of complex valued function of real variable, concept of curve
session 20: Lecture 05: Curves, Differentiable curves, Smooth curves, Contour, Opposite curves
session 21: Lecture 06: Simple closed curves, Orientation, Function of bounded variation
session 22: Lecture 07: Riemann Stieltjes integral, Definition of contour integral/line integral, Properties of contour integral/line integral
part 4: Week 04: Path independency, Cauchy-Goursat Theorem, Antiderivatives, Cauchy’s theorem and integral formula, Cauchy’s Estimate, Liouville’s Theorem
session 23: Lecture 01: Path independency and proof of Cauchy-Goursat theorem
session 24: Lecture 02: Concept of antiderivatives or primitives and related theorems and examples
session 25: Lecture 03: Consequence of Cauchy-Goursat Theorem and existence of antiderivatives
session 26: Lecture 04: Deformation of contours and Cauchy-Goursat theorem for multiply connected domain
session 27: Lecture 05: Cauchy's integral formula and its consequences
session 28: Lecture 06: Cauchy's integral formula for n-th derivatives and its consequences
session 29: Lecture 07: Cauchy’s Estimate and Liouville’s Theorem
part 5: Week 05: Application of Liouville’s Theorem, Fundamental Theorem of Algebra, Maximum Modulus Principle and its consequences, Schwarz’s Lemma, Morera’s Theorem
session 30: Lecture 01: Application of Liouville’s Theorem
session 31: Lecture 02: Fundamental Theorem of Algebra and Gauss Mean value Theorem
session 32: Lecture 03: Maximum Modulus Theorem and its proof
session 33: Lecture 04: Maximum Modulus Principle and its consequences
session 34: Lecture 05: Schwarz’s Lemma and Morera’s Theorem
part 6: Week 06: Power Series, Power series representation of an analytic function, Taylor series
session 35: Lecture 01: Sequence and Series of Functions
session 36: Lecture 02: Notion of power series and radius of convergence
session 37: Lecture 03: Power series with gaps, Analyticity of the Sum Function of a Power Series
session 38: Lecture 04: Proof of the Analyticity of the Sum Function of a Power Series
session 39: Lecture 05: Statement and Proof of Taylor's Theorem, Examples
session 40: Lecture 06: Taylor series, justification of Euler's formula, comparison between real and complex functions
part 7: Week 07: Laurent series, Zeros of Analytic Functions, Singularities: Classification of singularities
session 41: Lecture 01: Analyticity at the point at infinity and definition of singular points
session 42: Lecture 02: Laurent's theorem and an example of finding Laurent series
session 43: Lecture 03: Zeros of an analytic function, zeros of an analytic function are isolated, factorization
session 44: Lecture 04: Zeros of an analytic function, Identity and Uniqueness Theorem
session 45: Lecture 05: Application of Identity and Uniqueness Theorem, definition of isolated and non-isolated singularities
session 46: Lecture 06: Classification of isolated singularities: removable singularity, pole, and essential singularity
part 8: Week 08: More on singularities, Meromorphic functions, Residues, Argument principle
session 47: Lecture 01: Properties of removable singularity, pole, and essential singularity, singularity at Infinity
session 48: Lecture 02: Entire functions with isolated singularities and a meromorphic function
session 49: Lecture 03: Concept of residues, residue at removable singularity, residue formula for a pole of order m
session 50: Lecture 04: Residue Formula for Quotient of Analytic Functions, Residue at an Essential Singularity
session 51: Lecture 05: Residue at an isolated singularity infinity, Cauchy's residue theorem
session 52: Lecture 06: Cauchy's residue theorem and a few examples
session 53: Lecture 07: Argument principle and examples
part 9: Week 9: Rouche’s theorem, Open Mapping Theorem, Casorati-Weierstrass theorem, Evaluating definite integrals of trigonometric functions, improper integrals of rational functions
session 54: Lecture 01: Statement and proof of Rouche's theorem, and a few examples
session 55: Lecture 02: Statement and proof of Open Mapping Theorem
session 56: Lecture 03: Statement and proof of the Casorati-Weierstrass theorem
session 57: Lecture 04: Evaluation of definite integrals of trigonometric functions (Type-I)
session 58: Lecture 05: Definition of improper integrals of rational functions and the Cauchy principal value
session 59: Lecture 06: Evaluation of improper integrals of rational functions (Type-II)
part 10: Week 10: Evaluating improper integrals from Fourier Analysis, Indented Contour Integration: Indention around a Branch point
session 60: Lecture 01: Jordan's Lemma and its proof (concerning Type-III integrals)
session 61: Lecture 02: Evaluating improper integrals from Fourier Analysis (Type-III)
session 62: Lecture 03: Evaluating Indented Contour Integration (Type-IV(a))
session 63: Lecture 04: Evaluating Indented Contour Integration (Type-IV(b))
session 64: Lecture 05: Evaluating Indented Contour Integration (Type-IV(c))
part 11: Week 11: Integration along a Branch Cut (Type-V), Finding the Inverse Laplace Transform, Conformal mapping, Mobius transformations, and its properties
session 65: Lecture 01: Evaluating Keyhole Contour Integration (Type-V)
session 66: Lecture 02: Finding the Inverse Laplace Transform
session 67: Lecture 03: Conformal mapping and its properties
session 68: Lecture 04: Mobius transformations and their properties
session 69: Lecture 05: Finding Mobius transformations fixing points
part 12: Week 12: More on properties of Mobius transformations, Riemann Mapping Theorem, Weierstrass Factorization Theorem
session 70: Lecture 01: Finding Mobius transformations fixing points and the orientation principle
session 71: Lecture 02: Cross Ratio and Symmetry Principle
session 72: Lecture 03: Mobius transformation that map upperhalf plane onto the open unit disc
session 73: Lecture 04: Riemann Mapping Theorem and Weierstrass Factorization Theorem

Catalogue records

nptel: 111103002 (manifestation)
nptel_video: 111103002 (manifestation)