Contents:
Introduction : Introduction and overview of the course, lecture-wise description – The Algebra Geometry and Topology of the Complex Plane : Complex numbers, conjugation, modulus, argument and inequalities – Powers and roots of complex numbers, geometry in the complex plane, the extended complex plane – Topology of the complex plane: Open sets, closed sets, limit points, isolated points, interior points, boundary points, exterior points, compact sets, connected sets, sequences and series of complex numbers and convergence.

Complex Functions: Limits, Continuity and Differentiation : Introduction to complex functions – Limits and continuity – Differentiation and the Cauchy-Riemann equations, analytic functions, elementary functions and their mapping properties, harmonic functions – Complex logarithm multi-function, analytic branches of the logarithm multi-function, complex exponent multi-functions and their analytic branches, complex hyperbolic functions – Problem Session

Complex Integration Theory : Introducing curves, paths and contours, contour integrals and their properties, fundamental theorem of calculus – Cauchy’s theorem as a version of Green’s theorem, Cauchy-Goursat theorem for a rectangle, The anti-derivative theorem, Cauchy-Goursat theorem for a disc, the deformation theorem – Cauchy’s integral formula, Cauchy’s estimate, Liouville’s theorem, the fundamental theorem of algebra, higher derivatives of analytic functions, Morera’s theorem – Problem Session
Further Properties of Analytic Functions : Power series, their analyticity, Taylor’s theorem – Zeroes of analytic functions, Rouche’s theorem – Open mapping theorem, maximum modulus theorem.

Mobius Transformations : Properties of Mobius transformations – Problem Session
Isolated Singularities and Residue Theorem : Isolated singularities, removable singularities – Poles, classification of isolated singularities – Casoratti-Weierstrass theorem, Laurent’s theorem – Residue theorem, the argument principle – Problem Session

Other Resources

Course Curriculum

Mod-01 Introduction Details 39:20
Introduction to Complex Numbers Details 1:4:4
De Moivre’s Formula and Stereographic Projection Details 48:56
Topology of the Complex Plane Part-I Details 54:43
Topology of the Complex Plane Part-II Details 50:59
Topology of the Complex Plane Part-III Details 55:1
Introduction to Complex Functions Details 53:35
Limits and Continuity Details 49:22
Differentiation Details 59:51
Cauchy-Riemann Equations and Differentiability Details 53:39
Analytic functions; the exponential function Details 51:55
Sine, Cosine and Harmonic functions Details 57:15
Branches of Multifunctions; Hyperbolic Functions Details 51:7
Problem Solving Session I Details 51:6
Integration and Contours Details 48:43
Contour Integration Details 52:4
Introduction to Cauchy’s Theorem Details 41:19
Cauchy’s Theorem for a Rectangle Details 1:49
Cauchy’s theorem Part – II Details 50:32
Cauchy’s Theorem Part – III Details 48:1
Cauchy’s Integral Formula and its Consequences Details 56:6
The First and Second Derivatives of Analytic Functions Details 52:9
Morera’s Theorem and Higher Order Derivatives of Analytic Functions Details 50:36
Problem Solving Session II Details 55:39
Introduction to Complex Power Series Details 49:15
Analyticity of Power Series Details 48:54
Taylor’s Theorem Details 49:58
Zeroes of Analytic Functions Details 50:43
Counting the Zeroes of Analytic Functions Details 52:10
Open mapping theorem — Part I Details 51:17
Open mapping theorem — Part II Details 47:5
Properties of Mobius Transformations Part I Details 48:2
Properties of Mobius Transformations Part II Details 46:30
Problem Solving Session III Details 48:5
Removable Singularities Details 45:29
Poles Classification of Isolated Singularities Details 48:28
Essential Singularity & Introduction to Laurent Series Details 46:30
Laurent’s Theorem Details 45:45
Residue Theorem and Applications Details 51:56
Problem Solving Session IV Details 53:5

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