IIT Guwahati Course , Prof. P. A. S. Sree Krishna

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IIT Guwahati Course , Prof. P. A. S. Sree Krishna

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 - Cauchys theorem as a version of Greens 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, Taylors theorem - Zeroes of analytic functions, Rouches 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, Laurents theorem - Residue theorem, the argument principle - Problem Session

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 - Cauchys theorem as a version of Greens 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, Taylors theorem - Zeroes of analytic functions, Rouches 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, Laurents theorem - Residue theorem, the argument principle - Problem Session

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Complex Analysis by Prof. P. A. S. Sree Krishna,Department of Mathematics, IIT Guwahati.For more details on NPTEL visit httpnptel.iitm.ac.in

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- 1.Mod-01 Introduction
- 2.Introduction to Complex Numbers
- 3.De Moivres Formula and Stereographic Projection
- 4.Topology of the Complex Plane Part-I
- 5.Topology of the Complex Plane Part-II
- 6.Topology of the Complex Plane Part-III
- 7.Introduction to Complex Functions
- 8.Limits and Continuity
- 9.Differentiation
- 10.Cauchy-Riemann Equations and Differentiability
- 11.Analytic functions; the exponential function
- 12.Sine, Cosine and Harmonic functions
- 13.Branches of Multifunctions; Hyperbolic Functions
- 14.Problem Solving Session I
- 15.Integration and Contours
- 16.Contour Integration
- 17.Introduction to Cauchys Theorem
- 18.Cauchys Theorem for a Rectangle
- 19.Cauchys theorem Part - II
- 20.Cauchys Theorem Part - III
- 21.Cauchys Integral Formula and its Consequences
- 22.The First and Second Derivatives of Analytic Functions
- 23.Moreras Theorem and Higher Order Derivatives of Analytic Functions
- 24.Problem Solving Session II
- 25.Introduction to Complex Power Series
- 26.Analyticity of Power Series
- 27.Taylors Theorem
- 28.Zeroes of Analytic Functions
- 29.Counting the Zeroes of Analytic Functions
- 30.Open mapping theorem -- Part I
- 31.Open mapping theorem -- Part II
- 32.Properties of Mobius Transformations Part I
- 33.Properties of Mobius Transformations Part II
- 34.Problem Solving Session III
- 35.Removable Singularities
- 36.Poles Classification of Isolated Singularities
- 37.Essential Singularity & Introduction to Laurent Series
- 38.Laurents Theorem
- 39.Residue Theorem and Applications
- 40.Problem Solving Session IV

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