Lecture 6 of 73Week 01: Review of Complex Numbers, Topology of Sets, Complex Functions: Limit, Continuity
Lecture 06: More on topology and complex functions
Prof. Arup Chattopadhyay · IIT Guwahati
Credits and licence
- From
- Complex Analysis - I, IIT Guwahati (at NPTEL)
- Licence
- CC BY-NC-SA, as NPTEL states it. Non-commercial use only.
- Attribution
- NPTEL states: "Distributed under Creative Commons Attribution-ShareAlike / CC BY - NC - SA", beside a BY-NC-SA badge.
- Who can watch
- Tweens profiles and older. How courses are placed
All the lectures
- Lecture 01: Review of Complex Numbers
- Lecture 02: Properties of Complex Numbers
- Lecture 03: More properties, polar form and Trignometric form
- Lecture 04: Principle argument and powers of complex numbers
- Lecture 05: Topology of sets in the complex plane
- Lecture 06: More on topology and complex functions
- Lecture 07: Limit of complex functions
- Lecture 08: Stereographic projection and continuity of complex functions
- Lecture 01: Notion of differentiability and CR equation
- Lecture 02: Proof of CR equations and Examples
- Lecture 03: Necessary and Sufficient conditions for differentiabilty
- Lecture 04: Concept of analytic function and few results of analytic functions
- Lecture 05: Results of analytic functions, harmonic functions and harmonic conjugate
- Lecture 06: How to calculate harmonic conjugate
- Lecture 07: More of harmonic functions and singular points
- Lecture 01: Exponential Function
- Lecture 02: Other trigonometric functions and log function
- Lecture 03: Branch of log function, branch cut and branch point, complex exponents
- Lecture 04: Integration of complex valued function of real variable, concept of curve
- Lecture 05: Curves, Differentiable curves, Smooth curves, Contour, Opposite curves
- Lecture 06: Simple closed curves, Orientation, Function of bounded variation
- Lecture 07: Riemann Stieltjes integral, Definition of contour integral/line integral, Properties of contour integral/line integral
- Lecture 01: Path independency and proof of Cauchy-Goursat theorem
- Lecture 02: Concept of antiderivatives or primitives and related theorems and examples
- Lecture 03: Consequence of Cauchy-Goursat Theorem and existence of antiderivatives
- Lecture 04: Deformation of contours and Cauchy-Goursat theorem for multiply connected domain
- Lecture 05: Cauchy's integral formula and its consequences
- Lecture 06: Cauchy's integral formula for n-th derivatives and its consequences
- Lecture 07: Cauchy’s Estimate and Liouville’s Theorem
- Lecture 01: Application of Liouville’s Theorem
- Lecture 02: Fundamental Theorem of Algebra and Gauss Mean value Theorem
- Lecture 03: Maximum Modulus Theorem and its proof
- Lecture 04: Maximum Modulus Principle and its consequences
- Lecture 05: Schwarz’s Lemma and Morera’s Theorem
- Lecture 01: Sequence and Series of Functions
- Lecture 02: Notion of power series and radius of convergence
- Lecture 03: Power series with gaps, Analyticity of the Sum Function of a Power Series
- Lecture 04: Proof of the Analyticity of the Sum Function of a Power Series
- Lecture 05: Statement and Proof of Taylor's Theorem, Examples
- Lecture 06: Taylor series, justification of Euler's formula, comparison between real and complex functions
- Lecture 01: Analyticity at the point at infinity and definition of singular points
- Lecture 02: Laurent's theorem and an example of finding Laurent series
- Lecture 03: Zeros of an analytic function, zeros of an analytic function are isolated, factorization
- Lecture 04: Zeros of an analytic function, Identity and Uniqueness Theorem
- Lecture 05: Application of Identity and Uniqueness Theorem, definition of isolated and non-isolated singularities
- Lecture 06: Classification of isolated singularities: removable singularity, pole, and essential singularity
- Lecture 01: Properties of removable singularity, pole, and essential singularity, singularity at Infinity
- Lecture 02: Entire functions with isolated singularities and a meromorphic function
- Lecture 03: Concept of residues, residue at removable singularity, residue formula for a pole of order m
- Lecture 04: Residue Formula for Quotient of Analytic Functions, Residue at an Essential Singularity
- Lecture 05: Residue at an isolated singularity infinity, Cauchy's residue theorem
- Lecture 06: Cauchy's residue theorem and a few examples
- Lecture 07: Argument principle and examples
- Lecture 01: Statement and proof of Rouche's theorem, and a few examples
- Lecture 02: Statement and proof of Open Mapping Theorem
- Lecture 03: Statement and proof of the Casorati-Weierstrass theorem
- Lecture 04: Evaluation of definite integrals of trigonometric functions (Type-I)
- Lecture 05: Definition of improper integrals of rational functions and the Cauchy principal value
- Lecture 06: Evaluation of improper integrals of rational functions (Type-II)
- Lecture 01: Jordan's Lemma and its proof (concerning Type-III integrals)
- Lecture 02: Evaluating improper integrals from Fourier Analysis (Type-III)
- Lecture 03: Evaluating Indented Contour Integration (Type-IV(a))
- Lecture 04: Evaluating Indented Contour Integration (Type-IV(b))
- Lecture 05: Evaluating Indented Contour Integration (Type-IV(c))
- Lecture 01: Evaluating Keyhole Contour Integration (Type-V)
- Lecture 02: Finding the Inverse Laplace Transform
- Lecture 03: Conformal mapping and its properties
- Lecture 04: Mobius transformations and their properties
- Lecture 05: Finding Mobius transformations fixing points
- Lecture 01: Finding Mobius transformations fixing points and the orientation principle
- Lecture 02: Cross Ratio and Symmetry Principle
- Lecture 03: Mobius transformation that map upperhalf plane onto the open unit disc
- Lecture 04: Riemann Mapping Theorem and Weierstrass Factorization Theorem