Learn · The Geometry of Möbius Transformations
Lecture 11 of 56Week 3
Cauchy-Riemann revisited
Prof. Krishnendu Gongopadhyay · IISER Mohali
Credits and licence
- From
- The Geometry of Möbius Transformations, IISER Mohali (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. Coordinated by IIT Madras.
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- Tweens profiles and older. How courses are placed
All the lectures
- Introduction
- Complex numbers
- Euler's forumula
- Orthogonal matrices
- Euclidean isometries
- Classification of isometries
- An example
- Complex differentiability-I
- Complex differentiability-II
- Cauchy-Riemann Eqn
- Cauchy-Riemann revisited
- Complex logarithm
- Complex Integration
- Cauchy Integral Formula
- Power Series Expansion
- Zeros of holomorphic fn
- Open Maping thm
- Singularity
- Poles
- Essential singularities
- Residues
- The residue theorem
- The Riemann Sphere
- Meromorphic functions
- Rational functions
- degree of rational functions
- degree theorem and orders
- Mobius transformation
- Mobius transformation-II
- anti-automorphism
- Generators of Mobius group
- Circles on Riemann Sphere
- Mobius transformation & circles
- Cross Ratios
- Permutation and cross ratios
- Circles and Cross Ratios
- Group action
- Stabiliser of a disc & circle
- Standard forms
- classification of Mobius transformation
- Commutator
- Commutator and fixed points
- Commuting Motions
- Centralizers
- Conformality of stereographic projection
- Conformal automorphisms
- Conformal maps of spheres
- Rotations of Riemann sphere
- Subgroup of elliptic elements
- Subgroup of elliptic elements-II
- Finite group of rotations
- Elements of arc length
- hyperbolic upper half model
- Geodesics
- Hyperbolic isometries
- Concluding words