Conformal Transformations ( Complex Analysis)

Bilinear Transformations, Mobius Transformation, Fixed Points of Bilinear Transformation, Cross ratio of preservance

What you'll learn

Concept of Transformations or mapping from z plane to w plane.

Concept of Standard or Elementary Transformations including Linear Transformation

Bilinear or Linear Fractional Transformation including the concept of Mobius Transformations with all properties and examples.

Elliptic, Parabolic, Hyperbolic and Loxodromic Transformation including Assignments based on it.

Requirements

Basic knowledge of Complex Numbers

Description

A Conformal Mapping, also called a Conformal Map, Conformal Transformation, Angle-preserving transformation, or Biholomorphic map, is a transformation that preserves local angles. The Course 'Conformal Transformations'  describes about the mapping of points in the z plane to w plane including the other contents_Detailed concept of Transformations and Jacobian of Transformation.To determine the region in the w plane corresponding to the region given in z plane.Necessary and Sufficient Condition for w = f(z) to represent Conformal Mapping.Superficial magnification and Inverse points with respect to a Circle.Some Elementary Transformation as Translation Transformation, Rotation Transformation, Magnification Transformation, Rotation and magnification Transformation, Inversion Transformation, Linear transformation.Bilinear or Linear Fractional Transformation.Determinant of Transformation and its Normalized Form.Mobius Transformation and Critical Points.Resultant or Product of Transformation.Preservance of Cross ratio under bilinear TransformationTo Determine the Bilinear Transformation which maps the points in z plane to the points in w plane.Steiner Circles and Family of circles.Normal Form of Bilinear transformation and Fixed Points of Bilinear Transformation.Every Bilinear Transformation transforms circles or straight lines into circles or straight lines and inverse points into inverse points.Elliptical  Transformation, Hyperbolic Transformation, Parabolic Transformation & Loxodromic Transformation including all expected solved examples and Important Theorems.

Overview

Section 1: Conformal Mapping Introduction

Lecture 1 Definition of Transformations and Mapping, Jacobian of a Transformation with Eg.

Lecture 2 To Determine the Region in w plane for Circular Disc in z plane.

Lecture 3 Sufficient Condition for w = f(z) to represent Conformal Mapping

Lecture 4 Necessary Condition for w = f(z) to represent Conformal Mapping

Section 2: Some Elementary or Standard Transformations

Lecture 5 Standard Transformation TRANSLATION

Lecture 6 Coefficient of Magnification & Angle of Rotation

Lecture 7 Standard Transformation ROTATION

Lecture 8 Standard Transformation MAGNIFICATION

Lecture 9 Standard Transformation ROTATION & MAGNIFICATION

Lecture 10 Standard Transformation INVERSION

Section 3: Images under the Given Transformation

Lecture 11 Image of an Infinite Strip under Transformation w = 1/z

Lecture 12 Transformation maps Circle onto the Straight Line

Lecture 13 Transformation that maps Circle into the Cardioid in w plane.

Lecture 14 Linear Transformation explained with an Example

Section 4: Bilinear or Linear Fractional Transformations

Lecture 15 Bilinear Transformation and Inverse Transformation

Lecture 16 Solved example on Bilinear Transformation and Inversion Transformation

Lecture 17 Determine Bilinear Ttransformation if Mapping Points are Given.

Lecture 18 Alternative Method to Find Bilinear Transformation if mapping pts. are Given.

Lecture 19 Find the Bilinear Transformation if Mapping Points are Given…Assignment 1

Lecture 20 Find the Bilinear Transformation if mapping points are Given…Assignment 2

Lecture 21 Find the Bilinear Transformation if mapping Points are Given….Assignment 3

Section 5: Mobius Transformations and Fixed Points of Bilinear Transformation

Lecture 22 Every Mobius Transformation maps circles or St. Line into Circles or St. Line.

Lecture 23 Every Bilinear Transformation also transforms Inverse Points into Inverse Points

Lecture 24 Fixed Points of Bilinear Transformation

Lecture 25 Normal Form of Bilinear Transformation

Lecture 26 Bilinear Transformation having only One Fixed Point

Lecture 27 Determine the Form of Bilinear Transformation for Fixed Points

Section 6: Elliptic, Hyperbolic & Parabolic Transformations

Lecture 28 Concept about Elliptic, Hyperbolic , Parabolic and loxodromic Transformation

Lecture 29 Fixed Points and Normal form of Bilinear Transformation w = z/(z-2)

Lecture 30 Fixed Points and Normal form of Billinear Transformation w = 3z-4/(z-1)

Lecture 31 Fixed Points and Normal form of Bilinear Transformation w = z-1/z+1

This Course is a basic course offered to Undergraduate or Post Graduate Students of Engineering and Science background.