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.



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