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CONFORMAL MAPPINGS 1 (Complex Analysis)

Name: CONFORMAL MAPPINGS 1 (Complex Analysis)
Rating: 5.0 (1 reviews)

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

Created byJaswinder Kaur

Last updated 6/2023

English

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.

Course content

6 sections • 31 lectures • 4h 24m total length

Definition of Transformations and Mapping, Jacobian of a Transformation with Eg.13:54
Learn conformal mappings in complex analysis by defining a one-to-one correspondence between z and w planes through u and v, and compute the jacobian to relate areas.
To Determine the Region in w plane for Circular Disc in z plane.9:28
Sufficient Condition for w = f(z) to represent Conformal Mapping15:30
This lecture shows that w=f(z) is conformal when f is analytic and nonzero, preserving angle between curves in magnitude and sense; it notes isogonal mappings preserve magnitude but not sense.
Necessary Condition for w = f(z) to represent Conformal Mapping12:58

Standard Transformation TRANSLATION7:20
Coefficient of Magnification & Angle of Rotation7:42
Standard Transformation ROTATION6:25
Standard Transformation MAGNIFICATION6:22
Magnification is a homothetic transformation with positive factor c, stretching for c>1 or contracting for 0<c<1; in the example w=2z, the triangle maps to u+v=2.
Standard Transformation ROTATION & MAGNIFICATION12:28
Standard Transformation INVERSION10:47

Bilinear Transformation and Inverse Transformation12:11
Solved example on Bilinear Transformation and Inversion Transformation6:27
Determine Bilinear Ttransformation if Mapping Points are Given.12:44
Alternative Method to Find Bilinear Transformation if mapping pts. are Given.8:33
Derive the bilinear transformation mapping z1=2, z2=i, z3=-2 to w1=1, w2=i, w3=-1 using an alternative method, solving for a, b, c, d in w=(a z+b)/(c z+d).
Find the Bilinear Transformation if Mapping Points are Given...Assignment 14:47
Find the Bilinear Transformation if mapping points are Given...Assignment 23:31
Find the Bilinear Transformation if mapping Points are Given....Assignment 33:33

Every Mobius Transformation maps circles or St. Line into Circles or St. Line.14:56
Explore how Möbius transformations map circles or straight lines to circles or straight lines, using cross ratios to establish preservation under bilinear transformations.
Every Bilinear Transformation also transforms Inverse Points into Inverse Points5:46
Fixed Points of Bilinear Transformation6:47
Normal Form of Bilinear Transformation8:36
Derive the normal form of a bilinear transformation with fixed points alpha and beta, yielding w−alpha over w−beta = lambda (z−alpha)/(z−beta) where lambda = (d+c alpha)/(d+c beta).
Bilinear Transformation having only One Fixed Point6:01
Determine the Form of Bilinear Transformation for Fixed Points4:53

Concept about Elliptic, Hyperbolic , Parabolic and loxodromic Transformation19:04
Explore conformal mappings of complex planes via elliptic, hyperbolic, parabolic and loxodromic bilinear transformations, analyzing fixed points, normal forms, and the role of lambda.
Fixed Points and Normal form of Bilinear Transformation w = z/(z-2)4:06
The bilinear transformation w equals z^2 minus z has fixed points z equals 0 and z equals 1, and the normal form yields w/(w-1) = (1/2)·z/(z-1), a hyperbolic map.
Fixed Points and Normal form of Billinear Transformation w = 3z-4/(z-1)5:12
Fixed Points and Normal form of Bilinear Transformation w = z-1/z+15:49

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 Transformation

To 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.

Who this course is for:

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

CONFORMAL MAPPINGS 1 (Complex Analysis)

What you'll learn

Explore related topics

Course content

Conformal Mapping Introduction4 lectures • 52min

Some Elementary or Standard Transformations6 lectures • 51min

Images under the Given Transformation4 lectures • 29min

Bilinear or Linear Fractional Transformations7 lectures • 52min

Mobius Transformations and Fixed Points of Bilinear Transformation6 lectures • 47min

Elliptic, Hyperbolic & Parabolic Transformations4 lectures • 34min

Requirements

Description

Who this course is for: