Chap 8 Mapping by Elementary Functions 68 Linear
Chap 8 Mapping by Elementary Functions 68. Linear Transformations 1
2
69. The Transformation mapping between nonzero points of z and w planes. a reflection in the real axis An inversion with respect to unit circle 3
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5
6
Ex 1. 7
Ex 2. Ex 3. 8
70. Linear Fractional Transformation is called a linear fractional transformation or Mobius transformation. bilinear transformation linear in z linear in w bilinear in z and w 9
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Denominator=0 11
This makes T continuous on the extended z plane (Ex 10, sec 14). We enlarge the domain of definition, (5) is a one-to-one mapping of the extended z plane onto the extended w plane. 12
A linear fractional transformation 13
There is always a linear fractional transformation that maps three given distinct points, z 1, z 2 and z 3 onto three specified distinct points w 1, w 2 and w 3. Ex 1. 14
Ex 2: 15
71. An Implicit Form The equation 16
Ex 1. 17
18
Ex 2. 19
72. Mapping of the upper Half Plane Determine all 1 inear fractional transformation T that 20
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Ex 1. Ex 2. 23
73. Exponential and Logarithmic Transformations 24
Ex 1 25
Ex 2. any branch of log z , maps onto a strip 26
Ex 3. 27
74. The transformation Ex 1. (1 -to-1) 28
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30
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Ex 2. Ex 3. Ex 4. 32
75. Mapping by Branches of 33
Ex 1 Ex 2 34
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76. Square roots of polynomials Ex 1. 36
Ex. 2 37
- Slides: 37
