1 Chap 8 Mapping by Elementary Functions 68. Linear Transformations.

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Presentation transcript:

1 Chap 8 Mapping by Elementary Functions 68. Linear Transformations

2

3 69. The Transformation mapping between nonzero points of z and w planes. An inversion with respect to unit circle a reflection in the real axis

4

5

6

7 Ex1.

8 Ex2. Ex3.

9 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

10

11 Denominator=0

12 This makes T continuous on the extended z plane (Ex10, sec14). We enlarge the domain of definition, (5) is a one-to-one mapping of the extended z plane onto the extended w plane.

13 A linear fractional transformation

14 Ex1. There is always a linear fractional transformation that maps three given distinct points, z1, z2 and z3 onto three specified distinct points w1, w2 and w3.

15 Ex2:

An Implicit Form The equation

17 Ex1.

18

19 Ex2.

Mapping of the upper Half Plane Determine all 1inear fractional transformation T that

21

22

23 Ex1. Ex2.

Exponential and Logarithmic Transformations

25 Ex1

26 Ex2. any branch of log z, maps onto a strip

27 Ex3.

The transformation Ex1. (1-to-1)

29

30

31

32 Ex2. Ex3. Ex4.

Mapping by Branches of

34 Ex1 Ex2

35

Square roots of polynomials Ex1.

37 Ex.2