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

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3. 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

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7. 7 Ex1.

8. 8 Ex2. Ex3.

9. 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

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11. 11 Denominator=0

12. 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. 13 A linear fractional transformation

14. 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. 15 Ex2:

16. 16 71. An Implicit Form The equation

17. 17 Ex1.

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19. 19 Ex2.

20. 20 72. Mapping of the upper Half Plane Determine all 1inear fractional transformation T that

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23. 23 Ex1. Ex2.

24. 24 73. Exponential and Logarithmic Transformations

25. 25 Ex1

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

27. 27 Ex3.

28. 28 74. The transformation Ex1. (1-to-1)

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32. 32 Ex2. Ex3. Ex4.

33. 33 75. Mapping by Branches of

34. 34 Ex1 Ex2

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36. 36 76. Square roots of polynomials Ex1.

37. 37 Ex.2