Chapter 8. Mapping by Elementary Functions

1. Chapter 8. Mapping by Elementary Functions Weiqi Luo (骆伟祺) School of Software Sun Yat-Sen University Email：weiqi.luo@yahoo.com Office：# A313

2. Chapter 8: Mapping by Elementary Functions • Linear Transformations • The Transformation w=1/z • Mapping by 1/z • Linear Fractional Transformations • Mapping of the Upper Half Plane

3. 90. Linear Transformations • The Mapping where A is a nonzero complex constant and z≠0. We write A and z in exponential form: Then Expands or contracts the radius vector representing z by the factor a and rotates it through the angle α about the origin. The image of a given region is geometrically similar to that region.

4. 90. Linear Transformations • The Mapping where B is any complex constant, is a translation by means of the vector representing B. That is, if Then the image of any point (x,y) in the z plane is the point in the w plane The image of a given region is geometrically congruent to that region.

5. 90. Linear Transformations • The General (non-constant) Linear Transformation is a composition of the transformations and when z≠0, it is evidently an expansion or contraction (scaling) and a rotation, followed by a translation.

6. 90. Linear Transformations • Example The mapping transforms the rectangular region in the z=(z, y) plane of the figure into the rectangular region in the w=(u,v) plane there. This is seen by expressing it as a composition of the transformations

7. 90. Linear Transformations • Example (Cont’) Scaling and Rotation Translation (x,y)-plane (X,Y)-plane (u,v)-plane

8. 90. Homework • pp. 313 Ex. 2, Ex. 6

9. 91. The Transformation w=1/z • The Equation establishes a one to one correspondence between the nonzero points of the z and the w planes. Since , the mapping can be described by means of the successive transformations To make the transformation continuous on the extended plane, we let

10. 92. Mapping by w=1/z • The Mapping reveals that Similarly, we have that Based on these relations between coordinates, the mapping w=1/z transforms circles and lines into circles and lines

11. 92. Mapping by w=1/z • Consider the Equation represents an arbitrary circle or line （B2+C2>4AD） Circle: Line: Note: Line can be regarded as a special circle with a infinite radius. 

12. 92. Mapping by w=1/z • The Mapping by w=1/z If x and y satisfy then after the mapping by w=1/z, we get that (a circle or line in (x,y)-plane ) (also a circle or line in (u,v)-plane )

13. 92. Mapping by w=1/z • Four Cases Case #1: A circle (A ≠ 0) not passing through the origin (D ≠ 0) in the z plane is transformed into a circle not passing through the origin in the w plane; Case #2: A circle (A ≠ 0) through the origin (D = 0) in the z plane is transformed into a line that does not pass through the origin in the w plane; Case #3: A line (A = 0) not passing through the origin (D ≠ 0) in the z plane is transformed into a circle through the origin in the w plane; Case #4: A line (A = 0) through the origin (D = 0) in the z plane is transformed into a line through the origin in the w plane.

14. 92. Mapping by w=1/z • Example 1 A vertical line x=c1 (c1≠0) is transformed by w=1/z into the circle –c1(u2+v2)+u=0, or • Example 2 A horizontal line y=c2 (c2≠0) is transformed by w=1/z into the circle

15. 92. Mapping by w=1/z • Illustrations

16. 92. Mapping by w=1/z • Example 3 When w=1/z, the half plane x≥c1 (c1>0) is mapped onto the disk For any line x=c (c ≥c1) is transformed into the circle Furthermore, as c increases through all values greater than c1, the lines x = c move to the right and the image circles shrink in size. Since the lines x = c pass through all points in the half plane x ≥ c1 and the circles pass through all points in the disk.

17. 92. Mapping by w=1/z • Illustrations

18. 92. Homework • pp. 318 Ex. 5, Ex. 8, Ex. 12

19. 93. Linear Fractional Transformations • The Transformation where a, b, c, and d are complex constants, is called a linear fractional (Möbius) transformation. We write the transformation in the following form this form is linear in z and linear w, another name for a linear fractional transformation is bilinear transformation. Note: If ad-bc=0, the bilinear transform becomes a constant function.

20. 93. Linear Fractional Transformations When c=0 When c≠0 which includes three basic mappings It thus follows that, regardless of whether c is zero or not, any linear fractional transformation transforms circles and lines into circles and lines.

21. 93. Linear Fractional Transformations To make T continuous on the extended z plane, we let There is an inverse transformation (one to one mapping) T-1

22. 93. Linear Fractional Transformations • Example 1 Let us find the special case of linear fractional transformation that maps the points z1 = −1, z2 = 0, and z3 = 1 onto the points w1 = −i, w2 = 1, and w3 = i.

23. 93. Linear Fractional Transformations • Example 2 Suppose that the points z1 = 1, z2 = 0, and z3 = −1 are to be mapped onto w1 = i, w2 =∞, and w3 = 1.

24. 94. An Implicit Form • The Equation defines (implicitly) a linear fractional transformation that maps distinct points z1, z2, and z3 in the finitez plane onto distinct points w1, w2, and w3, respectively, in the finite w plane. Verify this Equation Why three rather than four distinct points?

25. 94. An Implicit Form • Example 1 The transformation found in Example 1, Sec. 93, required that z1 = −1, z2 = 0, z3 =1 and w1 = −i, w2 = 1, w3 = i.Using the implicit form to write Then solving for w in terms of z, we have

26. 94. An Implicit Form • For the point at infinity For instance, z1=∞, Then the desired modification of the implicit form becomes The same formal approach applies when any of the other prescribed points is ∞

27. 94. An Implicit Form • Example 2 In Example 2, Sec. 93, the prescribed points were z1 = 1, z2 = 0, z3 = −1 and w1 = i, w2 =∞, w3 = 1. In this case, we use the modification of the implicit form, which tells us that Solving here for w, we have the transformation obtained earlier.

28. 94. Homework • pp. 324 Ex. 1, Ex. 4, Ex. 6

29. 95. Mappings of The Upper Half Plane • Mappings of the Upper Half Plane We try to determine all linear fractional transformations that map the upper plane (Imz>0) onto the open disk |w|<1 and the boundary Imz=0 of the half plane onto the boundary |w|=1 of the disk y v 1 x u

30. 95. Mappings of The Upper Half Plane • Imz=0 are transformed into circle |w|=1 when points z=0, z=∞ we get that Rewrite where α is a real constant, and z0 and z1 are nonzero complex constants.

31. 95. Mappings of The Upper Half Plane when points z=1, we get that If z1=z0, then is a constant function Therefore, Finally, we obtain the mapping

32. 95. Mappings of The Upper Half Plane • Mappings of The Upper Half Plane w

33. 95. Mappings of The Upper Half Plane • Example 1 The transform in Examples 1 in Sections. 93 and 94 can be written

34. 95. Mappings of The Upper Half Plane • Example 2 By writing z = x + iy and w = u + iv, we can readily show that the transformation maps the half plane y > 0 onto the half plane v > 0 and the x axis onto the u axis. Firstly, when the number z is real, so is the number w. Since the image of the real axis y=0 is either a circle or a line, it must be the real axis v=0.

35. 95. Mappings of The Upper Half Plane • Example 2 (Cont’) Furthermore, for any point w in the finite w plane, which means that y and v have the same sign, and points above the x axis correspond to points above the u axis. Finally, since point on x axis correspond to points on the u axis and since a linear fractional transformation is a one to one mapping of the extended plane onto the extended plane, the stated mapping property of the given transformation is established.

36. 95. Mappings of The Upper Half Plane • Example 3 The transformation where the principal branch of the logarithmic function is used, is a composition of the function According to Example 2, Z=(z-1)/(z+1) maps the upper half plane y>0 onto the upper half plane Y>0, where z=x+iy, Z=X+iY;

37. 95. Mappings of The Upper Half Plane • Example 3 (Cont’)

38. 95. Homework • pp. 329 Ex. 1, Ex. 2