1. Chapter 8. Mapping by Elementary Functions
Weiqi Luo (骆伟祺)
School of Software
Sun Yat-Sen University
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. The Mapping 92. Mapping by w=1/z 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. Consider the Equation 92. Mapping by w=1/z
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. The Mapping by w=1/z 92. 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. Four Cases 92. Mapping by w=1/z
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. Example 3 92. Mapping by w=1/z
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. The Equation 94. An Implicit Form
defines (implicitly) a linear fractional transformation that maps distinct points z1, z2, and z3 in the finite z 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. Example 1 94. An Implicit Form
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. For the point at infinity
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. Example 2 94. An Implicit Form
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
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 x y u v 1

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 Planew

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