This document is a lecture summary for a Complex Variables course. It discusses different types of mappings, including linear transformations of the form w=az+b, quadratic mappings of the form w=z^2, and inversion mappings of the form w=1/z. Examples are provided to illustrate how each type of mapping transforms a region in the z-plane to the w-plane. It also covers power series, including identifying power series, finding the radius of convergence using tests like the ratio test and root test, and examples of applying these tests to geometric series.

1. Complex Variables
Lecture 3
11/01/2015
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2. Announcements
Name: Gayani S. Yapa
Contact Number : 0714433551
E mail Address : gayani21@yahoo.com
TMA #1 Due Date : 14/01/2015TMA #1 Due Date : 14/01/2015
TMA #2 will post to MYOUSL on 15/01/2015
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3. Mappings
Unit 3 – Session 1
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4. Learning Outcomes:
At the end of this session you should be able to:
• Understand what is meant by mapping
according to complex variables context
• Understand the effect of linear transformations
(w=az+b), quadratic transformation (w=z2) &
inversion (w=1/z).
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5. Mapping & Image
Mapping from plane to plane 5

6. Mappings by Elementary functions
1. Mappings of linear function
W=aZ+b – linear transformation
2. Mapping of quadratic functions
W=Z2
3. Mappings of the function
w=1/Z - Inversion
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7. Linear transformation W = aZ + b
To understand the linear transformation W = aZ + b,
let us first consider the simple mapping,
W = Z + (1 + 2 i) (1)
A rectangular region in z plane and its image in W plane under the
mapping defined by equation (1) is shown in Figure 1.
W- PlaneZ- Plane
* *
**
A B
CD
A B
CD
Y
x
3
4
5
6
7
8
1 2 3 4 5 6 77654321
6
5
4
3
2
1
x
Y
Figure 1 7

8. Transformation W = Z + (1 + 2i)
A*, B*, C*, D* are the images of the points A, B, C and D respectively.
A represents the complex number Z = 2 + i in Z plane. Then A* represents,
W = Z + (1 + 2i) = 2 + i + 1+ 2i = 3 + 3i
Z-plane W-plane
Z = (2 + i ) ; A=(2,1) W = (3 +3 i ) ; A*=(3,3)
Z = (3 + i ) ; B=(3,1) W = (4 +3 i ) ; B*=(4,3)
Z = ( ) ; C=(3,3) W = ( ) ; C*=(4,5)
Therefore, the effect of the above mapping is to shift each point in the region
ABCD by 1 unit along the real axis and 2 units along the imaginary axis.
The shape and the size of the image are identical to the region mapped
(ABCD).
It is only shifted.
This kind of mapping is called a translation.
Z = (3 + 3 i ) ; C=(3,3) W = (4 +5 i ) ; C*=(4,5)
Z = (2 + 3 i ) ; D=(2,3) W = (3 +5 i ) ; D*=(3,5)
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13. Mapping of quadratic functions
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17. *A*D
*B*C
θ=2π/3
X
Y
θ=π/6
θ=π/3
1 4 9 16
4321
θ=π/3
θ=π/6
Y
C
B
A
D
2MappingW=Z
(b) W- Plane
Z- Plane
X1 4 9 16
4321 X
(a) Z- Plane
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18. Mapping W=1/Z
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29. Power Series
Unit 3 – Session 2
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30. Learning Outcomes:
At the end of this session you should be able to:
• Identify a power series.
• Find the radius of convergence of some power
series.
• Apply ratio test & root test to check the
convergence behavior of a power series.
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31. Power Series
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32. Power Series (Contd.)
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33. Radius of Convergence
Z 0
R
C
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34. Convergence
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36. Example: Ratio Test
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38. Example: Root Test
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42. Geometric Series
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43. Example:
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44. Solution:
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45. Example:
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46. Solution:
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47. Thank You
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Thank You