engineering mathamatics

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