Complex Variables

Lecture 3

11/01/2015

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Announcements

Name: Gayani S. Yapa

Contact Number : 0714433551

E mail Address : gayani21@yahoo.com

TMA #1 Due Date : 14/01/2015 TMA #1 Due Date : 14/01/2015

TMA #2 will post to MYOUSL on 15/01/2015

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Mappings

Unit 3 – Session 1

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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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Mapping & Image

Mapping from plane to plane 5

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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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

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5

4

3

2

1

x

Y

Figure 1 7

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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Mapping of quadratic functions

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*A*D

*B*C

θ=2π/3

X

Y

θ=π/6

θ=π/3

1 4 9 164321

θ=π/3

θ=π/6

Y

C

B

A

D

2Mapping W = Z

(b) W- Plane

Z- Plane

X1 4 9 164321 X

(a) Z- Plane

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Mapping W=1/Z

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Power Series

Unit 3 – Session 2

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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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Power Series

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Power Series (Contd.)

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Radius of Convergence

Z 0

R

C

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Convergence

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Example: Ratio Test

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Example: Root Test

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Geometric Series

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Example:

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Solution:

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Example:

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Solution:

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Thank You

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Thank You