WR1_EMT451T_2012B_Memo_A1

EMT451T/MMA401T Major Test 1: 2012/09/19 of 5

QUESTION 1 [13]

1.1 Matrix

1 2 2

2 4 1

2 1 4

B

has the following eigenvectors:

1 2 30 1 1 ; 1 0 2 ; 3 1 1 .T T T

X X X

1.1.1 Write down the modal matrix M associated with B. Use the eigenvalues in the order given. (1)

0 1 3

1 0 1

1 2 1

M

1.1.2 Write down 1M , the inverse of M. (2)

51 13 6 6

1 1 12 2

1 1 13 6 6

2 5 1 0.333 0.833 0.1671

0 0 3 3 0 0.5 0.56

2 1 1 0.333 0.167 0.167

M

1.1.3 According to James (2011:40), 1M AM . Use this relationship to determine the

eigenvalues of B. (4)

1

2 5 1 1 2 2 0 1 31

0 3 3 2 4 1 1 0 16

2 1 1 2 1 4 1 2 1

M AM

30 0 01

0 18 06

0 0 6

5 0 0

0 3 0

0 0 1

1 2 35; 3; 1.


1.2 A specific SISO system is characterized by the fourth-order differential equation

4 32

4 33 4 5 td y d y dy

y edt dt dt

.

Assume all the initial conditions are known. Then the dynamic equations for this system may be

written as

andx Ax bu

.Ty c x

1.2.2 Determine the system matrix A. Clearly show your calculations. (5)

Let 1x y

2 1

dyx x

dt

2

3 22

d yx x

dt

3

4 33

d yx x

dt

2

4 4 2 1

2

4 1 2 3 4

3 4 5

4 0 3 5

t

t

x x x x e

x x x x x e

0 1 0 0

0 0 1 0

0 0 0 1

1 4 0 3

A

1.2.3 Write down the matrix b . (1)

0 0 0 5T

b

QUESTION 2 [11]

2.1 Verify that the function ( ) j zf z e , where z is a complex variable, is analytic by making use of the

Cauchy-Riemann equations. Hint sinjxe cox j x (6)

( )( ) j x j yf z e

j x ye

cos sinye x j x

cosyu e x and sinyv e x

siny

xu e x cosy

xv e x

cosy

yu e x siny

yv e x

x yu v and

y xv u

Thus, f satisfies the Cauchy-Riemann equations therefore f is analytic


2.2 If a function f of a complex variable z is analytic and ( ) ( , ) ( , )f z u x y jv x y , then

'( ) x x y yf z u jv v ju .

Use this fact and your results in 2.1 to prove that ' j zf z je .

' sin cosy yf z e x je x

sin cosye x j x

cos sinye j x j x

cos sinyj e x j x

j zje in 2.1 cos sinjz ye e x j x

QUESTION 3 [10]

3.1 Consider the function

( )1 2

zf z

z z

3.1.1 Write ( )f z as partial fractions. (2)

1 23 3( )

1 2 1 2

zf z

z z z z

3.1.2 Determine the Laurent series expansion of

( )1 2

zf z

z z

valid for 0 1 3.z

Hint: make use of the partial fractions in 3.1.1. (5)

1 2

1 2( ) ( ) ( )

1 2 3 1 3 2

zf z f z f z

z z z z

About 1

1 3 13

zz

2

2 2 1 2 1

13 2 3 1 3 33 1

3

f zzz z

2 3

2 3

1 12 11 ...

9 3 3 3

z zz

2 32 1 2 1 12

...9 27 81 243

z z z

1 2( )

3 1 3 2f z

z z

2 3

2 1 2 1 11 2...

3 1 9 27 81 243

z z z

z


3.1.3 Name the type of singularity in ( )f z at 1z and give a reason for your answer. (2)

1z is a simple pole.

The Laurent series has one term in the principle part of the series expansion.

3.1.4 Use your answer in 3.1.2 to write down the residue of the expansion. (1)

1

Res ( ), 13

f z

QUESTION 4 [11] Use Cauchy’s integral theorem to determine the following contour integrals:

4.1

3

3

C

zdz

z z j

along C: 2 1z (3)

Simple pole at 0z , third order pole at z j

No poles enclosed in C

According to Cauchy’s integral formula

3

30

C

zdz

z z j

4.2

3

3

C

zdz

z z j

along C: 2z (8)

0z and z j are enclosed in C

1 2

3 3

3

C

f z f zzI dz dz dz

zz z j z j

1 3

3zf z

z j

2

3zf z

z

1 3

3 30f

jj

2 2 2

3 3'

z zf z

z z

2 3

6''f z

z

2 3

6 6''f j

j j

3

3

33

zz

z j zI dz dzz z j

3 1 6

2 22!

j jj j

6 6 0


QUESTION 5 [7]

Consider the function

2

22( )

1 2

zf z

z z

5.1 Identify the zeros and singularities of ( )f z . Classify the singularities. (3)

2 2

2 22 1 2 ( )( ) 2

z z

z z z j z j z

Zero at 0z Simple poles at z j Pole of order two at 2z

5.2 Use Cauchy’s residue theorem to determine

2

22 1 2C

zdz

z z along : 3 1C z . (4)

Pole of order two at 2 lies on C,z z j not enclosed in C

2

2

( 2)Res[ ( ), 2] lim

z

zdf z

dz

2

21 ( 2)

z

z 2( 1)z

2 2

22 2

2 1 2lim

1z

z z z z

z

22 2

2lim

1z

z

z

4

25

2

22

4 82

25 251 2C

z jdz j

z z

WR1_EMT451T_2012B_Memo_A1

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