Electronics

DEPARTMENT OF ELECTRONIC ENGINEERING

MAIN EXAMINATION 2017

TITLE OF PAPER

COURSE NUMBER

TIME ALLOWED

INSTRUCTIONS:

MATHEMATICAL METHODS II ( PAPER ONE)

E470(1)

THREE HOURS

ANSWER ANY FOUR OUT OF FIVE QUESTIONS.

EACH QUESTION CARRIES 25 MARKS.

MARKS FOR DIFFERENT SECTIONS ARE SHOWN IN THE RIGHT-HAND MARGIN.

THIS PAPER HAS SEVEN PAGES, INCLUDING THIS PAGE.

DO NOT OPEN THE PAPER UNTIL PERMISSION HAS BEEN GIVEN BY THE INVIGILATOR.

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E470(1) MATHEMATICAL METHODS II (PAPER ONE)

Question one

Given the following complex function f (z) : 6 { 21 – ) z1 where z = x+ iy (a)

(b)

(c)

(i) find its u(x,y) and v(x,y) ,

(ii) check for its analyticity ,

( 3 marks)

( 5 marks)

(i) plot the mapped image o f a rectangular region in z – plane defined by

1 < x < 5 and – 3 < y < 0 onto f – plane ,

(ii) plot the mapped image of a ring region in z – plane defined by

1 s r s 2 and O s 0 s 2 ,r onto f – plane ,

( 3 marks)

( 3 marks)

Evaluate the value o f f2 f (z) d z ‘ i f z, = 1 – 2i and Z2 = – 3 + 4 i ‘ and I ,I

(i) i f l is the straight line from z1 to z2 , ( 7 marks)

(ii) use int command directly irrespective o f the integration path l ,then

compare this result with that obtained in (i) and make a brief remark.

( 4 marks)

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Question two

(a) Determine the value of a such that u(x ,y) = 7 x 2 + a y 2 – 2 x 1s a

hannonic and then find its conjugate hannonic v( x, y)

(b) Given the following complex function /(z) as:

6z+ 12 f ( z) = z 2 – 2 z + I 0

( 6 marks)

(i) find the two roots of the denominator of f ( z ) , i.e., z 1 and z2 • Replace the

denominator of / ( z ) by (z – z1) (z – z2 ) and then convert / ( z ) into its

partial fraction, ( 3 marks)

(Hint : can use roots(z 2 – 2 z + 10, /) to find its complex roots)

(ii) find its convergent series representation of f ( z ) about the expansion centre

z = 9 – 3 i for all the values of z in the region of 8 < lz – 9 + 3 ii < 10 ( 8 marks)

(iii) evaluate the value of {f (z) d z if I

clockwise sense ,

I z – 3 i I = 5 and in counter (3 marks)

( c) Find the centre and the radius of convergence of the following power series :

00 (-1 r ( 4 n) ! ( ·)n L ( ) ( )’

z+6+z n=O 5n 2 n ! n!

( 5 marks)

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Question two

(a) Determine the value of a such that u(x ,y) = 7 x 2 + a y 2 – 2 x is a

harmonic and then find its conjugate harmonic v(x,y) ( 6 marks)

(b) Given the following complex function f(z) as:

6z+ 12 f ( z) = z2 – 2 z + 10

(i) find the two roots of the denominator of f(z) , i.e., z 1 and z2 . Replace the

denominator of f ( z) by ( z – z1) ( z – z2 ) and then cop.vert f ( z ) into its

partial fraction, ( 3 marks)

(Hint : can use roots(z 2 – 2 z + 10, / ) to find its complex roots)

(ii) find its convergent series representation of f ( z) about the expansion centre

z = 9 – 3 i for all the values of z in the region of 8 < lz – 9 + 3 ii < 10 ( 8 marks)

(iii) evaluate the value of ff (z) dz if l

clockwise sense ,

I z – 3 i I 5 and in counter ( 3 marks)

(c) Find the centre and the radius of convergence of the following power series:

a; {-1 r ( 4 n) ! ( ·)n I ( ) ( )’

z + 6 + , n=O 5 n 2 n ! n!

( 5 marks)

3

..

Question three

(a) Given the following definite integral:

r2a sin(2 0) .li 5 – 3 cos( 0) + 2 sin( 0) d

0

(i) use int command to find its value , ( 3 marks)

(ii) convert it to a complex contour integral, evaluate the value of this contour

integral . Compare it with that obtained in (i) .

(b) Find the Cauchy principal value of the following integral :

r x-6 ———-dxa:, x4 + 3 x 3 + 4 x2 – 3 x – 5

4

( 9 marks)

( 10 marks)

…

Question four

(a) Given the following improper integral:

f .oo X – 4

– 0 0 x4 – x 3 + 6 x 2 – 7 x + 15 d x

(i) use int command to find its value , (3 marks)

(ii) convert it to a complex contour integral, evaluate the value o f this contour

integral . Compare it with that obtained in (i) .

(b) Given the following improper integral :

Joo cos(k x) ——‘– —-dx – 00 x6 + 9 x4 + 23 x2 + 15 and k > 0

( 7 marks)

(i) convert it to a complex contour integral , find the result o f this contour integral in

(ii)

terms o f k ,

evaluate the values o f the given integrals when k = 1. 4 .

( 10 marks)

( 2 marks)

(iii) use int command to find its value o f the given improper integral when

k = 1.4 and compare it with that obtained in (ii) .

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( 3 marks)

Question five

A pair of long, non-coaxial, circular cross-section conductors is statically charged such

that the inner conductor ( radius of and centred at ( x = ! ,y = 0) ) is at zero potential , i.e., <I> = 0 volt , while the outer conductor ( radius of I and centred at origin ) is

maintained at <I> = 40 volts as shown in the diagram below :

-0.5 0

z – b Use the linear fractional transformation of the form w- – – – to transform the above · – bz-1 given non-coaxial circles in z – plane ( z = x + i y) to two coaxial circles in w- plane ( w = u + iv) ,

z – b (a) show that w = b maps the unit circle in z plane onto the unit circle in z – I

w- plane for any real value of b ( 5 marks)

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(b)

Question five ( continued)

1 find the appropriate value of b such that the inner circle of radius maps

2

to a coaxial circle of radius r0 ( < 1) . Find also the value of r0 ( 10 marks) ( c) since the general solution for coaxial conductors can be written as

<I> = k1 ln { lwl) + k2 , determine the values of k1 and k2 from the given boundary conditions ( 5 marks)

( d) plot the equal potential surfaces <I> = 0 , <I> = 10 , <I> = 20 , <I> = 30 and

<I> = 40 in z – plane and show them in a single display .

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( 5 marks)

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