07a4bs02 mathematics -iii

8/9/2019 07a4bs02 Mathematics -III

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R07 SET-1Code.No: 07A4BS02

JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD

II B.TECH II SEM – REGULAR/SUPPLEMENTARY EXAMINATIONS MAY - 2010

MATHEMATICS –III

(METALLURGY & MATERIAL TECHNOLOGY)

Time: 3hours Max.Marks:80

Answer any FIVE questions All questions carry equal marks

- - -

1.a) Show that( ) ( )

( , )( )

m nm n

m n β

Γ Γ=

Γ +

b) Show that1

0

( , )(1 )

m

m n

xm n dx

x β

∞ −

+=

+∫ [8+8]

2.a) Find whether 2( ) 2 x iy

f z x y

−

= + is analytic or not.

b) Find the analytic function f(z) = u(r, θ ) + iv(r, θ ) such that u(r, θ ) = -r 3 sin 3 θ .

[8+8]

3.a) Find the real part of the principal value of ilog (1 + i)

b) Separate into real and imaginary parts of sech ( x + i y ) . [8+8]

4.a) Evaluate ∫ ng the path y = x and y = x1

2

0

( )

i

x iy d

+

+ z alo 2.

b) Evaluate, using Cauchy’s integral formula2

( 1)( 2)

x

C

e dz z z− −∫

, where c is the circle

z = 3. [8+8]

5.a) Expand f(z) = sinz in Taylor’s series about z =4

π

b) Determine the poles of the function f(z) =2

2( 1) ( 2)

z

z z− +. [8+8]

6.a) Find the residue of f(z) =3

4( 1) ( 2)( 3) z

z z z− − − at z = 1.

b) Evaluate2

3

2 5C

z

z z

−

+ +∫ where c is a circle given by

i) z = 1

ii) 1 z i+ − = 2

iii) 1 z i+ + = 2 [8+8]

7.a) State and prove Fundamental theorem of Algebra.

b) State and prove Liouville's theorem. [8+8]


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8.a) Find the invariant (fixed) points of the transformation1

1

zw

z

−=

+.

b) Determine the bilinear transformations whose fixed points are 1, -1. [8+8]

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MATHEMATICS –III




- - -

1.a) Evaluate

i)

1

4 2

0

(1 ) x x dx−∫

ii)4 2 2

0

( )a

x a x d − x∫

iii)

2

3 1/3

0

(8 ) x x dx−∫

b) Show that 2 ( 1/ 2) 1.3.5........(2 1).n

n n π Γ + = − , where n is a positive integer.[12+4]

2.a) Show that both the real and imaginary parts of an analytic function are harmonic.

b) Find the analytic function f(z) = u + iv if u + v =sin2

(cosh 2 cos 2 )

x

y x−. [6+10]

3.a) Find all zeros of

i) sinh z

ii) cosh z

b) Prove that sinh z , cosh z are periodic functions of imaginary

period 2 π i. [8+8]

4.a) Use Cauchy’s integral formula to evaluate2( 2)( 1)

z

C

edz

z z+ +∫ where c is the circle

z = 3.

b) Use Cauchy’s integral formula to evaluate2

3( 1)

z

C

edz

z −∫ where c is the circle z = 3/2.

[8+8]

5.a) Expand f(z) =2

1

6 z z− − about

i) z = -1.

ii) z = 1.

b) Find the poles of the functions.

i)2

1

( 2

z

z z

+

− )


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ii)2

2( 1) ( 2)

z

z z− + [8+8]

6.a) Find the residue of2

4 1

z

z + at these singular points which lie inside the circle z = 2.

b) Find the residue of2

21

z

z− at these singular points which lie inside the circle z = 1.5.

[8+8]

7. Show that all the roots of z5+3z2 = 1 lie inside the circle 3 4 z < and that two of its

roots lie inside the circle 3/ 4 z < . [16]

8.a) Determine and graph the image of z a a− = under the transformation w = z2.

b) Find the Bilinear map of the points z = -1, 0, 1 on to w= 0, i, 3i. Find the fixed points

of the transformation. [8+8]

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MATHEMATICS –III




- - -

1.a) Evaluate

i)6 2

0

x x e dx

∞−

∫

ii)4 3/ 2

0

xe x dx

∞−

∫

iii)24

0

3 x dx∞

−

∫ b)

Evaluate

i)

/ 2

6 7

0

sin cos d

π

θ θ θ ∫

ii)

/ 2

10

0

sin d π

θ θ ∫ [10+6]

2.a) Find where the function

i) w =1

z

ii) w =1

z

z − ceases (fails) to be analytic.

b) In a two dimensional fluid flow, the stream function ψ = tan-1(y/x), then, find

velocity potential function φ . [8+8]

3.a) Find the principle values of(1 3)

(1 3) i

i +

+

b) If u = log tan4 2

π θ ⎛ +⎜

⎝ ⎠

⎞⎟ then prove that tanh tan

2 2

u θ ⎛ ⎞ ⎛ ⎞=⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠. [8+8]

4. Use Cauchy’s integral formula to evaluate3(1 )

z

C

edz

z z−∫ where c is

a)1

2 z =

b)1

1

2

z − =

c) 2 z = [16]


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5.a) Expand in Taylor’s series f(z) =4 9

z

z + about the point z = 0.

b) Find the poles of the functions 2 2

iz ze

z a+

c) Find the poles of the functions2 3 2

z

z z− +. [16]

6. Evaluate2 2

0( 9)( 4)

dx

x x

∞

+ +∫ 2 using Residue theorem. [16]

7. Prove, by using Rouche's theorem, that the equation ez = azn has n roots inside the unit

circle. [16]

8. Show that the transformation1

w z z= + , converts that the radial lines θ = constant in

the z-plane in to a family of confocal hyperbolar in the w-plane. [16]


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MATHEMATICS –III




- - -

1.a) Show that when n is a positive integer .( ) ( 1) ( )nn n J x J x− = −

b) Show that 1/ 22

( ) cos J x xπ

− = x . [8+8]

2.a) Determine whether the function f(z) = 2xy + i (x2 - y2), is analytic or not.

b) Find the analytic function whose real part u = sin2

(cosh 2 cos 2 )

x

y x−. [8+8]

3.a) Find the principal value of : (1-i)1+i.

b) Find the real and imaginary parts of sec z. [8+8]

4.a) Evaluate ∫ along y = x(1,1)

2 2

(0,0)

(3 4 ) x xy ix d + + z 2.

b) Evaluate3

3

sin3

2

C

z zdz

z

π

−

⎛ ⎞

−⎜ ⎟⎝ ⎠

∫ with c: z = 2 Cauchy’s integral formula. [8+8]

5. Determine and classify all singularities of the given functions.

a)3

1

z z−

b)4

41

z

z+

c)1 1

co t z z

−

d)2

41

x

e z−

e)1 c

os z

z

−

f)/ 2 z z

e [16]−

6. Evaluate

2

0

cos2

5 4cosd

π θ

θ θ +∫ using calculus of residues. [16]

7. Prove that all the roots of


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a) 16z5 – z + 8 = 0 lie between the circles1

2 z = and z = 1.

b) z6 – 9z2 + 11 = 0 lie between the circles z = 1 and z = 3. [16]

8. Find the bilinear transformation which maps the points z = 1, i, -1 on to the pointsw = i, 0, -i. [16]

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07a4bs02 mathematics -iii

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