Mathematics Iii May2003 Nr 220202

Set No. Code No. 220202 II-B.Tech. II-Semester Examinations April/May, 2003

1

MATHEMATICS-III (Common to Electrical and Electronics Engineering, Mechanical Engineering, Electronics and Communication Engineering, Electronics and Instrumentation Engineering, Electronics and Control Engineering, Metallurgical and Material Technology, Mechatronics, Electronics and Telematics, Aeronautical Engineering.) Time: 3 hours

Max. Marks: 80 Answer any Five questions All questions carry equal marks --1

∫ ( log 1 / y )

1. a) Show that t Γn =

n−1

dy

0

1

b) Prove that

∫ 0

dx 1− x

n

=

π Γ (1 / n) n Γ (1 / n + 1 / 2)

2. a) Write down the power series expansion for J n ( x ) and hence show that

2 sin x. πx d n x J n ( x ) = x n J n −1 ( x ) b) Show that dx J 1/ 2 ( x ) =

[

3. a) b) 4. a) b)

]

Define analyticity of a complex function at a point P and in a domain D. Prove that the real and imaginary parts of an analytic function satisfy Cauchy – Riemann Equations. Show that w = zn (n , a positive integer) is analytic and find it’s derivative. Find an analytic function f(z) such that Real [f ' (z)] = 3x2 – 4y – 3y2 and f(1+i) = 0. Determine p such that the function  px  1   2 2 –1  y  2 f(z) = log (x +y ) + i tan be an analytic function e

Contd…2

Code No. 220202

.2..

Set No.1

5. a) Show that

∫

x dx – y dy √ ( x2 + y2 )

C

Is independent of any path of integration which does not pass through the origin. b) Evaluate

∫

dz

.

( z- 2i )2 ( z + 2i )2

c

c being the circumference of the ellipse x2 + 4( y - 2 )2 = 4

6. a) b) 7. a)

b)

n When n is an integer, show that J − n ( x ) = ( −1) J n ( x )

Prove that

d  J n ( x)  −n = − x J n +1 ( x ) n   dx  x 

Find the residue of Z 2 − 2Z f(z) = at each pole. ( Z + 1) 2 ( Z 2 + 1) 4 − 3z 3 Evaluate ∫ z ( z − 1)( z − 2) dz Where c is the circle | Z | = 2 c

8. a)

Find the image of the rectangle R: -π < x <π; 1/2 < y< 1, under the transformation w = sin z. b) Find the bilinear transformation that maps the points ,(∞ i, 0), into the points (0, i, ∞) . ###

Set No. Code No. 220202 II-B.Tech. II-Semester Examinations April/May, 2003

2

MATHEMATICS-III (Common to Electrical and Electronics Engineering, Mechanical Engineering, Electronics and Communication Engineering, Electronics and Instrumentation Engineering, Electronics and Control Engineering, Metallurgical and Material Technology, Mechatronics, Electronics and Telematics, Aeronautical Engineering.) Time: 3 hours Max. Marks: 80 Answer any Five questions All questions carry equal marks --π /2

Show that β(m,n) = 2

1.

∫ sin

θ cos2 n−1 θ dθ and

2 m −1

o

deduce that

π /2

π

o

o

n n ∫ sin θ dθ = ∫ cos θ dθ = 1 / 2

Γ ((n + 1) / 2) π 1/2 Γ ((n + 2) / 2)

n J n ( x ) − J n '( x ) = J n +1 ( x ) x 2 b) Show that ∫ J 3 ( x )dx = − J 2 ( x ) − J 1 ( x ) x

2. a) Prove that

3. a) b) c)

4. a) b)

For w = exp(z2), find u and v and prove that the curves u(x,y) = c1 and v(x,y) = c2 where c1 and c2 are constants cut orthogonally. Prove that the function f(z) = z is not analytic at any point. Find the general and the principal values of (i) log e (1+ 3 i) (ii) log e(-1). If the potential function is loge(x2+y2) find the complex potential function. Show that f (z) = z + 2 z is not analytic anywhere in the complex plane.

5. a) State and prove Cauchy’s integral theorem. b) State and prove Cauchy’s integral formula

Contd…2

Code No. 220202 6. a)

.2..

Set No. 2

Write down the power series expansion for J n ( x ) and hence show that

2 sin x. πx d n Show that x J n ( x ) = x n J n −1 ( x ) dx J 1/ 2 ( x ) =

b)

7. a) b)

[

Find the residue of Evaluate

Z −3

]

Ze z ( Z − 1) 3

∫ Z 2 + 2Z + 5

at its pole.

dz where C is the circle.

c

(i) | Z | = 1

8. a)

(ii) | Z+1-i | = 2 (iii) | Z+1+i | = 2

Show that the transformation w = the straight line 4y+3=0.

b)

2z+ 3 changes the circle x2 + y2 –4x = 0 into z−4

5 − 4z transforms the circle |z| = 1 into a circle of 4z − 2 radius unity in the w-plane. Show that the relation w=

###

Set No. Code No. 220202 II-B.Tech. II-Semester Examinations April/May, 2003

3

MATHEMATICS-III (Common to Electrical and Electronics Engineering, Mechanical Engineering, Electronics and Communication Engineering, Electronics and Instrumentation Engineering, Electronics and Control Engineering, Metallurgical and Material Technology, Mechatronics, Electronics and Telematics, Aeronautical Engineering.) Time: 3 hours

Max. Marks: 80 Answer any Five questions All questions carry equal marks --∞

1. a)

Show that

∫

3

x e − x dx =

0 π /2

b)

Prove that

∫ sin

2

π 3

θ cos4 θ dθ =

0

5π 256

2. a) Show that sin (x sin θ) = 2(J1 sin θ + J3 sin 3θ +.....) 0 ,−1 < x ≤ 0  b) If f(x) =  ,0 < x < 1.  x 3.

Show that the curves rn =α sec nθ and rn = β cosec nθ cut orthogonally.

4. a)

Prove that the function v =sinx coshy+2cosx sinhy + x2–y2+4xy satisfies Laplace equation. Determine the corresponding analytic function u+iv. If w = u+iv = z3 prove that the curves u = c1 and v = c2 where c1 and c2 are constants, cut each other orthogonally.

b)

5. a) Evaluate

∫ (x – 2y ) dx + (y

2

c

– x2) dy where C is the boundary of the first quadrant of the

circle x2 + y2 = 4 b) Evaluate Using Cauchy’s theorem

∫

c

z 3 e-z ( z – 1)

dz

where c is : | z - 1 | = ½

3

Contd…2

Code No. 220202 6. a) b)

.2.

1

Prove that

1 − 2tx + t

Show that

Cos 2θ

∫ 1 − 2aCosθ + a 2 0 2π

b) 8. a) b)

= P0(x) + P1(x) t + P2(x) t2 + ....

Prove that Pn’(x) = x Pn-1’ (x) + nPn-1 (x). π

7. a)

2

Set No. 3

Show that

=

πa 2 1− a

2

, (a2 < 1)

2π dθ = 2 + Cosθ 3 0

∫

Show that the function w=4/z transforms the straight line x=c in the z-plane into a circle in the w-plane. z −i Show that the transformation w = maps the real axis in the z-plane into the z+ I unit circle |w| =1 in the w-plane. ###

Set No. Code No. 220202 II-B.Tech. II-Semester Examinations April/May, 2003

4

MATHEMATICS-III (Common to Electrical and Electronics Engineering, Mechanical Engineering, Electronics and Communication Engineering, Electronics and Instrumentation Engineering, Electronics and Control Engineering, Metallurgical and Material Technology, Mechatronics, Electronics and Telematics, Aeronautical Engineering.) Time: 3 hours

Max. Marks: 80 Answer any Five questions All questions carry equal marks --∞

1. a)

Prove that

∫e

− y 1/ m

dx = mΓm

o b

b)

Show that

∫ ( x − a)

m

(b − x ) n dx = (b − a ) m+ n +1 β (m + 1, n + 1)

a

1

= P0(x) + P1(x) t + P2(x) t2 + .... 1 − 2tx + t 2 b) Prove that Pn’(x) = x Pn-1’ (x) + nPn-1 (x).

2. a) Prove that

3.

Determine whether the function 2xy + i(x2- y2) is analytic.

4. a) b)

Find all values of k, such that f(z)=ex(cos ky+ i sin ky) is analytic. Show that xy 2 ( x + iy ) f(z) = , z ≠0 x2 + y4 0 , z=0 is not analytic at z =0 although C –R equations are satisfied at the origin

Contd…2

Code No. 220202

.2.

Set No. 4

5. a) Evaluate log z dz

∫ ( z −1)

3

where c : | z - 1| = ( ½ ) , using Cauchy’s Integral Formula Obtain the Taylor Expansion of e( 1 + z ) in the powers of ( z –1 ). c

b)

6. a) b)

n J n ( x ) − J n '( x ) = J n +1 ( x ) x 2 Show that ∫ J 3 ( x ) dx = − J 2 ( x ) − J 1 ( x ) x Prove that

2π

7. a) b)

2π  Sin 2 dθ 2 2  ∫ a + bCosθ = b 2 a − (a − b )  , a > b>0. 0 Use method of contour integration to prove that Prove that 2π

dθ

∫ 1 + a 2 − 2aCosθ

0

8. a) b)

=

2π 1− a2

, 0 < a< 1

Determine the bilinear transformation that maps the points 1-2i, 2+i, 2+3i into the points 2+i, 1+3i, 4. Prove that the transformation w=sin z maps the families of lines x=constant and y=constant in to two families of confocal central conics. ###