Mathematics Iii Rr220202

Set No. 1

Code No: RR220202

II B.Tech II Semester Supplimentary Examinations, Aug/Sep 2007 MATHEMATICS-III ( Common to Electrical & Electronic Engineering, Electronics & Communication Engineering, Electronics & Instrumentation Engineering, Electronics & Control Engineering, Electronics & Telematics, Metallurgy & Material Technology, Aeronautical Engineering and Instrumentation & Control Engineering) Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ⋆⋆⋆⋆⋆ 1. (a) Prove that Γ(2n) =

22n−1 √ Γ(n)Γ(n π

(b) Prove that β(m, n) =

R1 0

2. (a) Prove that

R1

+ 12 ).

xp−1 +xq−1 dx (1+x)m+n

(x2 − 1) Pn+1 Pn1 dx =

[8+8] 2n(n+1) (2n+1)(2n+3))

−1

(b) Prove that J3/2 (x) =

q

2 πx

 sin x x

− cos x



[8+8]

3. (a) Find k such that f(x,y) = x3 + 3kxy 2 may be harmonic and find its conjugate [8+8] (b) If tan (π/6 + i α) = x+ iy prove that x2 + y 2 + √2x3 = 1 R z dz 4. (a) Evaluate Cos z−sin with c: | z | = 2 using Cauchy’s integral formula (z+i)3 c

(b) Evaluate

2+i R

2x + 1 + iy)dz along (1-i) to (2+i) using Cauchy’s integral for-

1−i

mula

[8+8]

5. (a) State and prove Taylor’s theorem. (b) Obtain Taylor series to represent the function

z 2 −1 (z+2) (z+3)

in the region |z| < 2 [8+8]

6. (a) Find the poles corresponding residues. R (b) Evaluate (zz−dz 2 +1) where C is |z+1| = 1 by residue theorem.

[8+8]

C

7. (a) Evaluate

R∞

cos xdx (x2 +16) (x2 +9)

using residue theorem.

−∞

(b) If a>e, use Rouche’s theorem to prove that the equation ez = a zn has n roots inside the circle |z| = 1 [8+8] 8. (a) Determine the region of the w- plane into which the region bounded by x=1,y=1,x+y=1, is mapped by w=z2 . show that the angles are preserved. 1 of 2

Set No. 1

Code No: RR220202

(b) Find the image of the domain x=0,x=/2 under the transformation w=cosz [8+8] ⋆⋆⋆⋆⋆

2 of 2

Set No. 2

Code No: RR220202

II B.Tech II Semester Supplimentary Examinations, Aug/Sep 2007 MATHEMATICS-III ( Common to Electrical & Electronic Engineering, Electronics & Communication Engineering, Electronics & Instrumentation Engineering, Electronics & Control Engineering, Electronics & Telematics, Metallurgy & Material Technology, Aeronautical Engineering and Instrumentation & Control Engineering) Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ⋆⋆⋆⋆⋆ 1. Evaluate the following using β − Γ functions (a)

R1

x7 (1 − x)5 dx

0

(b) (c)

π/2 R

0 ∞ R

sin5 θ cos7/2 θdθ

y −1/2 (1 − e−y )dy

[5+5+6]

0

2. (a) Show that Jn−1 (x) + Jn+1 (x) = q 2 (b) Prove that J−1/2 (x) = πx cos x.

2n J (x). x n

(c) Show that (n+1) Pn+1 (x) - (2n+1) x Pn (x) + n Pn−1 (x) = 0.

[5+5+6]

y 3. (a) If u=x2 -y2 , v=− x2 +y 2 then show that both u and v are harmonic, but u+iv is not analytic. 1 (b) Show that for the function f(z) = |xy| /2 , the C –R equations are satisfied at the origin [8+8] R ez sin 2z−1 dz 4. (a) Evaluate where c is | z | = 1/2 using Cauchy’s integral formula z 2 (z+2)2 c

(b) Evaluate

1+i R

(x − y 2 +ix3 )dz Along the real axis from z=0 to z=1 using Cauchy’s

0

integral formula R e−2z z2 dz (c) Evaluate (z−1) 3 (z+2) where c is | z + 2 | = 1 using Cauchy’s integral formula c

[5+5+6] 5. (a) Show that when | z + 1 | < 1, z −2 = 1 +

∞ P

(n + 1)(z + 1)n

n=1

(b) Expand f (z) =

1 z 2 −z−6

about (i) z = -1 (ii) z = 1.

6. (a) State and prove residue theorem. 1 of 2

[8+8]

Set No. 2

Code No: RR220202

(b) Using residue theorem Evaluate ∫ (4-3z)/ (z2 - z) dz Where C is the circle |z|=2. [8+8] 7. (a) Evaluate (b) Evaluate

R2π

0 Rα 0

dθ , a+b cos θ dx (1+x2 )2

a> b>0 using residue theorem.

using residue theorem.

[8+8]

8. (a) Show that the image of the hyperbola x2 -y2 =1 under the transformation w=1/z is r2 = cos 2θ. (b) Show that the transformation u = the straight line 4u+3=0.

2z+3 changes z−4

⋆⋆⋆⋆⋆

2 of 2

the circle x2 + y2 –4x = 0 into [8+8]

Set No. 3

Code No: RR220202

II B.Tech II Semester Supplimentary Examinations, Aug/Sep 2007 MATHEMATICS-III ( Common to Electrical & Electronic Engineering, Electronics & Communication Engineering, Electronics & Instrumentation Engineering, Electronics & Control Engineering, Electronics & Telematics, Metallurgy & Material Technology, Aeronautical Engineering and Instrumentation & Control Engineering) Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ⋆⋆⋆⋆⋆

1. (a) Evaluate

π/2 R

sin2 θ cos4 θdθ =

0

(b) Prove that (c) Show that

R∞ e−x2

0 ∞ R 0

√

x

dx =

R∞

5π 256

using β − Γfunctions.

4

x2 e−x dx using B-T functions and evaluate

0

xm−1 dx (x+a)m+n

= a−n β(m, n)

[5+6+5]

2. (a) Prove that J02 +2 (J12 + J22 + ....) = 1. (b) Prove that x4 =

8 p (x) 35 4

+ 47 P2 (x) + 15 P0 (x).

[8+8]

3. (a) State sufficient condition for f( z) to be analytic and prove it. (1+i√3 √ 3 i (b) Find all principal values of 2 + √2 4. (a) Evaluate

R c

(b) Evaluate

Cos z−sin z dz (z+i)3

2+i R

[8+8]

with c: | z | = 2 using Cauchy’s integral formula

2x + 1 + iy)dz along (1-i) to (2+i) using Cauchy’s integral for-

1−i

mula

[8+8]

in the region 5. Expand f (z) = (z−2)(z+2) (z+1)(z+4) (i) 1 < | z | < 4 (ii) | z | < 1.

[8+8] 2

6. (a) Find the poles and residues at each pole z3z−1 R z dz (b) Evaluate (zze2 +9) where c is |z | = 5 by residue theorem.

[8+8]

C

7. (a) Evaluate by residue theorem

R2π 0

dθ 2+Cosθ

(b) Use the method of contour integration to evaluate

R∞

−∞

1 of 2

x2 dx (x2 +a2 )3

[8+8]

Set No. 3

Code No: RR220202

8. (a) Find the image of the infinite strip bounded by x=0 and x=π/4 under the transformation w=cosz (b) Show that the transformation w=(5-4z)/(4z-2) transform the circle |z|=1 into a circle of radius unity in w- plane and find the centre of the circle. [8+8] ⋆⋆⋆⋆⋆

2 of 2

Set No. 4

Code No: RR220202

II B.Tech II Semester Supplimentary Examinations, Aug/Sep 2007 MATHEMATICS-III ( Common to Electrical & Electronic Engineering, Electronics & Communication Engineering, Electronics & Instrumentation Engineering, Electronics & Control Engineering, Electronics & Telematics, Metallurgy & Material Technology, Aeronautical Engineering and Instrumentation & Control Engineering) Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ⋆⋆⋆⋆⋆ 1. Evaluate the following using β − Γ functions. (a)

π/2 R

sin9/2 θ cos5 θdθ

0

(b)

R1

e−x x11/3 dx

R1

4 √x dx . 1−x2

3

0

(c)

0

[5+5+6]

2. (a) Prove that 1+ 12 P1 (cos θ) + (b) Prove that

1 2

1 3

P2 (cos θ) + .... = log



1+sin sin

θ 2

θ 2



xJn = (n + 1) Jn+1 − (n + 3) Jn+3 + (n + 5) Jn+5 − x2 Jn+6 [8+8]

3. (a) Determine the analytic function w = u+iv where u = f(0) =1.

2 cos x cosh y cos 2x +cosh 2y

given that

(b) If cosec ( π/4 + i α ) = u + iv prove that (u2 + v 2 ) = 2(u2 − v 2 ). [8+8] R πz 2 cos πz)2 dz where C is the circle |z| = 3 using Cauchy’s integral 4. (a) Evaluate (sin(z−1)(z−2) formula

C

(b) Evaluate

3=1+i R

(x2 + 2xy + i(y 2 − x))dz along y=x2

[8+8]

z=0

5. (a) Expand f(z)= sin z in a Taylor’s series about z = π4 and find the region of convergence (b) Expand

1 (1+z 2 )(z+2)

in

i. 1 < |z| < 2 ii. |z | > 2

[8+8]

6. (a) Determine the poles of the function and the corresponding residues (2z + 1)2 (4z 3 + z) 1 of 2

Set No. 4

Code No: RR220202 (b) Evaluate

R

(sin πz 2 +cos πz 2 )dz (z−1)2 (z−2)

R2π

dθ (5−3 sin θ)2

C

where C is the circle |z| = 3 using residue theorem.

[8+8] 7. (a) Evaluate (b) Evaluate

0 ∞ R 0

x sin mx dx x4 +16

using residue theorem.

using residue theorem.

[8+8]

8. (a) Find the image of the region in the z-plane between the lines y=0 and y=Π/2 under the transformation W = ez (b) Find the image of the line x=4 in z-plane under the transformation w=z2 [8+8] ⋆⋆⋆⋆⋆

2 of 2