Rr220202-mathematics-iii

Set No. 1

Code No: RR220202

II B.Tech Supplimentary Examinations, Aug/Sep 2008 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 ⋆⋆⋆⋆⋆ R1

1. (a) Show that

xm (log x)n dx =

0

(b) Show that β(m,n)=

R∞ 0

(c) Show that

R∞

(−1)n n! (m+1)n+1

where n is a positive interger and m>-1

y n−1 dy (1+y)m+n

2

x4 e−x dx =

0

√ 3 π 8

[6+5+5]

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]

3. (a) Find an analytic function whose imaginary part is e−x (x cos y + y sin y). (b) If z1 = a + ib and z2 = c-id are two complex numbers such that |z1 | = |z2 | = 1 and Re (z1 z2 ) = 0 then for the pair of complex numbers w1 = a + ic and w2 = b + id find Re ( w1 w2 ) and | w1 |. [8+8] R z dz where C: | z − 1 | = 1/2, using Caucy’s integral Formula. 4. (a) Evaluate log (z−1)3 C

(b) State and prove Cauchy’s Theorem. 5. (a) Expand log z by Taylor’s series about z=1 (b) Expand

(z 2

1 +1)(z 2 +2)

in positive and negative powers of z if 1 < |z| <

[8+8] √

2 [8+8]

1 6. (a) Find the poles and the residue at each pole of the function f(z) = (z2 +4) 2. R (3z−4) (b) Evaluate z(z−1) dz where c is the circle |z| =2 using 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) Find the image of the infinite strip 0
Set No. 1

Code No: RR220202

(b) Find the bilinear transformation which maps the points (–1, 0, 1) into the points (0, i, 3i). [8+8] ⋆⋆⋆⋆⋆

2 of 2

Set No. 2

Code No: RR220202

II B.Tech Supplimentary Examinations, Aug/Sep 2008 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) Show that

R1

xm (log x)n dx =

0

(b) Show that β(m,n)=

R∞ 0

(c) Show that

R∞

(−1)n n! (m+1)n+1

y n−1 dy (1+y)m+n

2

x4 e−x dx =

0

√ 3 π 8

[6+5+5]

2. (a) Using Rodrigue’s formula prove that (b) Prove that J0 (x) =

1 π

where n is a positive interger and m>-1

Rπ

R1

xm Pn (x)dx = 0 if m < n

−1

cos(x cos θ)dθ.

[8+8]

0

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

1−i

formula. 5. Expand

[8+8]

1 z(z 2 −3z+2)

for the regions

(a) 0 < |z| < 1 (b) 1 < |z| < 2 (c) |z| > 2

[16] iz

6. (a) Find the poles of the function (ze2 +1) and corresponding residues. R 1 z (b) Evaluate (z−1)(z−2) 2 dz Where c is the circle | z – 2 | = 2 using residue c

theorm. 7. (a) Evaluate by residue theorem

[8+8] R2π 0

dθ 2+cosθ

1 of 2

Set No. 2

Code No: RR220202 (b) Use the method of contour integration to evaluate

R∞

x2 dx (x2 +a2 )3

[8+8]

−∞

8. (a) Define conformal mapping. Let f(z) be an analytic function of z in a domain ′ D of the z-plane and let f (z)6=0 in D. Then show that w=f(z) is a conformal mapping at all points of D. (b) Find the bilinear transformation which maps the points (–i, o, i) into the point (–l, i, l) respectively. [8+8] ⋆⋆⋆⋆⋆

2 of 2

Set No. 3

Code No: RR220202

II B.Tech Supplimentary Examinations, Aug/Sep 2008 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

R1

x4 log

0

(b) Show that (c) Evaluate

R1

0 R∞ 0


1 3 x

dx using β − Γ functions

xm−1 +xn−1 dx (1+x)m+n

= 2β(m, n), m, n > 0

√ x4 a2 − x2 dx using β − Γ functions.

[5+6+5] 2

d y 2 2 2 dy 2. (a) Show that Jn (x) satisfies the differential equation x2 dx 2 +x dx +(x −n )y = 0

(b) Show that Pn (1) = 1 and Pn (x) = (-1)n Pn (x).

[8+8]

3. (a) State necessary condition for f ( z ) to be analytic and derive C-R equations in Cartesian coordinates. (b) If u and v are functions of x and y satisfying Laplace’s equations show that ∂v ∂v s+ it is analytic where s = ∂u − ∂x and t = ∂u + ∂y [8+8] ∂y ∂x 4. (a) Evaluate using Cauchy’s integral formula

R c

(b) Evaluate using Cauchy’s integral formula

R

(z+1)dz z 2 +2z+4

where c :| z + 1 + i | = 2

−

z f rom z = 0 to 4 + 2i along the

C

curve C given by i) z=t2 +it ii) Along the line z=0 to z=2: and there from z=2 to 4+2i

[8+8]

3

+1 5. (a) Expand as a Taylor series the function f(z) = 2z about z=1. z 2 +1

(b) Express f(z) =

z (z−1) (z−3)

in a series of positive and negative powers of (z-1) [8+8] 2

Z −2Z 6. (a) Find the residue of f(z) = (Z+1) 2 (Z 2 +1) at each pole. H 4−3z (b) Evaluate z(z−1)(z−2) dz where c is the circle | z | = c

3 2

using residue theorem. [8+8]

7. (a) Show that

R2π 0

dθ a+bsinθ

=

√ 2π a2 −b2

, a > b> 0 using residue theorem. 1 of 2

Set No. 3

Code No: RR220202 (b) Evaluate by contour integration

R∞ 0

dx 1+x2

[8+8]

8. (a) Find and plot the image of triangular region with vertices (0,0), (0,1), (1,0) under the transformation ω= (1-i) z+3. (b) Determine the image of the region 0< y <2,under the transformation in w= z1 ω = z1 . (c) Find the image of the rectangular region -1≤ x ≤ 3, −π ≤y≤ π in the z-plane under the transformation ω = ez [6+5+5] ⋆⋆⋆⋆⋆

2 of 2

Set No. 4

Code No: RR220202

II B.Tech Supplimentary Examinations, Aug/Sep 2008 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(n) =

R∞ 0

(b) Prove that (c) Prove that

[log x1 ]n−1 dx

R∞ x8 (1−x6 )dx

0 R∞ 0

(1+x)24

xn−1 dx (1+x)

=

= 0 using β − Γ functions π sin nπ

and show that

Γ(n)Γ(−n) = sinπnπ (0 < n < 1). R 2. (a) Prove that xJ12 dx = 21 x2 (J02 + J12 ) − xJ0 J1 ′

′

[5+5+6]

′

(b) Pn (x)=Pn+1 − 2x Pn (x) + Pn−1 (x)

[8+8]

3. (a) Find the analytic function whose imaginary part is f(x,y) = x3 y – xy3 + xy +x +y where z = x+iy.   2 ∂2 ∂ + |Real f (z)|2 = 2|f ′ (z)|2 where w =f(z) is analytic[8+8] (b) Prove that ∂x 2 ∂y 2 4. (a) Evaluate using Cauchy’s integral formula

R c

(b) Evaluate using Cauchy’s integral formula

R

(z+1)dz z 2 +2z+4

where c :| z + 1 + i | = 2

−

z f rom z = 0 to 4 + 2i along the

C

curve C given by i) z=t2 +it ii) Along the line z=0 to z=2: and there from z=2 to 4+2i

[8+8]

5. (a) Expand log(1+z) as Taylor series about z = 0 and determine the region of convergence for the resulting series. (b) State and prove Laurent theorem.

[8+8]

6. (a) Find the poles of f(z) and the residues of the poles which lie on imaginary axis (z 2 +2z) if f(z) = (z+1) 2 (z 2 +4) R e2z [8+8] (b) Evaluate (z+1)3 using residue theorem where c:|z|=2. C

7. (a) Evaluate

R2π 0

dθ , (a+b cos θ)2

a> b>0 using residue theorem. 1 of 2

Set No. 4

Code No: RR220202 (b) Evaluate

R∞ 0

dx (x2 +1)3

using residues.

[8+8]

8. (a) Find and plot the image of triangular region with vertices at (0,0), (1,0) (0,1) under the transformation w=(1-i) z+3. (b) If w =

1+iz find 1−iz

the image of |z| <1. ⋆⋆⋆⋆⋆

2 of 2

[8+8]