R05221801-mathematics-iii

Set No. 1

Code No: R05221801

II B.Tech Supplimentary Examinations, Aug/Sep 2008 MATHEMATICS-III (Metallurgy & Material Technology) Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ⋆⋆⋆⋆⋆ 1. (a) Prove that Γ(n) =

R1

[log . x1 ]n−1 dx

0

(b) Prove that

R∞ x8 (1−x6 )dx 0

(1+x)24

= 0 using β − Γ functions.

[8+8] 2

d y dy 2 2 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) Find an analytic function f(z) = u(r,θ)+ i v(r,θ) such that u(r,θ) = r2 cos2θ – r cosθ + 2.   is (b) Determine ‘p’ such that the function f(z) = 21 loge (x2 +y2 ) + i tan−1 px y an analytic function. [8+8] R Cos z−sin z dz 4. (a) Evaluate with C: | z | = 2 using Cauchy’s integral formula. (z+π/2)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) Find the Laurent series expansion of the function z 2 −6z−1 in the region 3< |z+2| <5. (z−1)(z−3)(z+2) 6. (a) Find the poles and the corresponding residues of the function R where c is |z| = 1 by residue theorem. (b) Evaluate (3z+5)dz (z 2 −1)2

[8+8] z3 . (z−1)(z−2)(z−3)

[8+8]

C

7. (a) Evaluate (b) Evaluate

R2π

0 R∞ 0

dθ (5−3cosθ)2 sin mx dx x

using residue theorem. using residue theorem.

[8+8]

8. (a) Find the image of 1< |z| <2 under the transformation w= 2iz+1. (b) Find the bilinear transformation which maps the points (–1, i, 1+i ) onto the points (0, 2i, 1-i). [8+8] ⋆⋆⋆⋆⋆ 1 of 1

Set No. 2

Code No: R05221801

II B.Tech Supplimentary Examinations, Aug/Sep 2008 MATHEMATICS-III (Metallurgy & Material Technology) 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)

π/2 R

7

sin5 θ cos 2 θ dθ .

[5+5+6]

0

2. (a) When n is a positive integer show that Jn (x) = (b) Show that x3 = 52 P3 (x) +

3 5

1 π

Rπ

cos(nθ − x sin θ)dθ.

0

P1 (x).

[8+8]

3. (a) Show that w = z n (n , a positive integer) is analytic and find it’s derivative. (b) If w = f(z) is an analytic function, then prove that the family of curves defined by u(x,y) = constant cuts orthogonally the family of curves v(x,y) = constant. (c) If α + iβ = tanh (x + i π/4) prove that α2 + β 2 =1. [6+5+5] R (z 2 − z−1) dz with C: |z − 1/2| = 1 using Cauchy’s integral theo4. (a) Evaluate z(z−i)2 (b)

rem. R (z 2 + 4) C

C

dz

z 2 (z+2)2

with C: | z + 2 | = 1 using Cauchy’s integral formula.

5. (a) Find the Laurent expansion of (b) Expand the Laurent series of

1 , (z 2 −4z+3)

z 2 −1 , (z+2)(z+3)

[8+8]

for 1 < | z | < 3.

for | z | > 3.

[8+8] 2

z 6. (a) Find the poles and the residues at each pole of f(z)z (z+2)(z−1) 2. R dz (b) Evaluate z sin z where c is |z| = 1 by residue theorem.

[8+8]

C

7. (a) Evaluate

Rπ

(b) Evaluate

0 R∞ 0

adθ a2 +sin2 θ dx (x4 +16)

using residue theorem. using residue theorem.

[8+8]

8. (a) Find the image of the straight lines x=0; y=0; x=1 and y=1 under the transformation w=z2 . (b) Show that the relation transformsw =

1 of 2

5−4z 4z−2

the circle |z| = 1 into a circle. [8+8]

Set No. 2

Code No: R05221801 ⋆⋆⋆⋆⋆

2 of 2

Set No. 3

Code No: R05221801

II B.Tech Supplimentary Examinations, Aug/Sep 2008 MATHEMATICS-III (Metallurgy & Material Technology) Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ⋆⋆⋆⋆⋆ 1. (a) Evaluate (b) Evaluate

R1

x4 log

0 R∞ 0

(c) Evaluate

−1 R 0

2. Prove that

R1

xdx (1+x6 )


1 3 x

dx using β − Γ functions.

using β − Γ functions.

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

Pm (x)Pn (x)dx =

−1



0 2 2n+1

if m 6= n . if m = n

[5+6+5] [16]

3. (a) Show that w = z n (n , a positive integer) is analytic and find it’s derivative. (b) If w = f(z) is an analytic function, then prove that the family of curves defined by u(x,y) = constant cuts orthogonally the family of curves v(x,y) = constant. (c) If α + iβ = tanh (x + i π/4) prove that α2 + β 2 =1.

[6+5+5]

4. (a) State and prove Cauchy’s integral theorem. R ez dz (b) Evaluate using Cauchy’s integral formula (z−1−i) 3 where C: | z − 1 | = 3. C

(c) Evaluate

1+i R 0

(x2 − iy) dz along y=x2 .

5. (a) Find the Laurent expansion of (b) Expand the Laurent series of

[5+6+5]

1 , (z 2 −4z+3)

z 2 −1 , (z+2)(z+3)

for 1 < | z | < 3.

for | z | > 3.

[8+8]

6. (a) Find the poles and residues at each pole of the function cosec2 z. R zeiz dz (b) Evaluate (z 2 +9)2 where c is the circle |z | = 4 by residue theorem. C

(c) Evaluate

R

C

7. (a) Evaluate (b) Evaluate

R2π

0 R∞

z 3 dz (z−1)2 (z−3)

dθ (5−3cosθ)2 sin mx dx x

where c is | z | = 2 by residue theorem.

[5+5+6]

using residue theorem. using residue theorem.

[8+8]

0

8. (a) Show that under the transformation w = (z-i)/ (z+i), real axis in the z-plane is mapped into the circle | w | = 1. Which portion of the z-plane corresponds to the interior of the circle? 1 of 2

Set No. 3

Code No: R05221801

(b) Prove that the transformation w= sin z maps the families of lines x=a and x=b into two families of confocal central conics. [8+8] ⋆⋆⋆⋆⋆

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Set No. 4

Code No: R05221801

II B.Tech Supplimentary Examinations, Aug/Sep 2008 MATHEMATICS-III (Metallurgy & Material Technology) Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ⋆⋆⋆⋆⋆

1. (a) Evaluate

π/2 R √

cotθ dθ.

0

(b) prove that Γ(n + 12 ) =

√

π Γ(2n+1) . 4n Γ(n+1)

(c) If m>0, n>0, then prove that n1 β(m, n + 1) =

1 β(m + 1, n) m

β(m,n) .[5+5+6] (m+n)

=

′ 2. (a) Prove that Pn′ (x) = x Pn−1 (x) + nPn−1 (x). q 2 sin x. (b) Show that J1/2 (x) = πx

(c) Show that

d dx

[xn Jn (x)] = xn Jn−1 (x).

[6+5+5]

3. (a) Determine the analytic function f(z) =u+iv given that 3u + 2v = y2 – x 2 + 16x. (b) If sin (α + iβ) = x + iy then prove that 4. (a) Evaluate

dz z 2 ez

R c

y2 x2 + sinh 2 cosh2 β β

= 1 and

2 x2 − cosy2 α sin2 α

= 1. [8+8]

where C is |z| = 1.

(b) Evaluate using Cauchy’s integral formula

1+i R

z 2 dz along y = x2 .

0

(c) Prove that

R

C

dz (z−a)

= 2πi where C is given by the equation |z − a| = r. [6+5+5]

5. (a) State and derive Laurent’s series for an analytic function f (z). (b) Expand

1 (z 2 −3z+2)

in the region.

i. 0 < | z – 1 | < 1 ii. 1 < | z | < 2.

[8+8]

6. (a) Find the poles and the corresponding residues of the function R (b) Evaluate (zzdz 2 +1) where C is |z+1| = 1 by residue theorem.

(z+2) . (z−2)(z+1)2

[8+8]

C

7. (a) Evaluate (b) Evaluate

R2π

0 ∞ R 0

dθ (5−3cosθ)2 sin mx dx x

using residue theorem. using residue theorem. 1 of 2

[8+8]

Set No. 4

Code No: R05221801 8. (a) Discuss the transformation w=cos z.

(b) Find the bilinear transformation which maps the points (l, i, -l) into the points (o,1,∞). [8+8] ⋆⋆⋆⋆⋆

2 of 2