R059210201 Mathematics Iii-gautam

Set No. 1

Code No: R059210201

II B.Tech I Semester Regular Examinations, November 2007 MATHEMATICS-III ( Common to Electrical & Electronic Engineering, Electronics & Communication Engineering, Electronics & Instrumentation Engineering, Electronics & Control Engineering, Electronics & Telematics, Electronics & Computer 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

(b)

R∞

sin9/2 θ cos5 θ dθ.

0 3

e−x x11/3 dx.

0

(c)

R1 0

4 √x dx . 1−x2

[5+5+6]

2. (a) Prove that J0 (x) = 1 − (b)

R1

xn Pn (x)dx =

x2 22

+

x4 22 .42

−

x6 +........... 22 .42 .62

2n+1 .(n!)2 (2n+1)!

[8+8]

−1

3. (a) Derive Cauchy Riemann equations in polar coordinates. (b) Prove that the function f(z) = z¯ is not analytic at any point. √ (c) Find the general and the principal values of (i) log e (1+ 3i) (ii) log e (–1). [5+5+6] R 2 −z−2) dz 4. (a) Find f(z) and f(3) if f(a)= (2z (z−a) where C is the circle |z| = 2.5 using C

Cauchy’s integral formula. R (b) Evaluate logzdz where C is the circle |z| = 1 using Cauchy’s integral formula.

C

[8+8] 5. (a) State and prove Laurent’s theorem. (b) Obtain all the Laurent series of the function

7z−2 (z+1)(z)(z−2)

about z= -2. [8+8] 2

(2z+1) 6. (a) Determine the poles and the corresponding residues of the function (4z 3 +z) . R πz 2 +cos πz 2 )dz (b) Evaluate (sin(z−1) where C is the circle |z| = 3 using residue theorem. 2 (z−2) C

[8+8]

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Set No. 1

Code No: R059210201 7. (a) Evaluate (b) Evaluate

R 2π

Cos2θ dθ using residue theorem. 0 5+4Cosθ R∞ x2 dx using residue theorem. −∞ (x2 +1)(x2 +4)

[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] ⋆⋆⋆⋆⋆

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Set No. 2

Code No: R059210201

II B.Tech I Semester Regular Examinations, November 2007 MATHEMATICS-III ( Common to Electrical & Electronic Engineering, Electronics & Communication Engineering, Electronics & Instrumentation Engineering, Electronics & Control Engineering, Electronics & Telematics, Electronics & Computer Engineering and Instrumentation & Control Engineering) Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ⋆⋆⋆⋆⋆ 1. (a) Define Gamma function and evaluate Γ(1/2). (b) Show that Γ(1/2)Γ(2n) = 22n−1 Γ(n) Γ(n+1/2). (c) Define Beta function and show that β(m,n) = β(n,m). 2. (a) Prove that (2n+1)(1–x2 ) P

′ n

[6+6+4]

(x)=n(n+1)[Pn+1 (x)-Pn−1 (x)].

′

(b) Prove that (1-x2 ) Pn (x)=(n+1)[x Pn (x)–Pn+1 (x)]. ′

(c) Show that nx Jn (x) + Jn (x) = Jn−1 (x).

[6+5+5] 3

f (z) = x xy(y−ix) 6 +y 2 3. (a) Test for analyticity at the origin for = 0

f or z 6= 0 f or z = 0

(b) Find all values of z which satisfy (i) ez = 1+i (ii) sinz =2. 4. (a) Evaluate

5+i R

[8+8]

z 3 dz using Cauchy’s integral formula along y = x.

−2+i

(b)

R

(x + y)dx + ix2 y dy along y=x2 from (0,0) to (3,9).

(c) Evaluate

2+i R

(x2 − y 2 + i xy) dz using Cauchy’s integral formula along y=x2 .

−1+i

[5+5+6] 5. (a) Find the Laurent series expansion of the function region 2 < |z| < 3. (b) Evaluate f (z) =

2 (2z+1)3

z 2 −1 z 2 +5z+6

about (i) z = 0 (ii) z = 2.

about z = 0 in the [8+8] 2

(2z+1) 6. (a) Determine the poles and the corresponding residues of the function (4z 3 +z) . R (sin πz2 +cos πz2 )dz (b) Evaluate where C is the circle |z| = 3 using residue theorem. (z−1)2 (z−2) C

[8+8] 7. (a) Evaluate

R2π

(b) Evaluate

0 ∞ R 0

dθ , (a+b cos θ)2 dx (x2 +1)3

a>0, b>0 using residue theorem.

using residue theorem. 1 of 2

[8+8]

Set No. 2

Code No: R059210201

8. (a) Show that the transformation w=(z+1/z) converts the straight line
π arg z = α |α| < 2 in to a branch of the hyperabola of eccentricity sec α. (b) Find the bilinear transformation which maps the points (0, 1, ∞) into the points (–1, –2, –i). [8+8] ⋆⋆⋆⋆⋆

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Set No. 3

Code No: R059210201

II B.Tech I Semester Regular Examinations, November 2007 MATHEMATICS-III ( Common to Electrical & Electronic Engineering, Electronics & Communication Engineering, Electronics & Instrumentation Engineering, Electronics & Control Engineering, Electronics & Telematics, Electronics & Computer Engineering and Instrumentation & Control Engineering) Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ⋆⋆⋆⋆⋆

1. (a) Show that β(m,n)=2

π/2 R

sin2m−1 θ cos2n−1 θdθ and deduce that

o

π/2 R

sinn θ dθ =

π/2 R

(b) Prove that Γ(n) Γ( (1-n)= R∞ 2 2 (c) Show that xm e−a x dx = 0

cos(x2 )dx =

o

2. Prove that

Γ 1 Γ( n+1 2 ) (2) Γ( n+2 2 )

o

o

R∞

1 2

cosn θ dθ =

R∞ o

R1

.

π . sin nπ 1 Γ 2am+1

m+1 2




and hence deduce that

p sin(x2 )dx = 1/2 π/2.

Pm (x)Pn (x)dx =

−1



0 2 2n+1

[5+5+6]

if m 6= n . if m = n

[16]

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. (b) Prove that ∂x 2 ∂y 2 [8+8] R 4. (a) Evaluate using Cauchy’s integral formula (z(z+1)dz 2 +2z+4) where C :| z + 1 + i | = 2. C

(b) Evaluate

R

−

z dz f rom z = 0 to 4 + 2i along the curve C given by

C

i. z=t2 +it ii. Along the line z=0 to z=2 and then from z=2 to 4+2i. 5. Expand f (z) =

(z−2)(z+2) (z+1)(z+4)

[8+8]

in the region.

(a) 1 < |z| < 4 (b) | z | < 1.

[8+8]

6. (a) Find the poles and the residues at each pole of f(z)= z2z+1 . R z dz where c is |z | = 5 by residue theorem. (b) Evaluate (zze2 +9) C

1 of 2

[8+8]

Set No. 3

Code No: R059210201

7. (a) Evaluate

R2π

(b) Evaluate

0 ∞ R 0

dθ , a+b cos θ dx (1+x2 )2

a>0, b>0 using residue theorem.

using residue theorem.

[8+8]

8. (a) Find the image of the infinite strip 0
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Set No. 4

Code No: R059210201

II B.Tech I Semester Regular Examinations, November 2007 MATHEMATICS-III ( Common to Electrical & Electronic Engineering, Electronics & Communication Engineering, Electronics & Instrumentation Engineering, Electronics & Control Engineering, Electronics & Telematics, Electronics & Computer 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) =

R1 0

(b) Prove that

[log . x1 ]n−1 dx

R∞ x8 (1−x6 )dx (1+x)24

0

= 0 using β − Γ functions.

[8+8] x

1

2. (a) Show that the coefficient of tn in the power series expansion of e 2 (t− t ) is Jn (x). R1 [8+8] (b) Prove that xPn (x)Pn−1 (x) = (4n2n 2 −1) . −1

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 (b) Prove that ∂x + |Real f (z)|2 = 2|f ′ (z)|2 where w =f(z) is analytic. 2 ∂y 2 [8+8] R 4. (a) Evaluate using Cauchy’s integral formula (z(z+1)dz 2 +2z+4) where C :| z + 1 + i | = 2. C

(b) Evaluate

R

−

z dz f rom z = 0 to 4 + 2i along the curve C given by

C

i. z=t2 +it ii. Along the line z=0 to z=2 and then from z=2 to 4+2i. 5. (a) Expand logz by Taylor’s series about z=1. (b) Expand

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

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

[8+8] √

2. [8+8]

2

6. (a) Find the poles and residue at each pole of the function (z4z−1) . R z ) dz where C is |z| = 1 by residue theorem. (b) Evaluate (z (1+e cos z+sin z)

[8+8]

C

7. (a) Show that (b) Evaluate

Rπ

0 R∞ 0

adθ (a2 +sin2 θ)

dx (x4 +1)

=

√ π 1+a2

, ( a > 0) using residue theorem.

using Residue theorem.

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[8+8]

Set No. 4

Code No: R059210201

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] ⋆⋆⋆⋆⋆

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