Set No. 2

R05

Code No: R05210201

(b) Evaluate

R2π 0 R∞ 0

dθ , a+b cos θ dx (1+x2 )2

a>0, b>0 using residue theorem.

using residue theorem.

[8+8]

ld .

1. (a) Evaluate

in

II B.TECH – I SEM EXAMINATIONS, NOVEMBER - 20102010 II B.Tech I Semester Regular Examinations,November MATHEMATICS-III Common to ICE, E.COMP.E, ETM, E.CONT.E, EIE, ECE, EEE Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ?????

2. (a) Show that the transformation w=z+1/z maps the circle |z| =c into the ellipse u=(c+1/c) cos θ, v =(c–1/c)sinθ. Also discuss the case when c=1 in detail.

3. (a) Prove that

√

1 1−2tx+t2

or

(b) Find the bilinear transformation which maps the points (2, i, –2) into the points (l, i, –l). [8+8] = P0 (x) + P1 (x) t + P2 (x) t2 + ....

(b) Write J5/2 (x) in finite form.

[8+8]

uW

4. (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 0 (z)|2 where w =f(z) is analytic. + (b) Prove that ∂x 2 ∂y 2 [8+8] 5. (a) Show that when | z + 1 | < 1, z −2 = 1 + (b) Expand f (z) =

(n + 1)(z + 1)n .

n=1

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

[8+8]

nt

R1

1 z 2 −z−6

∞ P

6. (a) Evaluate (b) Evaluate

0 R∞ 0

xdx (1+x6 )

−1 R

Aj (c) Evaluate

x4 log


1 3 x

dx using β − Γ functions.

using β − Γ functions.

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

[5+6+5]

0

7. (a) Show that

R

(z + 1) dz = 0 where C is the boundary of the square whose

C

vertices at the points z = 0, z = 1, z = 1+i, z = i. R 2 using Cauchy’s integral formula where c is |z| = 2 find (b) If F(a)= 3z +7z+1)dz (z−a) C

F(1) F(3) F 00 (1-i). 8. (a) Find the residue of f(z) =

[8+8] Z 2 −2Z (Z+1)2 (Z 2 +1)

1

at each pole.

R05

Code No: R05210201 (b) Evaluate

H c

4−3z z(z−1)(z−2)

Set No. 2

dz where c is the circle | z | =

3 2

using residue theorem. [8+8]

Aj

nt

uW

or

ld .

in

?????

2

Set No. 4

R05

Code No: R05210201

R

1. (a) Show that

in

II B.TECH – I SEM Regular EXAMINATIONS, NOVEMBER - 2010 II B.Tech I Semester Examinations,November 2010 MATHEMATICS-III Common to ICE, E.COMP.E, ETM, E.CONT.E, EIE, ECE, EEE Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ????? (z + 1) dz = 0 where C is the boundary of the square whose

C

ld .

vertices at the points z = 0, z = 1, z = 1+i, z = i. R 2 using Cauchy’s integral formula where c is |z| = 2 find (b) If F(a)= 3z +7z+1)dz (z−a) C

F(1) F(3) F 00 (1-i).

[8+8]

2. (a) Show that the transformation w=z+1/z maps the circle |z| =c into the ellipse u=(c+1/c) cos θ, v =(c–1/c)sinθ. Also discuss the case when c=1 in detail.

3. (a) Evaluate

x4 log

0 R∞

xdx (1+x6 )


1 3 x

dx using β − Γ functions.

using β − Γ functions.

uW

(b) Evaluate

R1

or

(b) Find the bilinear transformation which maps the points (2, i, –2) into the points (l, i, –l). [8+8]

0

(c) Evaluate

−1 R

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

[5+6+5]

0

2

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

3 2

using residue theorem.

nt

[8+8]

5. (a) Prove that

√

1 1−2tx+t2

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

(b) Write J5/2 (x) in finite form. R2π

Aj

6. (a) Evaluate (b) Evaluate

0 R∞ 0

dθ , a+b cos θ

dx (1+x2 )2

[8+8]

a>0, b>0 using residue theorem.

using residue theorem.

7. (a) Show that when | z + 1 | < 1, z −2 = 1 +

[8+8] ∞ P

(n + 1)(z + 1)n .

n=1

(b) Expand f (z) =

1 z 2 −z−6

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

8. (a) Find the analytic function whose imaginary part is f(x,y) = x3 y – xy3 + xy +x +y where z = x+iy. 3

[8+8]

R05

Code No: R05210201 (b) Prove that



∂2 ∂x2

+

∂2 ∂y 2



Set No. 4

|Real f (z)|2 = 2|f 0 (z)|2 where w =f(z) is analytic. [8+8]

Aj

nt

uW

or

ld .

in

?????

4

Set No. 1

R05

Code No: R05210201

in

II B.TECH – I SEM EXAMINATIONS, NOVEMBER - 2010 II B.Tech I Semester Regular Examinations,November 2010 MATHEMATICS-III Common to ICE, E.COMP.E, ETM, E.CONT.E, EIE, ECE, EEE Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ?????

(b) Evaluate

R2π 0 R∞ 0

dθ , a+b cos θ dx (1+x2 )2

a>0, b>0 using residue theorem.

using residue theorem.

or

2. (a) Evaluate

ld .

1. (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 ∂x2 + ∂y2 |Real f (z)|2 = 2|f 0 (z)|2 where w =f(z) is analytic. [8+8]

[8+8]

3. (a) Show that the transformation w=z+1/z maps the circle |z| =c into the ellipse u=(c+1/c) cos θ, v =(c–1/c)sinθ. Also discuss the case when c=1 in detail.

uW

(b) Find the bilinear transformation which maps the points (2, i, –2) into the points (l, i, –l). [8+8] 4. (a) Evaluate (b) Evaluate

R1

0 R∞ 0

(c) Evaluate

x4 log xdx (1+x6 )

−1 R


1 3 x

dx using β − Γ functions.

using β − Γ functions.

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

[5+6+5]

nt

0

2

Z −2Z 5. (a) Find the residue of f(z) = (Z+1) 2 (Z 2 +1) at each pole. H 4−3z dz where c is the circle | z | = (b) Evaluate z(z−1)(z−2)

Aj

c

6. (a) Prove that

√

1 1−2tx+t2

3 2

using residue theorem. [8+8]

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

(b) Write J5/2 (x) in finite form.

[8+8]

7. (a) Show that when | z + 1 | < 1, z −2 = 1 +

∞ P

(n + 1)(z + 1)n .

n=1 1 z 2 −z−6

about (i) z = -1 (ii) z = 1. [8+8] (b) Expand f (z) = R 8. (a) Show that (z + 1) dz = 0 where C is the boundary of the square whose C

vertices at the points z = 0, z = 1, z = 1+i, z = i. 5

R05

Code No: R05210201 (b) If F(a)=

R C

3z 2 +7z+1)dz (z−a)

Set No. 1

using Cauchy’s integral formula where c is |z| = 2 find

F(1) F(3) F 00 (1-i).

[8+8]

Aj

nt

uW

or

ld .

in

?????

6

Set No. 3

R05

Code No: R05210201

2

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

using residue theorem.

ld .

c

3 2

in

II B.TECH – I SEMRegular EXAMINATIONS, NOVEMBER - 2010 II B.Tech I Semester Examinations,November 2010 MATHEMATICS-III Common to ICE, E.COMP.E, ETM, E.CONT.E, EIE, ECE, EEE Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ?????

[8+8]

(b) Evaluate

R2π 0 R∞ 0

3. (a) Prove that

dθ , a+b cos θ dx (1+x2 )2 √

a>0, b>0 using residue theorem.

using residue theorem.

1 1−2tx+t2

[8+8]

or

2. (a) Evaluate

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

(b) Write J5/2 (x) in finite form.

[8+8]

uW

4. (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 0 (z)|2 where w =f(z) is analytic. + 2 ∂y 2 [8+8] 5. (a) Show that when | z + 1 | < 1, z −2 = 1 +

∞ P

(n + 1)(z + 1)n .

n=1

1 z 2 −z−6

nt

about (i) z = -1 (ii) z = 1. [8+8] (b) Expand f (z) = R 6. (a) Show that (z + 1) dz = 0 where C is the boundary of the square whose C

vertices at the points z = 0, z = 1, z = 1+i, z = i. R 2 (b) If F(a)= 3z +7z+1)dz using Cauchy’s integral formula where c is |z| = 2 find (z−a) C

F(1) F(3) F 00 (1-i).

Aj

[8+8]

7. (a) Show that the transformation w=z+1/z maps the circle |z| =c into the ellipse u=(c+1/c) cos θ, v =(c–1/c)sinθ. Also discuss the case when c=1 in detail. (b) Find the bilinear transformation which maps the points (2, i, –2) into the points (l, i, –l). [8+8]

8. (a) Evaluate (b) Evaluate

R1 0 R∞ 0

x4 log xdx (1+x6 )


1 3 x

dx using β − Γ functions.

using β − Γ functions. 7

Code No: R05210201

(c) Evaluate

−1 R

R05

Set No. 3

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

[5+6+5]

0

Aj

nt

uW

or

ld .

in

?????

8