R05
Code No: 33047
Set No - 1
1. (a) Evaluate
R c
(z 2 −2z−2) dz (z 2 +1)2 z
in
II B.Tech I Semester Supplimentary Examinations,Nov/Dec 2009 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 ????? where c is | z − i | = 1/2 using Cauchy’s integral for-
mula. R
(z 2 + 3z + 2) dz where C is the arc of the cycloid x = a(θ + sin θ),
ld .
(b) Evaluate
C
y = a (1 − cos θ) between the points (0,0) to (πa, 2a).
[8+8]
2. (a) Find the image of the domain in the z-plane to the left of the line x=–3 under the transformation w=z2 .
3. (a) Evaluate
π/2 R
sin2 θ cos4 θ dθ =
0
R∞ √
(c) Show that
R∞
2
x e−x dx = 2
π 32
R∞
using β − Γ functions. 4
x2 e−x dx using β − Γ functions and evaluate.
uW
(b) Prove that
or
(b) Find the bilinear transformation which transforms the points z=2,1,0 into w=1,0,i respectively. [8+8]
0
0
0
xm−1
(x+a)m+n
dx = a−n β(m, n).
[5+6+5]
4. (a) State necessary condition for f ( z ) to be analytic and derive C-R equations in Cartesian coordinates.
nt
(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 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]
Aj
6. (a) Use Rouche’s theorem to show that the equation z5 + 15 z + 1=0 has one root in the disc |z| < 23 and four roots in the annulus 32 < |z| < 2. R∞ cos xdx (b) Evaluate (a > 0) using residue theorem. [8+8] (a2 +x2 ) −∞
2z+1 7. (a) Find the poles and residue at each pole of the function (1−z 4) . R sin z (b) Evaluate z cos z dz where C is | z | = π by residue theorem. C
8. (a) Prove that (2n+1)(1–x2 ) P 2
0 n
(x)=n(n+1)[Pn+1 (x)-Pn−1 (x)].
0
(b) Prove that (1-x ) Pn (x)=(n+1)[x Pn (x)–Pn+1 (x)]. 1
[8+8]
R05
Code No: 33047
Set No - 1
0
(c) Show that nx Jn (x) + Jn (x) = Jn−1 (x).
[6+5+5]
Aj
nt
uW
or
ld .
in
?????
2
R05
Code No: 33047
Set No - 2
(b) Evaluate
R2π 0 R∞ 0
dθ (5−3cosθ)2 sin mx dx x
using residue theorem. using residue theorem.
2. (a) For the function f(z)= z=1.
2z 3 +1 z (z+1)
[8+8]
ld .
1. (a) Evaluate
in
II B.Tech I Semester Supplimentary Examinations,Nov/Dec 2009 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 ?????
find Taylor’s series valid in a neighbourhood of 1 z 2 (z−1)
about z=1 and find the region of
or
(b) Find Laurent’s series for f(z) = convergence.
[8+8] 2
(2z+1) 3. (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)
uW
C
[8+8]
4. (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. 2+3i R
(z 3 + z)dz along the line joining z=1–i to z=2+3i.
nt
5. (a) Evaluate
[6+5+5]
1−i
(b) Evaluate
R
C
e2z dz (z+1)4
where C is the circle |z − 1| = 2 using Cauchy’s integral
formula.
R
Aj
(c) Evaluate
C
(z+1) dz (z 3 +2z 2 )
where C is the circle |z| = 1 using Cauchy’s integral for-
mula.
[5+5+6] 0
0
6. (a) Establish the formula Pn+1 (x) − Pn−1 (x) = (2n + 1) Pn (x). (b) Prove that
d dx
[x−n Jn (x)] = −x−n Jn+1 (x).
(c) When n is an integer, show that J−n (x)=(–1)n Jn (x). 7. (a) Show that Γ(n) =
R1
(log 1/y)n−1 dy.
0
3
[6+5+5]
R05
Code No: 33047
(b) Prove that
R1 0
(c) Prove that
R1 0
Set No - 2
√
√ dx n 1−x
=
2 √x dx 1−x4
=
1 π Γ( n ) . 1 n Γ( n + 12 )
√
π Γ (3/4) Γ(1/4)
[5+5+6]
8. (a) Find the image of the circle |z|=2, under the transformation w=z+3+2i.
in
(b) The points (0, –i, –1) are mapped into the points (i, 1, 0). Find the image of the line y = mx under this transformation. [8+8]
Aj
nt
uW
or
ld .
?????
4
R05
Code No: 33047
Set No - 3
1. (a) Using Cauchy’s Integral Formula evaluate
R C
z 2 dz (z+1)(z−i)2
in
II B.Tech I Semester Supplimentary Examinations,Nov/Dec 2009 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 ????? where C is ellipse
9x2 + 4y2 =36.
ld .
(b) Integrate z2 along the straight line OM and along the path OLM consisting of two straight line segments OL and LM. Where O is the origin L is the point z=3 and M is z=3+i. R 3 2 +2z−1) dz where C is | z | = 3 using Cauchy’s integral formula. (c) Evaluate (z +z (z−1) 3 c
2. (a) Evaluate (b) Evaluate
R2π 0 R∞
x sin mx dx x4 +16
using residue theorem.
using residue theorem.
uW
0
dθ (5−3 sin θ)2
or
[6+5+5]
[8+8]
2
Z −2Z 3. (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
4. Prove that
R1
Pm (x)Pn (x)dx =
−1
0
2 2n+1
3 2
using residue theorem. [8+8]
if m 6= n . if m = n
nt
5. (a) If w = u + iv is an analytic function of z and u + v = f(z).
[16] sin 2x cos h 2y − cos 2x
(b) If sin (θ + iα) = cos α + i sin α, then prove that cos2 θ =sin2 α.
then find [8+8]
Aj
6. (a) Find the image of the infinite strip 1/4 < y< 1/2 under the transformation w=1/z. (b) Find the bilinear transformation which maps the points (1, –1, ∞) onto the points (1+i, 1–i, 1). [8+8]
7. Evaluate the following using β − Γ functions. (a)
π/2 R
sin9/2 θ cos5 θ dθ.
0
(b)
R∞
3
e−x x11/3 dx.
0
5
Code No: 33047
(c)
R1 0
R05
Set No - 3
4 √x dx . 1−x2
[5+5+6]
8. (a) State and prove Taylor’s theorem. [8+8]
in
(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)
Aj
nt
uW
or
ld .
?????
6
R05
Code No: 33047
Set No - 4
in
II B.Tech I Semester Supplimentary Examinations,Nov/Dec 2009 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 ????? 1. (a) Find the image of the straight lines x=0; y=0; x=1 and y=1 under the transformation w=z2 .
R1
2 √x dx 1−x5
0
(b) Prove that
R1
interms of β function.
(1 − xn )1/n dx =
0
(c) Prove that Γ
1 n
the circle |z| = 1 into a circle. [8+8]
2 n
Γ
Γ
3 n
1
2
1 [Γ( n )] . n 2Γ(2/n)
or
2. (a) Evaluate
5−4z 4z−2
ld .
(b) Show that the relation transformsw =
.........Γ
n−1 n
3. (a) Show that when | z + 1 | < 1, z −2 = 1 +
=
∞ P
(2
Q n−1 ) 2 n1/2
.
[5+5+6]
(n + 1)(z + 1)n .
uW
n=1
(b) Expand f (z) =
1 z 2 −z−6
about (i) z = -1 (ii) z = 1.
[8+8] z
4. (a) Find the poles and the residues at each pole of f (z) = 1−e . Where z=0 is a z4 pole of order 4 R (Z 2 + 2Z + 5) dz where c is the circle using residue theorem. (b) Evaluate (Z−3 ) c
nt
i. | z | = 1 ii. | z+1-i | = 2.
[6+10] 3
5. (a) Test for analyticity at the origin for
f (z) = x xy(y−ix) 6 +y 2 = 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.
Aj
6. (a) Prove that
√
1 1−2tx+t2
= P0 (x) + P1 (x) t + P2 (x) t2 + ....
(b) Write J5/2 (x) in finite form.
7. (a) Evaluate (b) Evaluate
R2π
0 R∞ 0
dθ , a+b cos θ dx (1+x2 )2
[8+8]
[8+8]
a>0, b>0 using residue theorem.
using residue theorem.
8. (a) Evaluate using Cauchy’s Integral Formula
[8+8] R C
7
(z+1) dz (z 3 −4z)
where C is |z + 2| = 23 .
R05
Code No: 33047 (b) Evaluate
R c
e3z dz (z+1)4
Set No - 4
where c is | z | = 3 using Cauchy’s integral formula.
[8+8]
Aj
nt
uW
or
ld .
in
?????
8
