Mathematics Iii May2004 Nr Rr 220202

Set No:

1

Code No: NR/RR-220202 II-B.Tech. II-Semester Regular Examinations, April/May-2004 NR-MATHEMATICS-III (Common to Electronics and Electrical Engineering, Mechanical Engineering, Electronics and Communication Engineering, Electronics and Instrumentation Engineering, Electronics and Control Engineering, Metallurgy and Material Technology, Mechatronics, Electronics and Telematics and Aeronautical Engineering) RR-MATHEMATICS-III (Common to Electronics and Electrical Engineering, Electronics and Communication Engineering, Electronics and Instrumentation Engineering, Electronics and Control Engineering, Metallurgy and Material Technology, Electronics and Telematics, Aeronautical Engineering and Instrumentation and Control Engineering) Time: 3 Hours Max. Marks: 80 Answer any FIVE questions All questions carry equal marks --∞

1.a)

Prove that

∫e

− y 1/ m

dx = mΓm

o b

b)

Show that

∫ ( x − a)

m

(b − x ) n dx = (b − a ) m+ n +1 β (m + 1, n + 1)

a

2.a) b) 3.a) b)

4.a)

b)

n J n ( x) + J n' ( x ) = J n +1 ( x) x P Show that n (1) = 1 and Pn(-1) = (-1)n Show that

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. If f(z) is an analytic function, show that  ∂2 ∂ 2   + | f ( z ) |2 = 4 | f ' ( z ) |2 .  ∂x 2 ∂y 2    ze z dz Evaluate 3 ∫ C ( z + a) where c is any simple closed curve enclosing the point z = -a . Evaluate ez z4  ∫  3 + ( z + i ) 2  C z where c : | Z | = 2 using Cauchy’s Integral Formula Contd…2

Code No. NR/RR-220202 5.a)

-2-

Set No.1

Determine the poles of the function z +1 z 2 ( z − 2)

b) 6.a) b)

7.a) b)

ez Obtain the Taylor series expansion of f ( z ) = about z = 2 z ( z + 1) State and prove Cauchy’s Residue Theorem. Z2 Find the residue of at these Singular points which lie inside the circle Z 4 +1 | Z | = 2. Show that the polynomial Z5 + Z3 + 2 Z + 3 has just one zero in the first quadrant of the complex plane. Show that the equation Z4 + 4( 1+i) Z + 1 = 0 has one root in each quadrant.

8.a)

Show that the function w=4/z transforms the straight line x=c in the z-plane into a circle in the w-plane. b) Discuss the transformation of w=ez. ^^^

Set No:

2

Code No: NR/RR-220202 II-B.Tech. II-Semester Regular Examinations, April/May-2004 NR-MATHEMATICS-III (Common to Electronics and Electrical Engineering, Mechanical Engineering, Electronics and Communication Engineering, Electronics and Instrumentation Engineering, Electronics and Control Engineering, Metallurgy and Material Technology, Mechatronics, Electronics and Telematics and Aeronautical Engineering) RR-MATHEMATICS-III (Common to Electronics and Electrical Engineering, Electronics and Communication Engineering, Electronics and Instrumentation Engineering, Electronics and Control Engineering, Metallurgy and Material Technology, Electronics and Telematics and Aeronautical Engineering, Instrumentation and Control Engineering) Time: 3 Hours Max. Marks: 80 Answer any FIVE questions All questions carry equal marks --∞

1.a)

Show that

∫x

m

1

n

e − ax dx = a

0

Γ((m + 1) / n) where n and m are positive

m+1 n

constants. π /2

b)

Prove that

∫

cos x dx =

π /2

0

2.a) b) 3.a) b) 4.a)

0

Show thatt Jn-1 (x) + Jn+1(x) =

dx cos x

=π

2n J n ( x) x

2 cos x πx

Prove that J-½(x) =

Define analyticity of a complex function at a point P and in a domain D. Prove that the real and imaginary parts of an analytic function satisfy Cauchy – Riemann Equations. Show that w = zn (n , a positive integer) is analytic and find it’s derivative.

∫ (z

Evaluate

C

b)

∫

e z dz 2

+ Π2

)

2

Where c is | Z | = 4.

Evaluate

dz

∫ z ( z + 4) dz 8

Where c is the circle | Z | = 2.

C

Contd…2

Code No. NR/RR-220202 5.a) b)

6.a) b)

Evaluate

-2-

dz

∫c Sinhz , where C is the circle

|z|=4

Obtain all the Laurent series of the function

Find the residue of Evaluate

Z −3

Ze z ( Z − 1) 3

∫ Z 2 + 2Z + 5

Set No. 2

7z − 2 about Z0 = -1 ( z + 1)( z )( z − 2)

at its pole.

dz where C is the circle.

c

(i) | Z | = 1 7.a)

b)

(ii) | Z+1-i | = 2 (iii) | Z+1+i | = 2

Use Rouche’s theorem to show that the equation Z5 + 15 Z +1=0 has one toot in 3 3 the disc | Z | < and four roots in the annulus < | Z | < 2. 2 2 State and prove fundamental theorem of algebra.

8.a) Find the bilinear transformation that maps the points 1, i, -1 into the points 2, i, -2 b) Prove that under the transformation w=1/z the image of the lines y=x-1 and y=0 are the circle u2+v2-u-v=0 and the line ν=0 respectively. ^^^

Set No:

Code No: NR/RR-220202 II-B.Tech. II-Semester Regular Examinations, April/May-2004 NR-MATHEMATICS-III (Common to Electronics and Electrical Engineering, Mechanical Engineering, Electronics and Communication Engineering, Electronics and Instrumentation Engineering, Electronics and Control Engineering, Metallurgy and Material Technology, Mechatronics, Electronics and Telematics and Aeronautical Engineering) RR-MATHEMATICS-III (Common to Electronics and Electrical Engineering, Electronics and Communication Engineering, Electronics and Instrumentation Engineering, Electronics and Control Engineering, Metallurgy and Material Technology, Electronics and Telematics and Aeronautical Engineering, Instrumentation and Control Engineering) Time: 3 Hours Max. Marks: 80 Answer any FIVE questions All questions carry equal marks ---

3

1

1.a)

Show that

∫ (1 + x)

p −1

(1 − x) q −1 dx = 2 p + q −1

−1 ∞

b)

Show that

∫x

4

2

e − x dx =

0

2.a) b) 3.a) b)

4.

Γ ( p )Γ ( q ) Γ( p + q )

3 π 8

Show that cos (x sin θ) = J0 + 2 (J2 cos 2θ + J4 cos4θ+....)

1π Prove that J0(x) = ∫ cos( x cosθ )dθ π0 Show that f(x,y) = x3y – xy3 + xy +x +y can be the imaginary part of an analytic function of z = x+iy. Find where the function w =

Evaluate

z2 − z −1

∫ z( z − i )

2

dz

z+2 fails to be analytic. z ( z 2 + 1)

with c : | Z – 1/2 | = 1 using Cauchy’s Integral Formula

C

5.a) Determine the poles of f ( z ) = b)

z cos z

Expand the Laurent series of z2 −1 , for | z | > 3. ( z + 2)( z + 3) Contd…2

Code No. NR/RR-220202 6.a)

b)

-2-

Set No. 3

Find the residue of Z 2 − 2Z f(z) = at each pole. ( Z + 1) 2 ( Z 2 + 1) Evaluate

4 − 3z

∫ z ( z − 1)( z − 2)

dz

Where c is the circle | Z | =

c

3 2

7.a) b)

State and prove Argument Principle Prove that one root of the equation Z4 + Z3 + 1 = 0 lies in the first quadrant.

8.a) b)

Under the transformation w=1/z, find the image of the circle |z-zi|=z. Find the bilinear transformation which maps the points (z, i, -z) into the points (l, i, -l). ^^^

Set No:

Code No: NR/RR-220202 II-B.Tech. II-Semester Regular Examinations, April/May-2004 NR-MATHEMATICS-III (Common to Electronics and Electrical Engineering, Mechanical Engineering, Electronics and Communication Engineering, Electronics and Instrumentation Engineering, Electronics and Control Engineering, Metallurgy and Material Technology, Mechatronics, Electronics and Telematics and Aeronautical Engineering) RR-MATHEMATICS-III (Common to Electronics and Electrical Engineering, Electronics and Communication Engineering, Electronics and Instrumentation Engineering, Electronics and Control Engineering, Metallurgy and Material Technology, Electronics and Telematics and Aeronautical Engineering, Instrumentation and Control Engineering) Time: 3 Hours Max. Marks: 80 Answer any FIVE questions All questions carry equal marks ---

4

1

1.a)

Show that

m ∫ x ( log x ) dx = n

0

( −1) n n! where n is a positive (m + 1) n +1

integer and m > -1 ∞

b)

2.a) b)

3.a) b) 4.a)

y n −1 dy Show that β(m,n) = ∫ m+ n 0 (1 + y ) Establish the formula Pn' +1 ( x ) − Pn' −1 ( x) = (2n + 1) Pn ( x)

1π 2 n When n is a +ve integer prove that Pn(x) = ∫ ( x ± x − 1 cosθ ) dθ π0  ∂2 ∂2  2 2 Prove that  2 + 2  | Re al f ( z ) | = 2 | f ' ( z ) | where w =f(z) is analytic. ∂y   ∂x Find k such that f(x,y) = x3 + 3kxy2 may be harmonic and find its conjugate. Evaluate using Cauchy’s Integral Formula

( z − 2) dz

∫ ( z − 1) ( z + 2) 3

3

C

with c : | Z | = 3 b)

Evaluate c∫ ( y2 + z2 ) dx +( z2 + x2)dy + ( x2 + y2)dz from ( 0,0,0 ) to ( 1,1,1 ) where C is the curve x = t, y=t2 , z=t3 in the parameter form. Contd…2

Code No. NR/RR-220202

-2-

Set No. 4

5.a)

Locate and classify the singularities of 4z + 3 f ( z) = 2 z ( z + 1)( z − 1) 3 b) Obtain the Taylor Expansion of e(1+z) in power of (z-1)

6.

x2dx (x +1)(x2+4) Show that the polynomial Z5 + Z3 + 2 Z + 3 has just one zero in the first quadrant of the complex plane. Show that the equation Z4 + 4( 1+i) Z + 1 = 0 has one root in each quadrant. Evaluate

-∞

∫

2

7.a) b) 8.a) b)

Determine the bilinear transformation that maps the points (1-2i, 2+i, 2+3i) into the points (2+i, 1+3i, 4). Prove that the transformation w=sin z maps the families of lines x=constant and y=constant in to two families of confocal central conics. ^^^