Mathematics Iii Nov2003 Nr 220202

Set No.

1

Code No: 220202 II B.Tech. II-Semester Supplementary Examinations, November-2003. MATHEMATICS-III (Common to Electrical and Electronics Engineering, Mechanical Engineering, Electronics and Communication Engineering, Electronics and Instrumentation Engineering, Electronic and Control Engineering, Metallurgy and Material Technology, Mechatronics, Electronics and Telematics, Aeronautical Engineering) Time: 3 hours

1.a) b)

2.a)

b) 3.a) b) c) 4.a)

Max. Marks: 80

Answer any FIVE questions All questions carry equal marks --Define Gamma function and evaluate Γ(½). Show that Γ(½)Γ(2n) = 22n-1 Γn Γ(n+½).

Write down the power series expantion for J n ( x ) and hence show that 2 J 1/ 2 ( x ) = sin x. πx d n x J n ( x ) = x n J n −1 ( x ) . show that dx

[

]

For w = exp(z2), find u and v and prove that the curves u(x,y) = c1 and v(x,y) = c2 where c1 and c2 are constants cut orthogonally. Prove that the function f(z) = z is not analytic at any point. Find the general and the principal values of (i) log e (1+ 3 i) (ii) log e(-1). Evaluate using Cauchy’s integral formula

∫

( z + 1) dz z2 + 2z + 4

c b)

where c is | z + 1 + i | = 2. Show that

∫

x dx – y dy √ ( x2 + y 2 )

C Is independent of any path of integration which does not pass through the origin. Contd…..2.

Code No: 220202 5.a) b)

:: 2 ::

Set No:1

State and derive Laurent’s series for an analytic function f (z). 1 Expand 2 in the region (i) 0 < | z – 1 | < 1 (ii) 1 < | z | < 2. ( z − 3 z + 2)

6.a)

Determine the poles of the function f(z)=tan z and the residue at each of the poles. b) Using residue theorem integrate ∫(4-3z)/ (z2 - z) dz.

7.a)

b) 8.a) b)

Use Rouche’s theorem to show that the equation Z5 + 15 Z +1=0 3 3 has one toot in the disc | Z | < and four roots in the annulus < | Z | < 2. 2 2 State and prove fundamental theorem of algebra. Define conformal mapping? Let f(z) be analytic function of z in a domain D of the z-plane and let f’(z)≠0 in D. Then show that w=f(z) is a conformal mapping at all points of D. @@@@@

Set No.

2

Code No: 220202 II B.Tech. II-Semester Supplementary Examinations, November-2003. MATHEMATICS-III (Common to Electrical and Electronics Engineering, Mechanical Engineering, Electronics and Communication Engineering, Electronics and Instrumentation Engineering, Electronic and Control Engineering, Metallurgy and Material Technology, Mechatronics, Electronics and Telematics, Aeronautical Engineering) Time: 3 hours

1.a)

Max. Marks: 80

Answer any FIVE questions All questions carry equal marks --Define Beta function and show that β(m,n) = β(n,m). 1

b)

Show that

∫ 0

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

xn 1− x2

dx =

2.4.6....(n − 1) where n is an odd integer. 13 . .5.... n

n J n ( x ) − J n '( x ) = J n +1 ( x ) . x 2 Show that ∫ J 3 ( x ) dx = − J 2 ( x ) − J 1 ( x ) . x Prove that

Find all values of z which satisfy (i) ez = 1+i (ii) sinz =2. Prove that the function f(z) = xy + iy is everywhere continuous but not analytic. Find the values of z for which the function f(z) = cosec z is continuous. Evaluate

∫

ez

dz

( z-1-i )3

C

where c : | z- 1 | = 3. b) 5.a)

b)

State and prove Cauchy’s integral theorem. Find the poles and zeroes of 1 z (e z − 1) Show that when | z + 1 | < 1, z −2 = 1+

∞

∑ (n + 1)( z + 1) n

n =1

.

Contd…..2.

Code No: 220202

:: 2 ::

Set No:2

6.

Evaluate the following integrals over a unit circle C. 1) ∫ (2z+1)2 /(4z3 + z) dz 2) ∫ ez sec∏ z dz.

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)

Prove that the transformation W=1/z transforms circles and straight lines in the z-plane into circles or straight lines in the w-plane. Find the bilinear transformation which maps the points (–i, o, i) into the point (–l, i, l) respectively.

b)

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Set No.

3

Code No: 220202 II B.Tech. II-Semester Supplementary Examinations, November-2003. MATHEMATICS-III (Common to Electrical and Electronics Engineering, Mechanical Engineering, Electronics and Communication Engineering, Electronics and Instrumentation Engineering, Electronic and Control Engineering, Metallurgy and Material Technology, Mechatronics, Electronics and Telematics, Aeronautical Engineering) Time: 3 hours

1.a)

Max. Marks: 80

Answer any FIVE questions All questions carry equal marks --Show that β(m1n) = Γ( m ) Γ( n ) / Γ( m + n) . π /2

b)

Show that

∫

tan θ dθ =

Γ (1 / 4) Γ (3 / 4)

0

2

.

1

2.a)

Using Rodrigue’s formula prove that

∫x

m

Pn ( x)dx = 0 if m < n.

−1

b)

2 2 2 Prove that J 0 + 2 ( J 1 + J 2 + ..........) =1.

3.a) b)

Determine whether the function 2xy + i(x2- y2) is analytic. Find an analytic function whose imaginary part is e-x(x cos y + y sin y).

4.a)

Evaluate z 3 − Sin 3 z dz ∫c π (z − )3 2 with c : | z| = 2

b)

Evaluate

∫

dz ez ( z - 1 )3

c

5.

.

where c : | z| = 2 , using Cauchy’s Integral Formula

Find the Laurent series expansion of the function z2 −1 (i) about z = 0 in the region 2 < | z | < 3 z 2 + 5z + 6 (ii)

z 2 − 6z −1 in the region 3 < | z + 2 | < 5. ( z − 1)( z − 3)( z + 2) Contd…..2.

Code No: 220202

:: 2 ::

Set No:3

6.

Evaluate the following integrals 1) ∫ 2z2 + 3 / z(z+1)(z+2) where C:│z│ = 1.6 (2 ∫ dz/ z2 (z+2) where C: │z-1│ = 2.

7.a)

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.

b) 8.a) b)

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. Find the bilinear transformation which maps the points( l, -i, -l) into the point (i, o, -i). @@@@@

Set No.

4

Code No: 220202 II B.Tech. II-Semester Supplementary Examinations, November-2003. MATHEMATICS-III (Common to Electrical and Electronics Engineering, Mechanical Engineering, Electronics and Communication Engineering, Electronics and Instrumentation Engineering, Electronic and Control Engineering, Metallurgy and Material Technology, Mechatronics, Electronics and Telematics, Aeronautical Engineering) Time: 3 hours

Max. Marks: 80 Answer any FIVE questions All questions carry equal marks --1

1.a)

Show that

∫ 0

π /2

b)

Prove that

∫

xn 1− x

2

dx =

1.35 . ....(n − 1) π where n is an even integer. 2.4.6.... n 2

cot θ dθ =

Γ (1 / 4) Γ (3 / 4) 2

0

2a) b) 3.a)

Express J5/2(x) in finite form. 2 3 3 Show that x = P3 ( x) + P1 ( x). 5 5 Find the following limits (i)

b)

.

lim z z → ∞ 2 − iz

If u=x2-y2,

v= −

(ii)

lim (3x + iy 2 ) z → 2i

y then show that both u and v are harmonic, but x + y2 2

u+iv is not analytic. 4.

Evaluate

∫

log z

dz

.

( z - 1)3

c

where c : | z - 1| = ( ½ ) , using Cauchy’s Integral Formula.

Contd…..2.

Code No: 220202 5.a)

b)

:: 2 ::

Determine the poles of the function z +1 1 − e2z (i) (ii) z 2 ( z − 2) z4 1 Expand f ( z ) = in the region (i) |z|<1 ( z − 1)( z − 2)

Set No:4

(ii) 1<|z|<2 (iii) |z|>2.

6.

Evaluate the following integrals over a unit circle 1) ∫ (2z +1) /(z+1)3 (z-1) dz c: |z-1| =2.5 wise sense). Π

7.a) b)

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

8.a) b)

Under the transformation w= ----- , find the image of the circle |z|=1 in the w-plane. l-iz Find the bilinear transformation which maps the points (l, i, -l) into the points (o,1,∞). @@@@@