Mathematics Iii Jan2003 Nr 220202

Set No.

1

Code No.220202

II-B.Tech. II-Semester –Supplementary-Examinations January-2003. MATHEMATICS-III (common to Electrical and Electronics Engineering, Mechanical Engineering, Electronics and Communication Engineering, Metallurgical Engineering, Electronics and Instrumentation Engineering, Electronics and Control Engineering, Metallurgy and Material Technology, Mechatronics, Electronics and Telematics, Aeronautical Engineering) Time: 3 hours

1.a) b) 2.a)

Max. Marks:80

Answer any FIVE questions All questions carry equal marks --2 1 dx 1 x dx = Π/4 Show that ∫0 ∫ 0 4 4 1 − x 1− x 2n-1 Prove that 2 Π (n) Π (n+ ½)= Π (2n) Π 2 x nJn-(n+2)Jn+2+(n+4) Jn+4……… And hence deduce that x 2 ……….. Jn=(n+1)Jn+1-(n+3) Jn+3+(n+5)Jn+5 Prove that Jn-1= 1

b)

Prove that

∫ pm ( x) pn ( x)dx = 0 if m ≠ n

−1

3.a) b)

2 if m=n. 2n + 1 Prove that Zn is analytic and hence find its derivative. If u(x, y) and v(x,y) are harmonic functions in a region R, prove that the function  ∂u ∂v   ∂u ∂v   −  +i  +  is an analytic function.  ∂y ∂x   ∂x ∂y 

4.a)

(i) Find the image of |Z| =2 under the transformation ω =3z. (ii) Find the points at which ω = cos hz is not conformal.

=

b)

5.a) b)

2 2 Find the conjugate harmonic of u = e x − y cos 2 xy . Hence find f(z) in terms of z.

State and prove Cauchy’s integral theorem. cos Π z 2 Evaluate ∫ 3 where C is |z| =3 by using Cauchy’s integral Formula. c ( z − 1)( z − 2) Contd…….2

Code No.220202 6.a)

Evaluate

-212 z − 7

Set No.1

∫ (2 z + 3)( z − 1) 2 dz where C is x +y =4. 2

2

c

b)

(i) Calculate the residue at z=0 of f(z)= ∞

∫0 7.a) b) 8.a) b)

dx

1+ ez z cos 2 + sin 2

(ii) Evaluate

2

(x + a2 )2

Expand log(1-2) when |z|<1 using Taylor series. ∞ 1 + 2 cos θ dθ Evaluate ∫ 0 5 + 4 cos θ Find the bilinear transformation which transform the points ∞ , i, o in the Z-plane into o, i, ∞ in the w-plane. Prove that P(z)= ao+a1z+a2z2+……+an zn, an ≠ 0 has exactly n(-roots) – zeros. ---

Set No.

2

Code No.220202

II-B.Tech. II-Semester –Supplementary-Examinations January-2003. MATHEMATICS-III (common to Electrical and Electronics Engineering, Mechanical Engineering, Electronics and Communication Engineering, Metallurgical Engineering, Electronics and Instrumentation Engineering, Electronics 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) 4.a)

b)

5.a) b)

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

∞ x m −1 (ii) ∫o (1 + x ) m + n dx ∫ Prove that (1-2xh+h2)-1/2 is the generating function of Legendre polynomial.

Evaluate (i)

1 m 1 x  log  0 x 

n

dx

J ′ Prove that (i) h Pn= xPn ′ − Pn ′ − 1 (ii) J 2 = J o ″ − o x 2 2 2 2 Prove that Jo +2(J1 +J2 +J3 +…..)=1 Derive Cauchy’s Riemann Equation in Cartesian co-ordinates. 1 Obtain Laurent’s expansion for f ( z ) = in (i) |z|<2 ( z + 2)(1 + z ) 2

(ii) |1+z|>1.

(i) Find the bilinear transformation which maps the points ∞ , i, o in the z-plane x into -1, -i, 1 in the w-plane. (ii) Show that u = 2 is harmonic. x + y2 sin 2 x Find f(z) =u+iv given that u+v= cosh 2 y − cos 2 x Evaluate

2+i 2 z dz o

∫

along the imaginary axis to i and o horizontally to 2+i.

e z dz Evaluate ∫ 3 if (i) 0 lies inside C and 1 lies outside C (ii) 1 lies inside C c z (1 − z ) and 0 lies outside C (iii) both lie inside C.

Code No.220202

-2-

Contd…..2 Set No.2

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

cos 3θ dθ 5 − 4 cosθ State and prove the sufficient condition for w=f(z) to be conformal at the point zo. State and prove Rouche’s theorem. ∞ x 2 dx Evaluate ∫− ∞ 2 x +1 x2 + 4 Evaluate

2Π

∫o

(

)(

)

Find the image of the triangular region in the z-plane bounded by the lines x=0, y=0 and x+y=1 under the transformation w=2z. ∞ Sinxdx (a > 0) . Evaluate ∫o x x2 + a2

(

)

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Set No. Code No.220202

3

II-B.Tech. II-Semester –Supplementary-Examinations January-2003. MATHEMATICS-III (common to Electrical and Electronics Engineering, Mechanical Engineering, Electronics and Communication Engineering, Metallurgical Engineering, Electronics and Instrumentation Engineering, Electronics and Control Engineering, Metallurgy and Material Technology, Mechatronics, Electronics and Telematics, Aeronautical Engineering) Time: 3 hours

1.a) b)

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

Prove that β (m, n) = Evaluate

∫∫∫ xyz ∨

Π ( m )Π ( n ) Π ( m + n)

dx dy dz taken over the volume V of the tetrahedron given by x

≥ 0, y ≥ 0, z ≥ 0 and x+y+z ≤ 1. 2.a) b)

State and derive the Rodrigue’s Formula of Legendre polynomial. 3x 2 − 1 Show that Pn(x)=1, P1(x)=x1 P2(x)= and hence express 2x2-4x+2 as a 2 Legendre polynomial.

3.a) b)

Derive the necessary conditions for f(z) to be analytic in Polar-co-ordinates. Find a and b if f(z) = (x2-2xy+ay2)+i (bx2-y2+2xy) is analytic. Hence find f(2) interms of z.

4.a)

Prove that any bilinear transformation maps the totality of circles and straight lines in the z-plane into the totality of circles and straight lines in the w-plane. state and Derive Cauchy’s integral formula.

b) 5.a)

(i) Obtain the Lauvent’s expansion of singular (1,3)

∫(0,0) (3x b)

points 2

and

hence

ez

in the neighbourhood of its ( z − 1) 2 find its residue. (ii) Evaluate

ydx + ( x 3 − 3 y 2 )dy along the curve y=3x. 2Π

Evaluate ∫o

dθ (a + b cosθ ) 2

( a > b > o) Contd……….2

Code No.220202 6.a) b)

7.a)

(i) Evaluate

-2-

∫ tan z dz where C is |z|=2. c

(ii) Expand

Set No.3 1 z 2 − 3z + 2

in 0<|z-1|<1 as

Laurent’s series. Find the image of the triangle with vertices at i, 1+i, 1-i in the z-plane under the transformation w=3z+4-2i. ∞ cos x dx Evaluate ∫o (1 + x 2 ) 2

b)

Show that all the roots of z5+3z2=1 lie inside the circle |z|< 3 4 and that two of its 3 root lie inside the circle |z|< . 4

8.a)

Locate the quadrants in which the roots of the equation z4+z3+4z2+2z+3=0 are situated. ∞ x4 dx Evaluate ∫ − ∞ x 6 −1

b)

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Set No.

4

Code No.220202

II-B.Tech. II-Semester –Supplementary-Examinations January-2003. MATHEMATICS-III (common to Electrical and Electronics Engineering, Mechanical Engineering, Electronics and Communication Engineering, Metallurgical Engineering, Electronics and Instrumentation Engineering, Electronics and Control Engineering, Metallurgy and Material Technology, Mechatronics, Electronics and Telematics, Aeronautical Engineering) Time: 3 hours

1.a)

b) 2.a) b)

Max. Marks:80

Answer any FIVE questions All questions carry equal marks --Prove that (i) π (n + 1) = nπ (n) and hence π (n + 1) =n! π / 2 dθ π /2 sin θ d θ = Π (ii) ∫o ∫ sin θ o 1 2n(n + 1) ( x) ( x) 2 Prove that ∫−1x p n +1 p n −1 dx = ( 2n − 1)(2n + 1)(2n + 3) Show that Pn(1)=1, Pn(-x) = (-1)nPn(x). Πx 1 J 3 ( x ) = Sinx − Cosx . Prove that 2 x 2

3.a) b)

4.a) b)

(i) If f(z)=4+iv is an analytic function then prove that the family of curves u(x,y)= C1 and v(x,y)=C2 are orthogonal (ii) Prove that u=log(x2+y2) is harmonic. Define bilinear transformation. Find the bilinear transformation that maps 1, i and -1 of the z-plane onto 0, 1 and ∞ of w-plane.

(

Determine the poles and residues of f(z)=

5.a) b)

)

If f(z) is analytic prove that ∂ 2 / ∂x 2 + ∂ 2 / ∂y 2 |f(z)|p=p2|f'(z)|2|f(z)|p-2. 7z − 2 (i) Find the Laurent’s expansion of f(z)= in 1<|z+1|<3. ( z + 1) z ( z − 2)

(ii)

z2 ( z + 2) z ( z − 1) 2

State and prove Cauchy’s Residue theorem and using it evaluate

∫z

2 1/ 2

e

dz

c

where C is |z|=1. Discuss the transformation w=cos hz. Contd……2

Code No.220202 6.a)

Evaluate

-2z 3 − 2z + 1

∫

( z − i) 2

c

b)

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

Set No.4

where C is |z|=2 by using Cauchy’s integral Formula.

Using Contour integration evaluate ∫

2Π

o

cosθ dθ 3 + sin θ

dx o x + a4 Prove that e4z=ez has a unique root in |z| ≤ 1 and that root is real and positive. Evaluate

∫

∞

4

x sin mx dx x2 + a2 Find the images of the following under the transformation w=ez. (i) the line y=x (ii) the segment of y-axis given by 0 ≤ y ≤ Π (iii) the left half of the strip 0 ≤ y ≤ Π and (iv) the right half of the strip 0 ≤ y ≤ Π . Evaluate

∫

∞

o

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