Mathematics Iii Jan2003 Or 220556

OR Code No.220556 II-B.Tech. II-Semester –Supplementary-Examinations January-2003. MATHEMATICS-III (common to Electrical and Electronics Engineering, Electronics and Communication Engineering, Metallurgical Engineering, Electronics and Instrumentation Engineering, Electronics and Control Engineering, Metallurgy and Material Technology, Electronics and Telematics.) Time: 3 hours

1.a) b) 2.a)

Max. Marks:70 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 Prove that 22n-1 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. ( z − 1 )( z − 2 ) c Contd…….2

Code No.220556

-2-

OR

6.a)

Evaluate

12 z − 7

∫ (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

(x2 + 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. ---