Mathematics Iii Nov2003 Or 220556

OR Code No: 220556 II B.Tech. II-Semester Supplementary Examinations, November-2003. MATHEMATICS-III (Common to Electrical and Electronics Engineering, Electronics and Communication Engineering, Electronics and Instrumentation Engineering, Bio-Medical Engineering, Electronic and Control Engineering, Metallurgy and Material Technology, Electronics and Telematics,) Time: 3 hours

1.a) b)

2.a)

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

Max. Marks: 70

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: 220556 5.a) b)

:: 2 ::

OR

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. @@@@@