Mathematics Iii May2004 Or 320360

Code No: 320360 III-B.Tech II-Semester Supplementary Examinations- May, 2004 MATHEMATICS - III (Common to Mechanical Engineering and Production Engineering) Time: 3 hours.

Max. Marks: 70 Answer any FIVE questions All questions carry equal marks ---

1.a)

1 π  Prove that Γ(n) Γ n +  = 2 n −1 Γ(2n) 2 2  b

b)

∫ ( x − a)

m

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

a

2.a)

Prove that function f(z) defined by x 3 (1 + i ) − y 3 (1 − i ) f ( z) = , z ≠0 x2 + y2 =0 , z =0 is continuous and the Cauchy Riemann equations are satisfied at the origin Yet f 1 (0) does not exist. b) If f(z) = u + iv is an analysis function, show that u and v satisfy Laplaces equation

3.a) b)

2  ∂2 ∂2  2 1 If f(z) is a regular function of z, prove that  2 + 2  f ( z ) = 4 f ( z ) ∂y   ∂x Determine the analytic function whose real part is e2x(x cos 2y – y sin 2y)

4.a)

State and prove Cauchy’s Integral Theorem

b)

Using cauchy’s Integral formula, evaluate. ∫

5.a) b)

e2z dz around the circle z − 1 = 2 ( z + 1) 4

State and prove Laurent’s Theorem Show that when | z +1| < 1. ∞

Z − 2 = 1 + ∑ ( n + 1)( z + 1)

n

n =1

6.a) b)

State and prove couchy’s Residue Theorem z −3 where c in the circle c z + 2z + 5 │z│=1 (ii) │z+1-i│=2 (iii)

Evaluate (i)

∫

2

│z+1+i│=2 (Contd..2)

OR

Code No: 320360

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

Evaluate

-2-

α

dx

∫ α 1+ x −

OR

4

State and prove Fundamental Theorem of Algebra Discuss the transformation W= sin Z Find the transformation which maps the points –1,i,1 of the z- plane onto 1, i,-1 of the w-Plane respectively.

#$#$#.