Mathematics Iii April2003 Or 320360

Code No.: 320360 III-B.Tech. II-Semester Examinations April 2003

OR

MATHEMATICS-III (common to Mechanical Engineering, Production Engineering, Mechanical Manufacturing Engineering.) Time: 3 Hours

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

1.a)

Prove that β(m,n) =

Γ ( m) Γ ( n ) Γ ( m + n)

1

b)

2.a)

b) 3.a)

b) 4.a) b)

(−1 ) n ∠n Prove that ∫ x (logx) dx = (m + 1) n +1 0 Where n is a positive integer and m>-1 m

n

Examine the nature of the function x 2 y 5 ( x + iy ) f(z) = , Z≠0 x 4 + y 10 f(0) = 0 in the region including the origin. If f(z) = u(xy) + iv(xy) be an analytic function prove that the function u(xy) and v(xy) satisfy Laplace’s equation. If w = f(z) is an analytic function of z, prove that  ∂2 ∂2   2 + 2  log f ' ( z ) = 0 ∂y   ∂x Find an analytic function f(z) whose real part u = ex (x cosy-y siny) State and prove Cauchy’s Integral Theorem 1 Find the expansion f(z) = 2 (1 + z )( z + 2) When (i) z <1(ii) 1< z < 2

5.a)

Evaluate

∫z

C

2

(iii) z >2

z+4 dz Where C is the circle + 2z + 5

(ii) z + 1 − i = 2 (iii) z + 1 + i = 2 zdz 1 Evaluate ∫ 2 Where C is z − 2 = ( z − 1 )( z − 2 ) 2 C (i) z = 1

b)

Contd…2

Code No.: 320360 6.a)

b)

-2-

OR

By the method of Contour integration, prove that Π Cos 2θ Πa 2 ∫0 1 − 2aCosθ + a 2 dθ = 1 − a 2 , (-1
7.a) b)

Discuss the transformation W = Sinz Find the bilinear transformation which maps the points -1, 0, 1 in to the points 0, -1, ∞ respectively.

8.a) b)

Prove that all the roots of Z7-5Z3+12 = 0 lie between the circle z = 1 and z = 2 State and prove Argment principle. ^^^