Mathematics Iii Nov2003 Or 320360

OR

Code No: 320360

III B.Tech. II-Semester Supplementary Examinations, November-2003. MATHEMATICS-III (Common to Mechanical Engineering, Production Engineering, Mechanical Manufacturing Engineering) Time: 3 hours

1.a) b)

Answer any FIVE questions All questions carry equal marks --1 π Γ ( 2 m) . Prove that Γ(m) Γ(m + ) = 2 m 2 2 −1 Prove that b

∫a ( x − a) 2.a)

b) 3.a)

b) 4.a) b)

Max. Marks: 70

m

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

Prove that the f(z) defined by x 3 (1 + i ) − y 3 (1 − i ) f ( z) = , ( z ≠ 0) x2 + y2 f(0)=0 is continuous and the Cauchy’s Riemann equations are satisfied at the origin, yet f '(0) does not exist. If f(z)=u(xy)+iv(xy) be analytic function, show that u(xy)=c1 and v(xy)=c2 form an orthogonal system of curves. If f(z) is a regular function of Z, prove that  ∂2 ∂ 2   + | f ( z ) | 2 = 4 | f ′( z ) | 2 .  ∂x 2 ∂y 2    Find the regular function whose real part is u = e2x(x cos 2y – y zsin 2y). State and prove Cauchy’s Integral formula. z+3 Expand f ( z ) = when z ( z 2 − z − 2) (i) |z|<1 (ii) 1<|z|<2 (iii) |z|>2.

Contd…..2

Code No:320360 5.a)

-2z −1

∫ ( z + 1) 2 ( z − 2) dz where c is |z-i| = 2.

Evaluate

c

b)

3z 2 + 2

∫ ( z − 1)( z 2 + 9) dz

Evaluate

where c is |z-2| = 2.

c

6.a)

Use the residue theory to show that 2π

dθ

∫

b)

=

2πa

where a>0, b>0, a>b. 2 3/ 2 ( a + b cos θ ) ( a − b ) 0 Apply the Calulus of residue to prove that ∞

∫

2

dx

− ∞ (1 +

x 2 )3

=

2

3π . 8

7.a) b)

Discuss the transformation ω = log z Find the bilinear transformations that maps the points 2, i, -2 into 1, i, -1 respectively.

8.a)

Prove that one root of the equations z4+z3+1 = 0 lies in the first quadrant. State and prove Rouche’s theorem.

b)

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OR