Mathematics Iii Nr 220202 Dec04

Code No: NR-220202 II B.Tech. II-Semester Supplementary Examinations, Nov/Dec-2004 MATHEMATICS-III (Common to Electrical and Electronics Engineering, Mechanical Engineering, Electronics and Communication Engineering, Electronics and Instrumentation Engineering, Electronics and Control Engineering, Metallurgy and Material Technology, Mechatronics, Electronics and Telematics and Aeronautical Engineering) Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks --1. a) Define Gamma function and evaluate Γ(½). b) Show that Γ(½)Γ(2n) = 22n-1 Γn Γ(n+½). c) Define Beta function and show that β(m,n) = β(n,m).

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2. a) b)

d ( xJ n J n +1 ) = x ( J n2 − J n2+1 ) . dx

Prove that

Express x3+2x2-x-3 in terms of Legendee polynomials.

3. a) b) c)

Derive Cauchy Riemann equation in polar coordinates. 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).]

4. a)

State and prove Cauchy’s integral theorem.

b)

e 2 dz z −1 = 3 Evaluate using Cauchy’s integral formula ∫ 3 where c: ( z − 1 − i ) c 1+ i

c)

Evaluate

∫ (x

2

− iy ) along y=x2.

0

5. a) b)

State and prove Taylor’s theorem. Find the Laurent series expansion of the function

z 2 − 6z − 1 in the ( z − 1)( z − 3)( z + 2)

region 3<|z+2|<5. 6. a) b)

State and prove Cauchy’s Residue Theorem. Z2 Find the residue of 4 at these Singular points which lie inside the circle Z −1 | Z | = 2. Contd…2

Code No: NR-220202 2π

7. a)

Show that

∫

0

dθ = a + bSinθ

.2. 2π

∫

0

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dθ a + bSinθ

=

2π a 2 − b2

, a > b> 0 using residue

theorem. ∞

b)

Evaluate by contour integration

dx

∫1+ x

2

.

0

8. a)

b)

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. Find the bilinear transformation which maps the points (–i, o, i) into the point (–l, i, l) respectively. ###