Mathematics Iii Set1 Rr220202

NR

Code No: RR220202

II B.Tech II Semester Supplimentary Examinations, Aug/Sep 2007 MATHEMATICS-III ( Common to Electrical & Electronic Engineering, Electronics & Communication Engineering, Electronics & Instrumentation Engineering, Electronics & Control Engineering, Electronics & Telematics, Metallurgy & Material Technology, Aeronautical Engineering and Instrumentation & Control Engineering) Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ????? 1. (a) Prove that Γ(2n) =

22n−1 √ Γ(n)Γ(n π

(b) Prove that β(m, n) =

R1 0

+ 12 ).

xp−1 +xq−1 dx (1+x)m+n

[8+8] 2

d y 2 dy 2 2 2. (a) Show that Jn (x) satisfies the differential equation x2 dx 2 +x dx +(x −n )y = 0

(b) Show that Pn (1) = 1 and Pn (x) = (-1)n Pn (x).

[8+8]

3. (a) Show that w = z n (n , a positive integer) is analytic and find it’s derivative. (b) If w = f(z) is an analytic function, then prove that the family of curves defined by u(x,y) = constant cuts orthogonally the family of curves v(x,y) = constant. (c) If α + iβ = tanh (x + i π/4) prove that α2 + β 2 =1 [6+5+5] R Cos z−sin z dz with c: | z | = 2 using Cauchy’s integral formula 4. (a) Evaluate (z+i)3 c

(b) Evaluate

R(

2 + i2x + 1 + iy)dz) along (1-i) to (2+i) using Cauchy’s integral

1−i

formula. 5. Expand

[8+8]

1 z(z 2 −3z+2)

for the regions

(a) 0 < |z| < 1 (b) 1 < |z| < 2 (c) |z| > 2

[16]

1 6. (a) Find the poles and the residue at each pole of the function f(z) = (z2 +4) 2. R (3z−4) (b) Evaluate z(z−1) dz where c is the circle |z| =2 using residue theorem. [8+8] c

7. (a) Show that

R2π 0

dθ a+bsinθ

=

√ 2π a2 −b2

, a > b> 0 using residue theorem.

(b) Evaluate by contour integration

R∞ 0

dx 1+x2

1 of 2

[8+8]

NR

Code No: RR220202

8. (a) Find the bilinear transformation which maps the points (–1, i, 1+i ) onto the points (0, 2i, 1-i) (b) Discuss the transformation of w=ez in detail. ?????

2 of 2

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