Code No: NR220202 NR

II B.Tech II Semester Supplementary Examinations, Aug/Sep 2006MATHEMATICS-III

( Common to Electrical & Electronic Engineering, Mechanical Engineering,Electronics & Communication Engineering, Electronics & InstrumentationEngineering, Electronics & Control Engineering, Mechatronics, Electronics

& Telematics, Metallurgy & Material Technology and AeronauticalEngineering)

Time: 3 hours Max Marks: 80Answer any FIVE Questions

All Questions carry equal marks? ? ? ? ?

1. (a) Evaluate/2

0

cot d.

(b) prove that (n+ 12) =

4n(2n+1)(n+1)

(c) If m>0, n>0, then prove that 1n(m,n+ 1) = 1

m(n+1, m) = (m,n)

m+n[5+5+6]

2. (a) Prove that1

1(x2 1)Pn+1P 1

n

dx = 2n(n+1)(2n+1)(2n+3))

(b) Prove that J3/2(x) =

2x

[

sinxx

cos x]

[8+8]

3. (a) Define analyticity of a complex function at a point P and in a domain D.Prove that the real and imaginary parts of an analytic function satisfy CauchyRiemann Equations.

(b) Show that the function defined by f(z) = x3(1+i)y3(1i)

x2+y2at z 6= 0 and f(0) = 0

is continuous and satisfies C-R equations at the origin but f (0) does not exist.[8+8]

4. (a) Evaluate

c

ez sin 2z1 dzz2(z+2)2

where c is | z | = 1/2 using Cauchys integral formula

(b) Evaluate1+i

0

(x y2+ix3)dzAlong the real axis from z=0 to z=1 using Cauchysintegral formula

(c) Evaluate

c

e2zz2 dz(z1)3(z+2) where c is | z + 2 | = 1 using Cauchys integral formula

[5+5+6]

5. (a) Expand cos h z as a maclaurins series if |z| < (b) Find the Laurent series expansion of the function z

2 1(z+2) (z+3)

if 2 < |z| < 3[8+8]

6. (a) Find the poles, of f(z) and the residues of the poles which lie on imaginary

axis if f(z) = (z2+2z)

(z+1)2(z2+4)

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Code No: NR220202 NR

(b) Evaluate

C

e2z

(z+1)3using residue theorem. [8+8]

7. (a) State and prove fundamental theorem of algebra.

(b) Evaluate =

0

x2dx(x2+1)2(x2+9)

using residue theorem. [8+8]

8. (a) Show that the transformation w=z+1/z maps the circle |z| =c into the ellipseu=(c+1/c) cos , v =(c-1/c)sin. Also discuss the case when c=1 in detail.

(b) Find the bilinear transformation which maps the points (2, i, -2) into thepoints (l, i, -l). [8+8]

? ? ? ? ?

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