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Posted Date: 25 Oct 2008 Posted By: prashanth beerelli Member Level: Diamond
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2008 Jawaharlal Nehru Technological University Metallurgy & Material Technology II B.Tech Supplimentary Examinations, Aug/Sep 2008,MATHEMATICS-III Question paper
Code No: R05221801 Set No. 1
II B.Tech Supplimentary Examinations, Aug/Sep 2008
MATHEMATICS-III
(Metallurgy & Material Technology)
Time: 3 hours Max Marks: 80
Answer any FIVE Questions
All Questions carry equal marks
? ? ? ? ?
1. (a) Prove that ??(n) =
1
R0
[log . 1
x ]n-1dx
(b) Prove that
1
R0
x8(1-x6)dx
(1+x)24 = 0 using ß - ?? functions. [8+8]
2. (a) Show that Jn(x) satisfies the differential equation x2 d2y
dx2 +xdy
dx +(x2-n2)y = 0.
(b) Show that Pn(1) = 1 and Pn (–x) = (–1)n Pn (x). [8+8]
3. (a) Find an analytic function f(z) = u(r,?)+ i v(r,?) such that
u(r,?) = r2cos2? – r cos? + 2.
(b) Determine ‘p’ such that the function f(z) = 1
2 loge(x2+y2) + i tan-1 px
y is
an analytic function. [8+8]
4. (a) Evaluate RC
Cos z-sin z dz
(z+/2)3 with C: | z | = 2 using Cauchy’s integral formula.
(b) Evaluate
2+i
R 1-i
(2x + 1 + iy)dz along (1-i) to (2+i) using Cauchy’s integral for-
mula. [8+8]
5. (a) State and prove Taylor’s theorem.
(b) Find the Laurent series expansion of the function
z2
-6z-1
(z-1)(z-3)(z+2) in the region 3< |z+2| <5. [8+8]
6. (a) Find the poles and the corresponding residues of the function z3
(z-1)(z-2)(z-3) .
(b) Evaluate RC
(3z+5)dz
(z2
-1)2 where c is |z| = 1 by residue theorem. [8+8]
7. (a) Evaluate
2
R0
d
(5-3cos)2 using residue theorem.
(b) Evaluate
1
R0
sin mx
x dx using residue theorem. [8+8]
8. (a) Find the image of 1< |z| <2 under the transformation w= 2iz+1.
(b) Find the bilinear transformation which maps the points (–1, i, 1+i ) onto the
points (0, 2i, 1-i).
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