Kerala University 2005 B.C.A Computer Application real complex analysis - Question Paper
Reg. No..........................................................(Pages : 3) K 5182
Name.................................
SIXTH SEMESTER B.CA DEGREE EXAMINATION, FEBRUARY/MARCH 2005
(Vocational Course)
Optional Subject : Mathematics Paper XIREAL AND COMPLEX ANALYSIS Time : Three Hours Maximum : 90 Marks
Section A
(Maximum : 40 marks)
Each question carries 5 marks.
1. If A is closed and G is open, prove that
(a) GA is open.
(b) AG is closed.
2. Show that the only limit point of S = ja + : n e nJ is a.
3. Prove that countable union of countable sets is countable.
1 ct
4. Given > 0 and on+ j = + V n e N. Show that {an) converges to -J2.
an 2
5. Prove that (0, II is uncountable.
6. State and prove Cauchys first theorem on limits.
7. Examine the convergence of the series :
v-V2
8. Show that the function fix) = sin x* is continuous and bounded on R, but not uniformly continuous on R.
(_ j)"-1
9. Show that the series ~-converges uniformly on R but not absolutely.
x +n
2 K 5182
10. Define limit point of a sequence. Find the limit superior and limit inferior of the following :
(a) sequence (a) where an = sin ,neN.
3
( l)ft
(b) sequence (an) where an =-, n eN.
11. Show that the exponential function E satisfies E(*+j) = E(x)E(j) for all x,y s R.
12. State and prove Weierstrass M-test.
(8 x 5 = 40 marks)
Section B
(Maximum : 40 marks)
Each question carries 5 marks.
13. Show that the function f(z) = \ z |2 is differentiable at the origin, but not analytic there.
14. Find the equation of the circle described on the line joining 1 + i and 1 - i as diameter.
15. If a function is analytic, prove that it is independent of z.
16. State and prove Liouvilles theorem.
17. State and prove Cauchys Integral formulae.
X 1
18. Expand about z = 1 in :
(a) Taylor's series.
(b) Laurents series.
19. State and prove Cauchys residue theorem.
2 1 2n
20. Using contour integration along the unit circle, show that f-dQ = , a > |W.
21. Using contour integration, evaluate
22. Find the bilinear transformations which maps the points -i,o,i into - l,i, 1 respectively.
1 + 2 2+1
23. Show that both the transformations w --, w =- map the left half plane Re (z) < 0
1-2 2-1
onto |w| < 1.
24. Discuss the transformation w = Vz.
Section C
Answer all the five questions.
Each question carries 2 marks.
25. If a > 0, show that lim - = 0.
n_oo (1 + a)
26. Define interior of a set and prove that it is always open.
27. Show that z = 0 is an essential singularity of the function sin
28. If f (z) and f (z) are analytic in a region, show that f (z) is constant in that region.
29. Find an analytic function with real part 2 xy.
(5 x 2 = 10 marks)
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