Madurai Kamraj University (MKU) 2007 B.Sc Mathematics Real and Complex analysis - Question Paper
Real and Complex analysis
OCTOBER 2007
(6 pages)
6251/M31
Paper VI REAL AND COMPLEX ANALYSIS
(For those who joined in July 2003 and after)
Time : Three hours Maximum : 100 marks
SECTION A (8 x 5 = 40 marks)
Answer any EIGHT questions.
All questions carry equal marks.
1. (0,1] GiwasBf)0)60 OT6ffrp rSlpaja.
Prove that (0,1] is uncountable.
2. @0 (y3(LpinijDujiT65r QLDilifl Q&JciflaDiLi ajsoijiup.
Q6U6lfllL)li) (tp(J0DLDUjrT65TGgj OT65Tp rlp6ffi.
Define a complete metric space. Prove that discrete space is complete.
3. f : [0, l] - R / (re) = x2 crstTp Gucwijujpffiffiu-
Ulll*|-0LJlijl65T #IJ[T65I Qm_irff#lLL|S5)l_UJ ffmTL| OT6ffrp
Prove that : [0, l]-R defined by fix) = 2e2 is uniformly continuous.
4. @ss)L_iLirT65T loul GgrbjDjSao er(Lp<l
State and prove intermediate value theorem.
5. $0 c5))L_ITUJn-6ST QlDlllflffi Q6U6rfl)lLI ajQDIJUjp.
6ULp<5LDIT6ffr (olLDL_fl@L_6ST ah.Uf.UJ R l_ITUJIT65rg] 0)60 OT6imp Ilp64.
Define a compact metric space. Prove that R with usual metric is not compact.
6. 60 i_iia;ilujrrT GlLDL_ifl Qaj6rfluS)6T Gldsu ajcnijiuanjD Qa:tLjuJuuilii|.0<50Lb 60 Qrri_iTff#lujrr6BT ffrriTLj ijrTor
Qrri_iTffi#liL|0!)L_ujrTUJ croup jflp&jffi.
Prove that a continuous function defined on a compact metric space is uniformly continuous.
7. Zx iBjDpii) Z2 OT65TU65T rGjSfpLD ffisouGluRfr6iT
Z i . .
= -=r- CT6SrpLD %2
OT6afta) Z1xZ2=Z1xZ2 sretrpLb
Jlp6L|<E.
If Zx and Z2 are any two complex numbers, prove
8. f(z) = ze z CTeiiTn) ffrrrn51(i)0 C-R 5LD65Turr(5iffloaT
ffrflurriT<K<5.
Verify C-R equations for the function
2e~z
9. -1, 0,1 GupsDjD (jpaDoiGuj -1, - i, 1 gisurjiiffilOTT Gldso
LDrrppih QffujiL|Lb 10 GiTGaml ffirr6wr<$..
Find the bilinear transformation that maps -1, 0,1 onto -1, - i, 1 respectively.
10. C OT65rui 1-sb Q<5(Ti_r&j<l l-d> (ipii).iL|Lb | z | = 1 OT6imD
|0DIJ6mll_Lb OTdiflQ) j] Z | Z dz 65T LDluflDU ffirTOTTS.
c
Evaluate j] z | z dz where C is the semicircle
c
| z | = 1 starting at 1 and ending at 1.
11. UJ(T)<K65Bfl )l|.UU6S)L_ GjbfD65) CT(lgl rlp6L|ffi.
State and prove fundamental theorem of algebra.
12. --;-Tuf61 IjDULjU L|6TTeifl<Se5)en' 6WTI_j5ll5gl
(2 sin z -1)2
ajanuu@iffi.
Determine and classify the singular points of
1
Answer any SIX questions.
All questions carry equal marks.
13. R" = {(a, x2, , xn): x;e R for 1 <i < n} er&na. p > 1 mriipLb x, y e R\ d-m 6Li0oijuj0D(D
nVp
crssflo) d, Rn-U 90 QlJDlllflffi
d (x, y) =
Zl ** -yi\
t=i
tnTfaoinni jfs)p6L|5>.
Let R" = {(acj, x2, , jc) : x;e R for 1 < i < n\. Let p > 1 and x, y e R". If d is defined as
Vp
n
Zl -yt\
d (*, y) = Rn.
, prove that d is a metric in
i=1
14. Quiuifl65T C<SL_L_ffiifl GrDjDao <r(Lpl rglp&a.
State and prove Baires category theorem.
15. 6nija>Q&jrr0 il|Dj5gj lani_Qaj6ifliL|m R-ffi@
ewiDLjQurTrTUj l0ffi@Lb OT6rp jlp64<$.
Prove that any open interval is homeomorphic to
rp t m
16. R 60 a_6rrQajsifl Qrn_iTL|6iTisrrmjj I0lju(d@
G5)eillUIT65I)lh GutTlLDrT65TglLDIT6BT rS)uijj<5CT)6ffT 0 0Di_Qojcifl CT6ifruC CTwp rSlpGna;.
Prove that a subspace of R is connected if and bnly if it is an interval.
17. Qs)Qsfl Gurrija) GajDrDo) ot(I$ rpes.
State and prove Heine-Borel theorem.
18. Z-y LDfDjpm Z2, ZZ + aZ + aZ + /3 = 0 otostid qjL-i_>D3)u Qurrpgj &,<S6)&),\g L|6TT6rflffi6rrrr l0uurf)@ GjCS)6UUJrT65Tg|Lb Gun-gJLDIT6ST)LD(T65T r1u0O65T
ZjZ2 +ccZl+aZ2+P = 0 otottuG siwp rilpas.
Prove that Z1 and Z2 are inverse points with respect to a circle ZZ + aZ + oZ + j3 = 0 if and only if ZjZ2 + orZ+cifZ2 + )S = 0.
19. u = 3xly + 2x2 - yz - 2y2 ctottjt) <?rriTL| fi0 l65)<Fff
fflT(TL| OT0JTp rlp6ffi. U_irrG6pQl>IT0 )6S)& e5)655TuSl0D6STlLJLb <$rr65ffr<K.
Prove that the function u = 3x2y + 2x2 - y3 -2y2 is harmonic. Also find a Hormonic conjugate.
20. Quauj co@ sulLl-qst Gii>a) UMTjbjDib QffUJiLjih @0 GjsiTGaml LDrrfDjDe?) anmra.
Find the bilinear transformation that maps the real axis onto the unit circle.
5 6251/M31
21. Ol_UJ60lfl65T G<(f)(D0D|$ OT(Lp;l (SlpGlJ. State and prove Taylors theorem.
7 dx
22. LD<luiSl(hlffi : -
o>!+a2f
Evaluate : ~jj .
0 \X + a j
6
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