askiseis migadikis analysis
DESCRIPTION
Ασκήσεις μιαγαδικής ανάλυσηςTRANSCRIPT
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EvpfJoAtapo(: JR = OU'VOAO 1'W'V rtpct.Yf!Ci."t'LXW'\1 ct.pL'I')fLW'V,
C = OU'VOAO 1'W'V f!LYct.OLXW'\1 ct.pL'\')f!W'V,
lDJ = { Z E C : JzJ < 1 }, o ct.'llotX1'0<; f!O'Vct.Otct.(o<; o(oxo<; 0'1'0 C, D( a, r) = { z E C : Jz - aJ < r} o ct.'VOLX1'0<; o(oxo<; f!€ x€v1'po a Xct.L ct.X1'L'Vct. r > 0, Re(z) = rtpct.yf!ct.nxo f!€po<; 1'0U z, Im(z) = cpct.v1'ct.01'LXO f!€po<; 1'ou z,
1. fpct.tjJ€1'€ 01'7)'11 f!OPCfl7J a + f3i 1'0U<; f!LYct.OLXOU<;
1 (1+i)(2-i) (3- 2i)(2 + i).
2. Auon<: 1'7)'11 €~LOW07j z2 - iz + 1 = 0.
3. ~<:L~€1'€ 01'1 JJzJ-JwJJ ::; lz + wJ ::; Jzl + JwJ ytct. xa.'\')g z, wE C.
4. ~<:1~€1'€ 01'1 JRe(z)l ::; lzl ::; IRe(z)l + IIm(z)l Xct.1 llm(z)l ::; lzl ::; IRe(z)l + IIm(z)l y1ct. xa.'\')g z E C.
5. Ilt:ptypct.tjJ€1'€ yEWfLE1'ptXct. 1'Ci. OU'VOACi.
6.
7.
8.
A= {z E C: Jz -11 < ~}, B = {z E C: Im ( z- (~ + 2i)) = 0},
r = {z E C: lm ( z -1(~ ~ 2i)) = 0}, ~ = {z E C: lzl2
- 3lzl + 2 = 0}.
E = {z E C: Re(z2) = 1}, Z = {z E C: Re(z) · Im(z) = 1},
lz-11 H = {z E C: -
1 -.
1 = 1}, 8 = {z E C: lzl = Arg(z)}.
z-z
Av z, wE c XCi.1 lzl < 1, lwl < 1, od~€1'E 01'1 ~~~~< 1.
ftct. a, b E C, b # 0, rt<:ptyp&tj;t:1'E y<:Wf!E1'ptx& 1'ct. o6v 1-.a.:
z-a z-a A= {z E C: Im(-b-) = 0}, II= {z E C: Im(-b-) > 0}.
Bpd1't: 1'ct. 07Jf!dct. ouoowp<:uo<:w<; "CW'II a.xol..ou'\')twv:
1 ( ·2n)n Zn = in+ ;;;,' Wn = 1 + ~
9. ~d~t:1'E 01'L 7J ouv&p1'7J07J f(z) = JiXYT, (z = x + iy), tXct.'Vortmd n~ ouv'\')~x<:~ ChauchyRiemann o"Co z = 0, ct.f.A.& 0€'11 €Xt:L f!LYct.OLX~ rtct.p&ywyo 01'0 z = 0.
10. ~d~€1'€ 01'1 7J E~LOW07J 1'0U YE'VLXEUf!€'\IOU XUXAOU 01'0 €1tLrtEOO EL'VCi.L
alzl 2 + f3Re(z) + "'(lm(z) + J = 0, a, /3, "'(, J E R
Ilo1'E a.uw~ o xux'Ao<; Et'Vct.t gu'l')e;1a.;
11. Eo"CW f(z) = ~-(a.) ~<:t~E1'€ 01't ct.'\1 S €1'\lct.t <:vct.c; xuxt..o~ o"Co <:rtme:oo, 07JA. S = { z E C : lz - al = r} ytct.
xa.rtmo a E C xa.1 r > 0, 1'01'€ f(S) €1'11ct.1 xuxA.o~ ~ cu'\')<:Lct..
(~) ~<:L~E1'€ on a.v L El'Vct.t w'\')<:tct. mo Ertme:oo 1'01'€ f(L) E1'11ct.t xux'Aoc; ~ w'\')e:tct..
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1. BpE~-rE -r~<; cr-rEpE:Oypctcp~XE<; E~XO'VE<; -rwv 01]flE~wv 2 + 3i, 1 + i, i
2. BpEt-rE -roue; flLY<XOtxouc; ctpL't)flOU<; 1tOU <XV"ttcr-rmxouv a-ret 1t<Xp<XX<X"t'W 01JflEL<X 1'7]<; crcpcttpct<; -rou
Riemmann flEOW UJ<; mEpE:OypctcptXY)<; 1tpo~OAY)<; (1/v'B, 1/4, 3/4), (0, -/3/4, 1/4),
3. 8E"tOUflE f ( Z) = f (X + iy) = x2+y4 ' Z O X <XL ypctcpOUflE f = U + iv. .6.E~~E"tE O"tt:
{
xy2 (x+iy) =/=
0, z=O. ( 1) Ot flEptxE<; 1tctpctywym -rwv u, v we; 1tpoc; x, y U1tctpxouv cr-ro z0 = 0, (2) Ot cruvfi7]XE<; Cauchy-Riemann ~X<X'V01tOLouv-r<XL cr-ro z0 = 0, (3) Tct opt<X limz_.o /(zl=~(O) U1tctpxouv o-rctv z -+ 0 X<X"t<X flY)XO<; 01tOL<Xcr07]1tD-rE EUilEt<X<; 1tOU
m:pvct <X1to -ro 0, X<XL Y) "tLflY) -rou optou E~'V<XL Y) tOL<X y~ct xctilE -rnot<X EUilEt<X, (4) H f(z) OE'V EL'V<XL fl~Y<XOLX<X 1t<Xpctywytatf1.Y) mo z0 = 0 (T1toO. ilEwpEtcr-rE 1t<Xp~OAE<; x = y2
)
4. BpE(-rE -rL<; E~XO'VE<; "tW'V 1t<Xp<XXchw crU'VOAW'V flEOW "t7]<; EXilE"tLX~<; cruv&p-rY)OY)<; j ( z) = ez. <X. A= {z: -2 < Re(z) ~ -1}. ~- B = {z: 0 < Im(z) < ~}. y. C = {z: -1 < Re(z) < 1,-1 < Im(z) < 1}.
5. BpEt-rE OAE<; 1'~<; "tLflE<; -rou z E C yt<X "t'L<; 01tOLE<; ez = 1 + i
6. BpEt-rE OAE<; -r~<; "tLflE<; -rou z E C yt<X -rt<; o1tOLE<; cos( z) = 2
7. .6.Et~E-rE o-rt U1t<XPXEL to E lR wa-rE I cos( it) I > 100 y~ct It I > to.
8 . .6.Et~E-rE o·n: (ct) cos(z + w) = cos(z) cos(w)- sin(z) sin(w) yt<X xctilE z, wE C, (~) sin2 (z) + cos2 (z) = 1 y~ct xctilE z E C.
() ~n+1n! t ~ I Z. n. n=l
11. A 'V OL OUV<XflOOEtpE<; 2.::;:::'=1 anzn X <XL 2.:::=1 bnzn EXOUV <XX"tL'VE<; cruyxA.~crY)<; R 1 X<Xt R2 <XV-"t~(HOLX<X, OEL~E"tE o-rt Y) <XX"tL'V<X cruyxA.~crY)<; "t7]<; crEtpct<; 2.:::=1 (an + bn)zn EL'V<XL R ! R 2: min(R1, R2).
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1. Ea-cw u( X' y) : n --+ ~ tJ.E n ClVOLJCCO <JUVEJCCLXO UTI:O<JUVOAO -cou c. y n:oi}noUtJ.E 01"L un:crpxouv
OL tJ.EpLXEt;; n:ctpctywym ~~ XClL ~; XClL tJ.Y)OEVL~OV1"ClL <JE xcri}E O"Y)tJ.ELO -cou n. To-cE Y) u EtvCll
o-ccri}EpY) ouvcrp-cY)OY).
2. Av f : n --+ <C ELVCll ClVClAU1"LX1J 01"0 ClVOLX1"0 <JUVEX1"LXO n XCll Y) u(z) = Re(f(z)) EtvClt
mcri}c:p1J ouvctp1"Y)0"1J -co-cc: 1J f ELVClL o-ccri}c:p1J cruvcrp-cY)OTJ.
3. Av f : n--+ <C ELVCll ClVClAU1"LX1J 0"1"0 ClVOLX1"0 <JUVEX1"LXO n ){Cll 1J lf(z) I ELVClL o-ccri}EpY) ouvctp1"Y)O"Y)
1"01"E Y) j ELVClL <J"'CCli}EpY) <JUVClp1"Y)0"1J.
5. A v an ELVClL i}nLXOL ctpLi}tJ.OL XClL limn--+oo a~:1 = L OEL~E'"CE 01"L Y) ClX1"LVCl cruyxALOY)t;; 1"Y)t;; oc:tpctc;;
L:~=O anzn ELVClL R = 1/ L.
6. BpEL-cE OACl -cct z E <C ym -cct on:mct 1J <JELpct 2::~=1 ;,2 ( ~~ i) n cruyXALVEt.
(p) BpEl1"E EVCl ClVOLX1"0 UTI:O<JUVOAO n -cou <C 01"0 OTI:OLO Y) cruvctpcY)OY) f(z) = 2::~=1 n\ (~~ir ElVCll ClVClAU1"LXYJ. (1n:oOEl~Y): 1J O"Elpct ()E-v ELVCll OUVCltJ.OOElpct, Y) ouvi}c;crY) ClVClAU1"lXWV ELVClt
ClVClAU1"lXYJ).
7. l'n:oAoytonc: -co j_./z2 + ~) dz, (ct) <J1"1)V XCltJ.TI:AUA1J 1(t) XCltJ.TI:UAr) 1( t) = 1 +it, 0 ::; t ::; 1.
01"L
1 f(z) dz = 1 f(z) dz 'Yl 'Y2
0 ::; t ::; 27r,
{t + it2
, 0 < t < 1 9. 1n:o)..oytcrE1"E -co f (izi 2 + 2:Z) dz 0"1"1JV XCltJ.TI:AUATJ l(t) = In ·!II. l - -
2 'Y v 2ez 4 < t ::; .
(p) 0"1"Y)V
10. y TI:OAOytanE 1"0 OAOXA YJPWtJ.Cl J'Y ( z + z~ 1) dz on:ou I ELVClL Y) n:EpltJ.E-cpoc;; -cou n:ctpctAA Y)AOYPCltJ.
flOU tJ.E XopucpEt;; i, -i, 2 + i 2 - i
1
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1. Bpe:ne: tJ.La. n:ctpettJ.E-cptxon:onpY) 'Y)s n:e:ptp.e-cpou -cou -cnpetyuwou 111:: xopu<pe:c;;
1 + i, 1 - i, -1 + i, -1 - i,
OYJA. ~pe:m: XettJ-1tUAYJ 1(t) : [0, 1] ---1- C -rYJc;; o~tmw; TO txvoc;; e:tvru YJ c.v A.oyw ~te:pttJ.npoc;;.
2. Eo-rw 1(t) = Reit, 0 ~ t ~ 27r (-lE R > 0. ~Et~ne: o-n: (cr) J
7 zkdz = 0 ytet xa:&e k crxe:pmo, k of -1.
(~) J7 ~dz = 27ri (ytet k = -1).
(y) ~U[.mE:pctvns on J7
P(z)dz = 0 ytet xa-tte: ~toA.uw·vuflO P(z).
5. ~E:L~ETE Oct f7
1dz = 0 Xetl f7
zdz = 0 0\:Y)V Xet(-lTI:UAY} I cYj<; ctOXT)OTj<; 1 (-lE etn:' e:u'l'h:;tac;; u~toA.oyto{-lo (xwptc; i:YJV XPYJOYJ etvcmapaywywv ). t
6. Bpt:ne: -ro oAoXAYJpW(-let J'Y sin(z )dz o~tou 1 ewat -ro Y)(-ltXUXAto cou OXYJ(-l<Xcoc;;
7. ~Et~E1"E OTt J~?r ecos(IJ) cos(O + sin(O))dO = 0.
__ b_
8. Av f(z) e:tvett crxe:pettet ouvap-rYJOTJ, a E <C xett M = max{lf(z)l : lzl = 2lai} or::ti;:ETE OTt If (a) I :::; 2M. ~YJ(-lElWOYF etpyo-re:pet ( etpXY) (-lE:YLOLOU) 'I'Ja OEti;:OU(-lE: o·n OLY)V 1tpetY(-l<X"tlXOTYJT<X lf(a)! ~ M.
1
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1. Eo-cc.u n c <C IXVOLKt:O, I (lLIX AELIX XIX(lTI:UAT] a-co n xoo fn : n -+ <C IXXOAOU11LIX ouve:xc.uv
OUVIXfYC:TJOEWV l:El:OLIX C.UO"'CE fn -+ f O(lOLO(lOfl<j)IX 01:0 n (~ IXXO(llj ALYOl:EflO fn -+ f O(lOLO(lOfl<j)IX
TI:IXVC.U Ol:T)V I) YLIX XIXTI:OLIX OUVIXfll:T)OT) f : !1 --7 <C. ~EL~El:E Ol:L
lim 1 fn(z) dz = 1 f(z) dz. n--+ CXJ 'Y 'Y
2. A VIXTI:l:U~El:E l:T)V OUVIXfll:T)OT) f ( Z) = z-=-2 OE YEW(lEl:flLXT) OELfliX (lE XEVl:flO "'CO a = 3. IJOLIX ELVIXL
1) IXXl:LVIX oyXALOT)<;j
3. BpEL-cE -co IXV<Xn:-cuy(l<X oe: OUVIX(lOOELfliX 1::1)<; IXXEp<XLIX<; oUVIXfll:TJOTJ<; f(z) = z 3 (lE xe:v-cpo -co a= 2. K<Xve:-ce: -co LOLO YLIX l:TJV g( z) = 2 - 3z - z2 + 2z3 - z4 (lE xe:v-cpo -cc.up<X -co a = -1.
4. Av p(z) ELVIXL n:oA.uc.uW(lO P<X11(lOU n -co-ce ~zl=l ~<:} dz = 0 YLIX x!X11e: m > n + 1.
5. Bpe:ne: -co IXVIXn:-cuy(lot oe: OUVot(lOOELflot 1::1)<; iXXEpotLIX<; OUVIXfll:T)OT)<; f(z) = ez, (at) (lE xe:v-cpo -co
a= 0. (p) (lE xe:v-cpo -co a= 2.
6. Bpe:ne: -co <Xvotn:-cuy(l<X oe: OUVIX(lOOELpiX l:TJ<; IXXEpotLot<; oUVIXfll:T)01]<; f(z) = sin(z), (at) (lE xe:v-cpo
-co a = 0. (p) (lE xe:v-cpo -co a = 1r /2.
7. ML<X IXXEfl<XIIX ouv1Xfll:1J01J A.e:ye:-cotL IXfll:Lot <XV f( -z) = f(z) yLot xot11e z E <C. ML<X -ce:-coLot n:.x.
ELVIXL 1] oUVIXfll:1]01] f(z) = ez + e-z. ~WOEl:E IXAA.ot TI:IXfliXOELY(l<Xl:IX XIXL OEL~El:E Ol:L otv f(z) ELVotL
IXfll:Lot -co-ce: m ouv-ce:A.e:o-ce:<; (lE n:e:pLnou<; OELXl:E<; o-co IXVotn:-cuy(l<X 1::1]<; f (lE xe:v-cpo -co 0 ELVIXI OAOL
[lT)OEV.
8. Av f(z) ELVotL IXXEpotLot ouv<Xp-c1J01] (.11) o-ciX11Ep1] XotL a E <C OEL~E-ce: o-ct UTI:IXflXEL n = n(a) E N WOl:E f(nl(a)-=/= 0.
9. Av f(z) ELVIXL IXXEpotiiX OUV1Xfll:1]01] XotL [lE lf(z)l :::; log(2 + lzl) YLIX Xot11E Z E <C OEL~El:E Ol:L 1] f ELVIXL Ol:ot11Ep1].
10. Av f : <C -+ <C ELVIXL OAO[lO<j)T) XIXL limz--+= f~z) = 0 OEL~El:E Ol:L 1] f ELVIXL Ol:IX11Ep1].
11. Av f(z) dv<XL IXXEpot(ot (lE Re(f(z)) ;;:::: 0 yLot xcX.11e: z E <C -c6-ce: 1J f dv<XL m<X11Ep1J. (Tn:ooe:L~1]: 11e:c.up1)0El:E l:T)V e-f(z)).
12. Eo-cw f(z) IXXEP<XLIX. Tn:o11e:-co[.!E Ol:L UTI:IXflXEL M E R c.uo-ce: Re(f(z)) :=:; M yt<X X<X11e: z E <C. ~EI~El:E Ol:L 1) f ELVotL Ol:IX11Ep1).
13. Tn:o11e:-cO[lE Ol:L f : <C-+ <C E:LVIXL OAO(l0p<p1] xoo UTI:IXflXEL 11E-cLX1] o-c<X11Ep1X c (t)Ol:E lf(z)l :::; cex YL<X xot11E z = x + iy E <C. ~EL~e:-ce: o-cL UTI:IXflXEL c E <C (lE lei :::; C wme: f(z) = cez yt<X X<X11e:
z E <C.
14. Eo-cc.u f(z) IXXEfliXL<X OUVIXfll:1]01], (.11] o-c<X11Ep1]. ~e:L~El:E Ol:L UTI:<XflXEL IXXOAOU11L<X { Zn} o-co <C -ce:-cm<X
c.uo-ce: limn-+= f ( Zn) = 0 ( un:oOEL~1J: <XV 1) f EXEL pL~<X -co-ce: <p<Xve:po. A v oe:v EXEL flL~<X -co-ce: 1)
1 If E:LV<XL E:n:LOT)<; IXXEflOO<X X.A. n:.)
15. Eo-cc.u f ( z) !XXe:poo<X ouv<Xpl:1JOTJ· ~e:L~El:E o-cL <XV -co n:t:OLO 1:L(lWV 1::1)<; f "n:!XfJIXA.e:meL" e:v<X {11) xe:vo
IXVOLXl:O OUVOAO, l:Ol:E <XUl:O (-co TI:EOIO l:L(lC.UV) ELV<XL (lOVOOUVOAO. ~T)A., <XV UTI:IXflXEI !1 IXVOLXl:O,
[11] xe:vo, (lE n c <C \ f(<C) l:Ol:E 1) f ELV<XL ow11e:p1). Tn:oo.: niX a E n UTI:<XflXEL 8 > 0 (t)Ol:E
D(a,8) c n. To-ce: lf(z) -al2: 8 YL<X X<X11E z E <C <Xp<X 1J g(z) = 1/(f(z) -a) ELV<XL IXXEfl<XLIX X<XL
<pp<XY(lEV1].
1
Acrxijcrc:~c;; M~y~o~xwv :Euv~fYtijcrc:c.>v, E~~~vo EE;~{-l-TjVO 2011 AaxYjaev:;, <t>u/>.. 6
1. Bpen:e l:et. et.vet.n:l:UYtJ.Ct.l:et. ae OUWLtJ.OO"etpet. Xet.t l:t<; et.Vl:tO"l:OtXe<; et.Xl:tve<; auyxAtO"Yj<; let. TYjc f(z) = ~ + ~~!, [le xev1:po (et.) 1:0 zo = 1, (~) 1:0 zo = i
1~ TYjc f(z) = z3 + ~ [le xev1:po 1:0 zo = 1.
1y TYjc g(z) = l_;z3 [le xev1:po 1:0 zo = 0.
2. Yn:o/>.oytae1:e 1:et.
1 z2 + z - 1 . 1 zez + iz2
· (2a) ( _
2) dz, !'(t) = e't,o ~ t ~ 21r. (2b) _ dz, !'(t) = 2e2t,O ~ t ~ 21r.
1 zz I Z 1
1 sin(z2 + 1) . 1 cos(1rz) · (2c) 2 3 2
dz, l'(t)=(3/2)e't,o::;t::;27r. (2d) 2 dz, 1 (t)=i+e't,o::;t::;27r. 1 z-z+ 1 z+1
3. Yn:of..oytane 1:0 J, ~(:~z{) dz mt<; n:epmcwaet<;: (1) O"l:YjV Xet.[ln:UAYj izl = 1/3, (2) O"l:YjV XCI.f111:UA7) jz- 1j = 1/3, (3) O"TrjV Xet.[ln:UA1J jz- ij = 1/3.
4. Yn:et.pxet et.vet.AuctXYj auvet.p1:1J0"1) f : D(O, 2) -+ <C wme f(~) = ~ ytet. xet.\')e n EN xet.t f(-1) = 1;
0. Me XP1J0"1) 1:7)<; Ct.PXYlC l:OU l:Ct.Ul:LO"[lOU (xet.t xwpt<; n:pet.~et<;) oet~ne Ol:t sin2 (z)+cos2 (z) = 1 ytet. xet.\')e z E <C.
6. Emw f OAOf!OpcpYj 0"1:0 n = { z E <C : ~ < izl < n ytet. l:YjV on:otet. taxuet:
lf(z)l :::;2 ytct jzj = 1,
!f(z)! :::;18 yta !zl = 3.
(et.) .6.et~ne mt !f(z)i :::; 2lzl2 ytet. xet.t')e z [le 1 :::; lzl :::; 3. (Yn:oo.: 0ewpYjane l:YjV cruvet.p"tl)O"Y] f (z) / 2z2
).
(~) Av sn:mAC:OV .f(2) = 8 OetC:sce on f(z) = 2z2 ytet. xet.\')e zEn.
7. Em:w f ol>.of!opcpYj mov Otaxo ]])) ytet. 1:1JV on:otet. taxuet If ( z) - z21 :::; J1=lZT ytet. xet.\')e
z E ]]))_ .6.et~ne o1:t f(z) = z2 ytet. xet.\')e z E ]])).
8. Emw n Ct.VOtX"tO cruvex-rtx6 xat f : n -+ <C OAO[lOpcpYj. Yn:o\')E"tO[le 6n umipxet a E n [.Le "!YjV tOtO"!YJl:Ct.: Av {zn} elVC!l Ct.XOAOU\')tet. 0"1:0 n [le Zn-+ a, "!Ol:e lf(zn)l ~ lf(zn+l)l yta xet.\')e n . .6.et~e1:e Ol:t 1) f c:Lvet.t met.\')epYJ.
9. Eacw n aVOlX"tO 0"\JVEX"tlXO XC!l cppet.yfleVo XCI.t f : n -+ <C cruvexYJc 0"1:0 IT, OAOf!OpcpY) mo n, Xet.t fl1J al:et.\')ep1J. Av M = sup{jj(z)j : z E an} oetC:e-re on !f(z)i < M ytet. xet.\')e zEn.
10. Ecrcw n = {z E <C: -~ < Im(z) < ~}, )(C(t f(z) = eez . .6-etC:ne Ol:t If(()! = 1 yta xa\')e ( E an, et.A/>.a Y] f oev etVCt.t cppet.yf!eVYj a·w n ( et.pet. Y] un:o\')ec;Y] Ol:L "tO n etvat cppayf!eVO O"l:YjV n:pOYJYOUf1eV1J Ct.O"XYjO"Yj Oev f!n:opet Vet. n:et.pet.Aetcp\')et).
11. Eacw n C(VO\Xl:O cruvex-rtXO )(at cppet.yf!eVO )(Ct.t f : IT -+ <C auvexYjc 0"1:0 IT, OAO[lOpcpY] mo n, Xet.t flY] aca\')ep7J. Av u(z) = Re(f(z)) xet.t M = max{u(z) : z E an} oet~e1:e o-ct u(z) < M yta xa\')e zEn (Yn:oo.: \')ewpY]me "!YjV g(z) = ef(z)).
12. Eacw n Ct.VOlX"tO auvex-ctXO xat f : n -+ <C OAO[lopcpYJ. Yn:o\')sl:Of!e Ol:t un:apxet a E n wcr1:e lf(a)i :::; l.f(z)l ytet. xa\')e zEn . .6.et~ne oct me f(a) = 0 me Y] f c:tvet.t ma\')c:p7J mo n.
Aax~ae:Lc; MLycxoLx~v :Euvcxp't~ae:wv, EcxpLvo E~CXJ:lTJ'VO 2011 AaxY)aE:tc;, <PuA. 7
1. Ea-rw <P : liJ) ~ c <XVcxAU~LXaYJ r€ I <P( z) I :::; 1 yt<X xcx'i}g z E JI)). ~EL~€1'€ O"tL <X. A v a E II)) 1'01'€ I c/J(z) q,( ) < I z-~ I yt<X xcx'i}g z =/= a.
1-cp(a)cp(z) - 1-az
A. Av z E II)) -ro-re: l"''(z)l < 1-ltf>(z)l2 ~ 'f' - 1-lzl2
2. Ea-rw <P : II)) ~ C cxvW..u-rLXYJ ~e: I <P( z) I :::; 1 ytcx xcx'i)e: z E II)), ytcx "tY)V 07WL<X taXUE:L ¢(0) = 0 X<XL 1/>'(0) = 0. ~EL~€1'€ O"tL jlj>(z)j ~ lzl 2 yt<X X<X'i)€ Z E II)).
3. ~E:L~€1'€ o-rt un:cxpxE:L auvcxp-rYJaYJ f(z) oAo~opcpYJ mov ~ovcxotcxto otaxo II)) -rE:-rmcx wmE: f(z) 4 + 1 = z4 ytcx xcx'i}g z E II)). IIoaE:c; -re:-rmE:c; auvcxp-rYJaE:tc; un:cxpxouv;
4. E~E:-rcxane: cxv ~n:ope:t vex opta'i}E:L 1) log( ez - i) acxv oAo~opcpYJ auvcxp-rY)aY) a-rov ~OV<XOL<XLO OLaxo II)).
5. E~e:-rcxane: cxv ~n:opE:t vex opta'i}e:t YJ log( cos( z)) acxv oAo~opcpYJ auvcxp-rYJaYJ a-rov ~OV<XOL<XLO Otaxo II)).
6. BpE:t"t€ -ro <XV<X7nuy~cx Laurent 1'1)<; auvcxp-rY)aY)c; f(z) = z+ 1~z a-roue; O<XX"tUAtouc;: (ex) ~(0, 0, 1), (~) ~(0, 1, oo), (y) ~(1, 0, 1), (o) ~(1, 1, oo)
7. Bpe:L"te: -ro cxvcxn:-ruy~cx Laurent 1'1)<; auvcxp-rY)aY)c; f(z) = z3 sin(~) mov ocxx-ruAto ~(0, 0, oo).
8. BpE:t-re: -ro cxvcxn:-ruy~cx Laurent 1'1)<; auvcxp-rY)aY)c; f ( z) = 1!:2 ae: oAouc; -roue; ocxx-ruALOuc; n:ou OYJ~toupyouv-rcxt cxn:o -rex cxvw~W..cx a1)~€L<X 1'1)<;.
9. Bpe:t-re: X<XL X<Xp<XX"tY)ptae:-rE: -rtc; ~E:~ovw~E:ve:c; cxvw~iAtE:c; -rwv auvcxp-rYJae:wv
ez- 1 z2- 5z + 6 sin('rrz)
(a) f(z) = --;z-+ z _ 2 , (b) g(z) = z3 (cos(1/z)-1), (c) h(z) = z _ 1 10. Xcxpcxx-rY)ptae:-re: "tY)V cxvw~W..tcx a-ro 0 ytcx -rtc; auvcxp-rY)ae:tc;
(a) f(z) = sin(z) _ ~' (b) g(z) = z + (1- z)(1- ez) z2 z z2(1- z)
(Tn:oo. BpE:t"t€ "tY)V aE:tpcx Laurent).
11. Av f(z) Etvett crxe:pcxtcr auvcrp-rY)aY) xcrt ytcx xcrn:mo k E N auvcxp-rY)aY) g(z) =
zk/(1/z) E:LV<XL cppcxy~E:VYJ a€ xcrn:oto ocxx-ruAto ~(0,0,£5) ~€ t5 > 0 OE:L~€1'€ on 1) f(z) E:LV<XL 1tOAUWVU~O.
12. Ea-rw f(z) <XXE:poocx auvcxp-rYJaYJ n:ou oe:v E:tvoo n:oAuwvu~o. ~e:L~€1'€ o-rt ytcx xcx'i)e: c E C un:cxpxe:L <XXOAou'i}tcx ~LY<XOLXWV {zn} ~e: Zn ~ 00 Wa"t€ f(zn) ~ C.
13. Tn:o'i)E:-rou~E: o-rt OL f(z), g(z) E:xouv oumWOYJ cxvw~W..tcx mo 0. Tt E:LOouc; cxvw~cxAL<X ~n:opE:t vex E:XE:L 1) f(z) + g(z) mo 0; ~wanE: n:cxp<XOE:ty~cx-rcx.
14. Bp€L"t€ X<XL xcrp<XX"tY)ptae:-r€ OA<X "t<X <XVW~cxA<X a1)~€L<X "tY)<; f(z) = eiL1.
1
Acrx-wlcre:~o<; M~oycto~oxwv ~uvctp1:-wlcre:wv, Ectpwo Ei;ctp.YjVO 2011 Aax'Y)crt:t<;;, <f>uf.. 8
1. Av 'Y) f(z) Etvcrt of.otJ.opcpYJ xcrt tJ.YJ tJ.YJOEVLXYJ amv otaxo D(O, r), r > 0, -ct E:toou<;; crvwtJ.crf.tcr e::xe::t YJ g ( z) = f ( z) e ~ m:o 0;
2. EaTw f : D(zo, r) -+ C OAO!lOPqJYJ tJ.E r > 0. Ae:!le: o-ct 'Y) f sxe:t pt~cr T<X~YJ<; m a-co Zo crv !lrcope:t vcr ypcrq;e::t w<;;
f(z) = (z- z0 )mh(z)
!LE h(z) of.o!lopcpYJ aTov D(z0 , r) xcrt h(z0 ) # 0. ~e:t~ne: oTt YJ f(z) e:xe:t pt~cr TIX~YJ<;; m aTO Zo <XV X<Xl !lOVOV crv YJ g(z) = !Cz) EXEL TCOAO T<X~YJ<; m aTO Zo.
3. Av m f(z) xcrt g(z) c:xouv ICOAO a-co zo, Tt e::toou<; crvw!lcrAtcr !liCopst vcr e::xouv ot f(z)g(z) xcrt f(z)jg(z) a-co zo;
4. Bpe::rrc: TIX !lE!lOVW!lEVIX IXVW!lcrAIX aYJ!lEtcr TY)s f(z) = sin~~)
5. Bpe::tTE TIX !lE!lOVWtJ.EVIX IXVW!lcrAIX a'Y)tJ.El!X TWV aUV!XpT'Y)aEWV X!XL VIX TIX TIX~LVO!l'Y)ae:TE
( 1)
2 zcos(z)-z 1 (z+1)3 -1
(1)f(z)= z+- , (2)f(z)= . 3() , (3)f(z)= 4 + 2, (4)f(z)= 3 z smz z z z
6. Bpe::tTe:: TO Res(!, 0) xcrt Res(!, 1) ytcr T'Y)V f(z) = z(;~i)2
7. Bpe::ne:: TO Res(g, zo) ae:: of.cr -rcr crvwtJ.crf.cr aYJ!lEtcr zo T'Y)<; f(z) = (H~2)2
8. Bpe::rrc: -co Res(g, z0 ) a-ccr crvw!lcrAcr aYJ!lEtcr ytcr -ct<;
1 z3 -1 1 1 2 1 2 1
f(z) = z4 cos(-), g(z) = -4--, h(z) = z2e---z- -e72, ¢(z) =sin (-)+cos (-).
z z -1 z z z
9. A v f ( z) e::tvcrt OAO!lopcpYJ a-co ~ ( 1, 0, 1) !lE <XICAO ICOAO a To z0 = 1 xcrt g ( z) = zf(z2
) ~pE:tTe:: -co Res(g, 1) auvcrpT'Y)ae::t TOU Res(!, 1).
10. Bpe::tTE TIX OAOXAYJPW!liXTIX
1 1 1 2 . 1) 1 1 (1) 4 2 dz, (2) z sm(- dz, (3) ( 2 _ 1)( 2 16
) dz, lzl=2 z + z lzl=1 z 'Y z z z +
01COU a-co TEAEUTIXLO "( ElV!Xl YJ 1CEpl!le:Tpo<;; TOU -cnpcryWVOU !lE xopucpe::<;; ( ±2, ±2~.
11. Bpsm: -ccr of.oxf.YJpWtJ.<XTcr
1 ez 1 1 1 sin(z) + sin(l) (1) -- dz, (2) -4
- dz, (3) z dz lzl=2 z- 1 lz-11=1 z - 1 lzl=l z
12. Kcr-ccraxe::ucrane:: auvcrp-c'Y)a'Y) f(z) OAO!lOP({JYJ aTo C \ {0, 1} tJ.E TCOAO am T<X~Y)s 2 a-co 0, !LE ouatWOYJ crvw!lcrf.tcr am 1, xcrt !lE Res(!, 0) = 1 xcrt Res(!, 1) = 0.
1
l ao _co_s_( x--,-) dx = _1r_v'2_2 e-~ (cos - 1- + sin - 1-) -oo 1 + x4 2 y'2 y'2
Acrx~cre:~c; M~ycx.o~xwv :Euvcx.p-c~cre:wv, Ecx.p~vo E~cx.p.. YJVO 2011 AoXT)GE:t<;;, <PuA.. 9
1. Net urcoA.oytae:-re: -ro oA.oxA.Y)pc..>tJ.ct J7 jgj dz o-rctv (1) f(z) = z4 + 1 Xctt 'Y(t) =
2eit, 0 :$ t :$ 21f. (2) f(z) = (l~:)3 Xctt 'Y(t) = 2eit, 0 _-::; t _-::; 21f.
2. Bpe:rre: 1:0 TCAT){}O<;; TWV pt~WV l:Y)s f(z) = 3z4 - ez TCOU ~pLaXOVTctL tLE:Gct GTOV
otaxo D(O, 1).
3. ~E:L~E:TE: OTL OAE<;; m pt~E<;; -rou rcoA.uwvut-tou P( z) = z 7 + 2z4 - 5z + 10 ~ptaxov-rctL
t-te:crct a-ro auvoA.o { z : 1 :$ I z I < 2}.
10. ~E:L~E:TE: OTL
_ _;____:_dx = --e v'2 cos- +sm-100 cos(x) 1r.J2 _ _L ( 1 . 1 )
-oo 1 + x4 2 J2 J2
1
Acrx~cre:1.c; M~oyoto~oxwv ~uvotp.:l)cre:<..>v, Eotpwo Escxp.T)vo 2011 Aoxr)OEL(, <Pu"A. 10
1 A 1: Joo cos(3x) d _ 211" . uE:L<.,E1"E: 01"L _ 00 (l+x2)2 X - e3
2 · 6.c:t~E:TE: 01"L f~oo sin:2(x) dx = ~ ( T rcoo. f (z) = 1 ~~2iz )
3. BpE:LTE: 1"0 OAOXA1JpW~()( J~1r 2+;inO d() ·
4 B "\ "\ f21r sin2 (} d(} . pE:t1"E: 1"0 0A0XA1)pW[l()( Jo 5+3 cos 0 .
5. Bpe:LTE 1"0 OAOXAY)pW[l()( J0
21r a~~::o d(), orcou a E IR, a > 1.
6. Bpc:tTE: 1"0 OAOXA1JPW[l()( f0
21r (2+c~s0)2 d(}.
7. 6.c:t~E:1"E: 01"l
r1r d(} d(} - { 1::2 ' 0 < a < 1 lo 1-2acos8+a2 - a;:_1, 1<a<oo.
8. Bpc:nc: 1"0 OAOXA1JPW~()( fo21r 7-6cos~+2sin0 d(}
9. Bpc:nc: To OAOXA1JPW[l()(
OTCOU /(t) = 7eit, 0 ~ t ~ 27r
10. BpE:LTE: 1"0 OAOXA1JPW[l()(
1 1+z d 1'1-cosz z
AcrxT)cre:t.c; p.e:yotAu.:e:pT)<; oucrxoAI.ot<;
11. EoTw P(z) rcoAUWVU[lO ~c: ouvTE:AE:OTE( ()(JCO TO C. 6.c:t~nc: OTL OAE:( m pt~E:( TOU
P'(z) Mptc:xovT()(L 01"1JV xupT1J it1JX1J TOU ouvo"Aou TWV pt~wv TOU P(z). (Bva JwoA.o K C CC A.~yemz x:vpr:o av yza x:afJ~ Duo JT]JlEW a, bE K, oAox:Arypo
ro evfJvyparro r:rTJra [a, b] = {t1a + t2b : 0 ~ t1, t2 ~ 1, t1 + t2 = 1} nEpzexccaz JT:O K. H x:vpr:T] fJT]KT] C(A) Evor; JvvoAov A c C ezvaz T] 'W11TJ oAwv r:wv x:vpr:wv uvvoAwv nov nepzexovv r:o A. Er:T]v nEpmr:WUT] nov A =
{a!,~' ... 'an} El11Ul nEnEpaurEvO T:OT:E npoKVnT:El OT:l T] KVPT:TJ fJT]KT] El11Ul :w uvvoAo C(A) = {2:::~=1 tkak : ti E [0, 1]re t1 + t2 + · · · + tn = 1} oAwv r:wv x:vpr:wv uvvovaurwv ()WI){ElWV r:ov A).
12. Emw f : JD) -t- C ()(VcrAUTLX1J. 8c:wpou~c: T1JV ouv()(p1"1J01J m(r) : [0, 1) -t- IR, rcou
opt~E:T()(L W( m(r) = maxlzl:::;r lf(z)l. 6.c:t~nc: on 1J m(r) ELV()(l <JUVEX1J( mo [0, 1).
1
