Majhmatik gia thn Plhroforik kai tic...
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Transcript of Majhmatik gia thn Plhroforik kai tic...
Majhmatik� gia thn Plhroforik kai tic ThlepikoinwnÐec.
Mèroc G: Eisagwg sth Migadik An�lush.(Pr¸th Morf Shmei¸sewn)
I. G. STRATHS.M�ioc 2005
Perieqìmena
1 Oi MigadikoÐ ArijmoÐ 51.1 To S¸ma twn Migadik¸n Arijm¸n . . . . . . . . . . . . . . . . . . . . . . . 61.2 To Migadikì EpÐpedo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.3 Polikèc Suntetagmènec . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.3.1 ParadeÐgmata - Ask seic . . . . . . . . . . . . . . . . . . . . . . . . 101.3.2 Topologik� jèmata sqetik� me to migadikì epÐpedo . . . . . . . . . . 141.3.3 AkoloujÐec - Ask seic . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.4 Seirèc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.4.1 ParadeÐgmata � Ask seic . . . . . . . . . . . . . . . . . . . . . . . . 17
2 182.1 Kat�taxh Sunìlwn sto Migadikì EpÐpedo . . . . . . . . . . . . . . . . . . . 182.2 SuneqeÐc Sunart seic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2.1 Stereografik Probol . . . . . . . . . . . . . . . . . . . . . . . . 202.3 Sunart seic miac migadik c metablht c z . . . . . . . . . . . . . . . . . . . 20
2.3.1 Analutik� Polu¸numa . . . . . . . . . . . . . . . . . . . . . . . . . 202.3.2 Dunamoseirèc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.3.3 Analutikèc Sunart seic . . . . . . . . . . . . . . . . . . . . . . . . 23
2.4 Oi Stoiqei¸deic Sunart seic . . . . . . . . . . . . . . . . . . . . . . . . . . 282.4.1 H ekjetik sun�rthsh . . . . . . . . . . . . . . . . . . . . . . . . . 282.4.2 Oi trigwnometrikèc sunart seic . . . . . . . . . . . . . . . . . . . . 292.4.3 H Logarijmik Sun�rthsh . . . . . . . . . . . . . . . . . . . . . . . 312.4.4 Oi sunart seic zλ, λz, λ ∈ C . . . . . . . . . . . . . . . . . . . . . . 322.4.5 Oi antÐstrofec trigwnometrikèc sunart seic . . . . . . . . . . . . . 32
2.5 GewmetrÐa twn stoiqeiwd¸n sunart sewn . . . . . . . . . . . . . . . . . . . 352.5.1 f1(z) = z2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352.5.2 f2(z) =
√z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.5.3 f(z) = sin z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352.6 Olokl rwsh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.6.1 Orismèno olokl rwma . . . . . . . . . . . . . . . . . . . . . . . . . . 362.6.2 KampÔlec . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372.6.3 Olokl rwma migadik¸n sunart sewn miac migadik c metablht c . . . 37
2.7 To je¸rhma Cauchy-Goursat . . . . . . . . . . . . . . . . . . . . . . . . . . 40
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2.7.1 To je¸rhma Cauchy . . . . . . . . . . . . . . . . . . . . . . . . . . 402.7.2 To je¸rhma Cauchy-Goursat . . . . . . . . . . . . . . . . . . . . . . 40
2.8 Apl� kai Pollapl� Sunektik� SÔnola . . . . . . . . . . . . . . . . . . . . 412.9 O oloklhrwtikìc tÔpoc tou Cauchy . . . . . . . . . . . . . . . . . . . . . . 42
3 443.1 Memonwmènec AnwmalÐec Analutik¸n Sunart sewn . . . . . . . . . . . . . . 44
3.1.1 1. Kat�taxh Memonwmènwn Anwmali¸n � Arq tou Riemann � Je¸-rhma Casorati – Weierstrass . . . . . . . . . . . . . . . . . . . . . . 44
3.1.2 An�ptugma Laurent . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4 Oloklhrwtik� Upìloipa 484.1 DeÐkthc Strof c kai to Je¸rhma Oloklhrwtik¸n UpoloÐpwn tou Cauchy . 484.2 Efarmogèc tou jewr matoc Oloklhrwtik¸n UpoloÐpwn tou Cauchy ston
upologismì oloklhrwm�twn kai seir¸n . . . . . . . . . . . . . . . . . . . . 504.2.1 Upologismìc Oloklhrwm�twn . . . . . . . . . . . . . . . . . . . . . 50
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Kef�laio 1
Oi MigadikoÐ ArijmoÐ
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'Hdh apì ton 16o ai¸na emfanÐzontai arijmoÐ thc morf a+b√−1, (a, b ∈ R). O Cardan
touc qrhsimopoÐhse gia th lÔsh exis¸sewn 2oυ kai 3oυ bajmoÔ. To 18o ai¸na o Euler èkaneqr sh twn migadik¸n arijm¸n gia th lÔsh diaforik¸n exis¸sewn.
Oi migadikoÐ arijmoÐ eÐqan {ftwq f mh} wc to 1830, ètuqan ìmwc eurÔterhc apodoq ckurÐwc q�rh sth gewmetrik touc anapar�stash kai ston Gauss. O pr¸toc pl rhc kaiausthrìc orismìc ofeÐletai ston (sÔgqrono tou Gauss) Hamilton.
Sqèsh twn migadik¸n me th Fusik : Mhqanik twn Reust¸n, Hlektromagnhtismìc,Jermìthta, klp.
1.1 To S¸ma twn Migadik¸n Arijm¸nTo s¸ma twn migadik¸n arijm¸n C eÐnai to sÔnolo twn diatetagmènwn zeug¸n pragmatik¸narijm¸n (a, b) me prìsjesh kai pollaplasiasmì pou orÐzontai wc ex c:
(a, b) + (c, d) = (a + c, b + d)
(a, b)(c, d) = (ac− bd, ad + bc)
Idiìthtec'Estw zk = (ak, bk) tuqìntec migadikoÐ arijmoÐ. Tìte
• z1 + z2 = z2 + z1 antimetajetikìthta thc prìsjeshc
• z1 + (z2 + z3) = (z1 + z2) + z3 prosetairistikìthta thc prìsjeshc
• (0, 0) oudètero stoiqeÐo thc prìsjeshcantÐjetoc tou z = (a, b), eÐnai o −z = (−a, −b)
'Estw λ, µ ∈ R, z, w ∈ C. Tìte
• λ(µz) = (λµ)z
• (λ + µ)z = (λz + µz)
• λ(z + w) = λz + λw
• z1z2 = z2z1 antimetajetikìthta tou pollaplasiasmoÔ
• z1(z2z3) = (z1z2)z3 prosetairistikìthta tou pollaplasiasmoÔ
• z1(z2 + z3) = z1z2 + z1z3 epimeristikìthta tou pollaplasiasmoÔ wc proc thnprìsjesh
• (1, 0) oudètero stoiqeÐo tou pollaplasiasmoÔ
antÐstrofoc tou z = (a, b) 6= (0, 0), eÐnai o 1z
=(
aa2+b2
, −ba2+b2
)
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Sunep¸c to C eÐnai èna s¸ma (antimetajetikìc daktÔlioc me antÐstrofo pollaplasia-smoÔ.)
Mia idiìthta tou R pou den metafèretai sto C eÐnai ekeÐnh thc di�taxhc. MporeÐ eÔkolana deiqjeÐ ìti to i = (0, 1) den mporoÔme na to qarakthrÐsoume wc arnhtikì jetikì, qwrÐcna upopèsoume se antÐfash.
ParathroÔme ìti mporoÔme na antistoiq soume touc migadikoÔc arijmoÔc thc morf c(a, 0) me touc pragmatikoÔc arijmoÔc a. FaÐnetai amèswc ìti aut h antistoiqÐa diathreÐ ticarijmhtikèc pr�xeic pou orÐsame, ki ètsi den dhmiourgeÐtai sÔgqush an antikatast soume to(a, 0) me to a. M' aut thn ènnoia lème ìti to sÔnolo twn migadik¸n arijm¸n thc morf c(a, 0) eÐnai isìmorfo me to R. 'Etsi, lème ìti to (0, 1) eÐnai h tetragwnik rÐza tou −1,afoÔ (0, 1) · (0, 1) = (−1, 0) = −1.SumbolÐzoume me i to (0, 1).ParathroÔme ìti k�je migadikìc arijmìc gr�fetai wc ex c:
(a, b) = (a, 0) + (0, b) = a + bi
kai aut thn teleutaÐa graf ja qrhsimopoioÔme sto ex c.
Epistrèfontac sto jèma twn tetragwnik¸n riz¸n, up�rqoun dÔo migadikèc tetragwnikècrÐzec tou −1: to i kai to −i. Epiplèon, up�rqoun dÔo migadikèc tetragwnikèc rÐzec k�je mhmhdenikoÔ migadikoÔ arijmoÔ a + bi. Pr�gmati:
(x + iy)2 = (a + bi) ⇔{
x2 − y2 = a2xy = b
⇔{
4x4 − 4ax− b2 = 0y = b
2x
opìte
x = ±√
a +√
a2 + b2
2
kai ètsi
y =b
2x= ±
√−a +
√a2 + b2
2· sgn(b), ìpou sgn(b) =
{1, b ≥ 0−1, b < 0
ParadeÐgmata
i) Oi tetragwnikèc rÐzec tou 2i eÐnai oi 1 + i kai −1− i
ii) Oi tetragwnikèc rÐzec tou −5− 12i eÐnai oi 2− 3i kai −2 + 3i
ParathroÔme, tèloc, ìti opoiad pote deuterob�jmia exÐswsh me migadikoÔc suntelestècdèqetai lÔsh sto C. Pr�gmati:
az2 + bz + c = 0a, b, c ∈ C, a 6= 0
⇔(
z +b
2a
)2
=b2 − 4ac
4a2⇔ z =
−b±√b2 − 4ac
2a
AntÐjeta, ìpwc gnwrÐzoume, to x2 + 1 = 0, p.q., den èqei rÐza sto R. Dhl. to C eÐnaialgebrik¸c kleistì (afoÔ to anwtèrw isqÔei gia k�je poluwnumik exÐswsh.
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1.2 To Migadikì EpÐpedo(a, b) ↔ a + ib�xonac twn x ↔ pragmatikìc �xonac�xonac twn y ↔ fantastikìc �xonac
(prìsjesh)(pollaplasiasmìc)Fti�qnoume èna trÐgwno me dÔo pleurèc to 1 kai to z1. Met� fti�qnoume èna ìmoio
trÐgwno, me ton Ðdio prosanatolismì, kai to z2 na antistoiqeÐ sto 1. Tìte to di�nusma pouantistoiqeÐ sto z1 eÐnai to z1z2.
ParathroÔme ìti pollaplasiasmìc epÐ i eÐnai gewmetrik� isodÔnamoc me strof 90◦ an-tÐjeta me th for� thc kÐnhshc twn deikt¸n tou rologioÔ.
T¸ra, an z = x + iy, èqoume touc ex c ìrouc:Rez := x: to pragmatikì mèroc tou z.Imz := y: to fantastikì mèroc tou z, Imz ∈ R.z := x− iy: o suzug c tou z.|z| := √
z · z =√
x2 + y2: h apìluth tim , mètro, nìrma tou z.Argz := θ: to ìrisma tou z, opìtesin θ = Imz
|z| , cos θ = Rez|z|
ParadeÐgmata
(i) Rez > 0
(ii) {z : z = z}(iii) {z : −θ < argz < θ}(iv) {z : |z + 1| < 1}
(v){
z :∣∣∣argz − π
2
∣∣∣ < π2
}= {z : Imz > 0}
Prìtash 1.2.1 'Estw z, w ∈ C, tìte:1. z + w = z + w
2. zw = zw
3.(
zw
)= z
w, w 6= 0
4. z = z
5. Rez = 12(z + z)
6. Imz = 12i
(z − z)
7. zz = (Rez)2 + (Imz)2
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8. −|z| ≤ Rez ≤ |z|, −|z| ≤ Imz ≤ |z|9. zz = |z|2, |zw| = |z||w|
10. |z + w| ≤ |z|+ |w|: trigwnik anisìthta
11.∣∣∣|z| − |w|
∣∣∣ ≤ |z − w|
12. |z| ≤ |Rez|+ |Imz|
13. |z1w1 + · · · zkwk|2 ≤(|z1|2 + · · ·+ |zk|2
)(|w1|2 + · · ·+ |wk|2
)
anisìthta Cauchy - Schwarz
to � = � isqÔei ⇔ ∃λ, µ ∈ C : (λ, µ) 6= (0, 0)kai λzj = µwj, j = 1, . . . , k
14. argz = −argz
15. arg(zw) = argz + argw (mod2π)
16. arg zw
= argz − argw (mod2π)
1.3 Polikèc Suntetagmènec'Enac mh mhdenikìc migadikìc arijmìc prosdiorÐzetai pl rwc apì thn apìluth tim tou kaito ìrism� tou.An z = x + iy me |z| = r kai argz = θ, tìte
x = r cos θ, y = r sin θ kai
z = r(cos θ + i sin θ)
Ta r, θ lègontai polikèc suntetagmènec tou z kai h prohgoÔmenh sqèsh dÐnei thn polik morf tou z.Aut h morf eÐnai polÔ qr simh se upologismoÔc, afoÔ an
z1 = r1(cos θ1 + i sin θ1)
z2 = r2(cos θ2 + i sin θ2)
tìtez1z2 = r1r2[cos(θ1 + θ2) + i sin(θ1 + θ2)]
z1
z2
=r1
r2
[cos(θ1 − θ2) + i sin(θ1 − θ2)]
zn = rn(cos nθ + i sin nθ), n ∈ Z (tÔpoc tou deMoivre)
H teleutaÐa aut sqèsh eÐnai idiaÐtera qr simh sthn epÐlush exis¸sewn thc morf c zn = z0.
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Par�deigma: H eÔresh twn kubik¸n riz¸n thc mon�dac.
z3 = 1 ⇔ r3(cos 3θ + i sin 3θ) = 1(cos 0 + i sin 0)
⇔ r = 1, 3θ = 0 (mod2π)
⇔ z1 = cos 0 + i sin 0 = 1
z2 = cos2π
3+ i sin
2π
3= −1
2+ i
√3
2
z3 = cos4π
3+ i sin
4π
3=
1
2− i
√3
2
H polik morf twn tri¸n aut¸n riz¸n deÐqnei ìti eÐnai oi korufèc enìc isopleÔrou trig¸noueggegrammènou sto monadiaÐo kÔklo. OmoÐwc, oi n-ostèc rÐzec enìc z ∈ C eÐnai oi korufèckanonikoÔ polug¸nou me n pleurèc pou eÐnai eggegrammèno ston kÔklo kèntrou 0 kai aktÐnacr
12 .
Suqn� qrhsimoipoieÐtai oi tÔpoc tou Euler: eiθ = cos θ + i sin θ
Parat rhsh
'Estw z = x + iy = r(cos θ + i sin θ) = r(cos(θ + 2kπ) + i sin θ + 2kπ)).To sÔnolo twn gwni¸n θ + 2kπ : k ∈ Z eÐnai to argz.KÔria tim tou orÐsmatoc, Argz, eÐnai ekeÐno to ìrisma pou an kei sto (−π, π]. IsqÔei
argz = (Argz)(mod2π).An λ ∈ R, sumbolÐzoume me argλz ekeÐnh thn tim tou argz gia thn opoÐa isqÔei λ <
argλz ≤ λ + 2π.
EÔresh tou Argz
z = x + iy, x2 + y2 = 0JewroÔme ton z∗ = |x|+ i|y| kai brÐskoume to Argz∗ = φ,
φ = arctan|y||x| ,
(0 ≤ φ ≤ π
2
).
En suneqeÐa brÐskoume se poiì tetarthmìrio brÐsketai o z.
1.3.1 ParadeÐgmata - Ask seic1. Na brejeÐ to arg 3π
2(−1− i)
LÔshz := −1− i ⇒ z∗ = 1 + i ⇒Argz∗ = arctan(1
1) = π
4⇒
Argz = π4− π = −3π
4kai argz = 2kπ − 3π
4, k ∈ Z.
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T¸ra 3π2
< 2kπ − 3π4≤ 3π
2+ 2k ⇒ k = 2, dhlad
arg 3π2(−1− i) = 4π − 3π
4=
13π
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2. Na brejeÐ h anagkaÐa kai ikan sunj kh ¸ste oi z1, z2, z3 ∈ C na eÐnai suneujeiakoÐ.LÔsh
a) 'Estw ìti ta z1, z2, z3 eÐnai suneujeiak�. H eujeÐa pou dièrqetai apì ta z1, z2
èqei exÐswsh
z = z1 + τ(z1 − z2), τ ∈ R, h opoÐa epalhjeÔetai kai apì to z3
z3 − z1
z2 − z1
∈ R.
b) 'Estw ìti up�rqei τ ∈ R : z3 − z1 = τ(z1 − z2). Tìte
z3 = z1 + τ(z1 − z2)
z2 = z1 − (z1 − z2)
z1 = z1 + 0(z1 − z2)
ap' ìpou èpetai ìti ta z1, z2, z3 brÐskontai epÐ thc eujeÐac
z = z1 + τ(z1 − z2), τ ∈ R
3. Poiìc eÐnai o gewmetrikìc tìpoc twn shmeÐwn z = x + iy tou migadikoÔ epipèdou pouikanopoioÔn thn exÐswsh
|z − z1||z − z2| = k, k : staj., z1, z2 ∈ C.
LÔsh'Estw zj = aj + bji, j = 1, 2. Tìte
|x + yi− (a1 + b1i)||x + yi− (a2 + b2i)| = k ⇒ (x− a1)
2 + (y − b1)2 = k2(x− a2)
2 + k2(y − b2)2 ⇒
(1− k2)x2 + 2(k2(a2 − a1)x + (1− k2)y2 + 2(k2b2 − b1)y = k2(a22 + b2
2)− (a21 + b2
1).
a) k = 1 Ã2(b1 − b2)y = 2(a2 − a1)x + a2
1 + b21 − a2
2 − b22,
pou parist�nei eujeÐa (kai m�lista th mesok�jeto tou eujugr�mmou tm matocpou en¸nei ta z1, z2).
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b) k 6= 1 Ã(x + k2a2−a1
1−k2
)2
+(y + k2b2−b1
1−k2
)2
= A, ìpou
A :=k2(a2
2 + b22)− (a2
1 + b21)
1− k2+
(k2a2 − a1
1− k2
)2
+
(k2b2 − b1
1− k2
)2
,
pou parist�nei kÔklo kèntrou
z0 =a1 − a2k
2
1− k2+ i
b1 − b2k2
1− k2
kai aktÐnac R =√
A.
4. Na apodeiqjeÐ ìti, an a, b ∈ R kai ζ ∈ C, h exÐswsh
azz + ζz + ζz + b = 0
parist�nei eujeÐa ìtan {a = 0 & ζζ > 0} kai kÔklo peperasmènhc, mh mhdenik caktÐnac ìtan {a 6= 0 & ζζ > ab}.LÔsh'Estw ζ = γ + iδ kai z = x + iy, (γ, δ, x, y ∈ R). Tìte
ζz + ζz = 2Re(ζz)
= 2Re{(γ + iδ)(x + iy)}= 2(γx− δy)
ki ètsiazz + ζz + ζz + b = a(x2 + y2) + 2γx− 2δy + b = 0
(i) An a = 0 Ã 2γx− 2δy + b = 0,pou parist�nei eujeÐa ìtan γ2 + δ2 > 0 ⇔ ζζ > 0.
(ii) An a 6= 0 Ã x2 + y2 + 2γax− 2 δ
ay + b
a= 0
Ã(x + γ
a
)2
+(y + δ
a
)2
= r2, ìpou r2 = γ2+δ2−aba2 ,
pou parist�nei kÔklo ìtan r” > 0, dhlad ìtan ζζ = γ2 + δ2 > ab.
5. a) Na lujeÐ h exÐswsh z8 = 1
b) Na lujeÐ h exÐswsh z5 = −32
UpenjumÐzoume ton tÔpo tou De Moivre:An z = r(cos θ + i sin θ), tìte
zn = rn(cos nθ + i sin nθ).
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Gia na lÔsoume thn exÐswshzn = w
upojètoume ìti z = r(cos θ + i sin θ), w = %(cos φ + i sin φ)
kai èqoumer = n
√ρ kai nθ = φ + 2kπ Ã
z = n√
ρ
{cos
(φ + 2kπ
n
)+ i sin
(φ + 2kπ
n
)}
LÔsh
a) AfoÔ 1 = cos 2kπ + i sin 2kπ, èqoume
z = cos2kπ
8+ i sin
2kπ
8, k = 0, 1, 2, 3, 4, 5, 6, 7.
= 1,1√2
+i√2, i, − 1√
2+
i√2, −1, − 1√
2− i√
2, −i,
1√2− i√
2
b) 'Eqoume
−32 = 32{cos(π + 2kπ) + i sin(π + 2kπ)}, k ∈ Z= 25{cos(π + 2kπ) + i sin(π + 2kπ)}
z = 2
{cos
(π + 2kπ
5
)+ i sin
(π + 2kπ
5
)}, k = 0, 1, 2, 3, 4.
6. An |z| = 1, n.d.o.∣∣∣w1z+w2
w2z+w1
∣∣∣ = 1, gia opoiousd pote w1, w2 ∈ C.LÔsh'Eqoume
w1z + w2
w2z + w1
zz=|z|2=1=
w1z + w2
(w2 + w1z)z
opìte∣∣∣w1z + w2
w2z + w1
∣∣∣ =|w1z + w2||w2 + w1z||z|
|z|=|z|=
|w1z + w2||w1z + w2||z| =
|w1z + w2||w1z + w2||z| = 1
7. Migadikìc SumbolismìcExÐswsh EujeÐac: z = w1 + w2τ, w1, w2 ∈ C, tau ∈ RExÐswsh KÔklou: |z − w| = r, w ∈ C, r > 0
ExÐswsh 'Elleiyhc: |z−w|+|z+w| = 2a, w ∈ C, a > 0, estÐec:±w, meg�loc hmi�xonac:a.
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8. z = reiθ
zeia = reiθeia = rei(θ+a)
Dhlad , z epÐ eia shmaÐnei strof tou z kat� th jetik for� kat� gwnÐa a.
9. i) 'Estw p + qi rÐza thc a0zn + a1z
n−1 + · · · + an = 0 me a0 6= 0, a1, . . . , an ∈R, p, q ∈ R. Tìte h p− qi eÐnai epÐshc rÐza (jèse p + qi = reiθ kai met� gr�yeth suzug exÐswsh).
ii) z2 + (2i − 3)z + 5 − i = 0 ⇒ z =−(2i−3)±
√(2i−3)2−4·1·(5−i)
2·1 ⇒ z = 2 − 3i, z = 1 + i.OQI suzugeÐc rÐzec. (Ed¸ oi suntelestèc eÐnai migadikoÐ arijmoÐ.)
1.3.2 Topologik� jèmata sqetik� me to migadikì epÐpedoAkoloujÐec kai Seirèc
Orismìc 1.3.1 zn → zor⇔ |zn − z| → 0 (sugklÐnei sto R)
ParathroÔme ìti zn → z ⇒ Rezn → RezImzn → Imz
Orismìc 1.3.2 {zn} akoloujÐa Cauchy ⇔ ∀ε > 0 ∃N ∈ Z : n, m > N ⇒ |zn− zm| < ε
Prìtash 1.3.1 H {zn} eÐnai sugklÐnousa ⇔ h {zn} eÐnai Cauchy.
Orismìc 1.3.3 Mia seir�∑∞
k=1 zk sugklÐnei an h akoloujÐa twn merik¸n ajroism�twn{sn} sugklÐnei, ìpou sn = z1 +z2 + · · ·+zn. To ìrio, tìte, thc {sn} lègetai ìrio thc seir�c.
Idiìthtec
(1) To �jroisma kai h diafor� sugklinous¸n seir¸n sugklÐnei.
(2) AnagkaÐa sunj kh gia th sÔgklish thc∑∞
k=1 zk eÐnai: zn → 0 ìtan n →∞.
(3) Ikan sunj kh gia th sÔgklish thc∑∞
k=1 zk eÐnai h sÔgklish thc∑∞
k=1 |zk| (opìte h∑∞k=1 zk lègetai apolÔtwc sugklÐnousa).
ParadeÐgmata
(1) zn → 0 an |z| < 1 afoÔ |zn − 0| = |z|n → 0
(2) nn+i
→ 1 afoÔ∣∣∣ nn+i
− 1∣∣∣ =
∣∣∣ −in+i
∣∣∣ = 1√n2+1
→ 0
(3) H∑∞
k=1ik
k2+isugklÐnei, afoÔ
∣∣∣ ik
k2+i
∣∣∣ = 1√k4+1
kai afoÔ h∑∞
k=11√
k4+1sugklÐnei.
13
(4) H∑∞
k=11
k+iapoklÐnei, afoÔ 1
k+i= k−i
k2+1kai afoÔ h
∑∞k=1 Re
(1
k+i
)apoklÐnei.
Pr�gmati,∑∞
k=1 Re(
1k+i
)=
∑∞k=1
kk2+1
.
Jèse ak = 1k+1
, bk = kk2+1
.Tìte bk ≥ ak ≥ 0 gia k ≥ 1.Ex�llou h
∑∞k=1
1k+1
apoklÐnei. Apì to Krit rio SÔgklishc apoklÐnei kai h∑∞
k=1k
k2+1.
1.3.3 AkoloujÐec - Ask seic1. 'Estw xn = 1+r cos a+r2 cos 2a+· · ·+rn cos na, r ∈ (0, 1). Na brejeÐ to limn→∞ xn.
LÔshJètw yn = 1 + r sin a + r2 sin 2a + · · ·+ rn sin na
kai zn := xn + iyn = 1 + r(cos a + i sin a) + · · ·+ rn(cos na + i sin na)
Jètw w := r(cos a + i sin a) kai parathr¸ ìti |w| = r < 1.Apì ton tÔpo tou de Moivre èqw:
zn = 1 + w + w2 + · · ·+ wn =1− wn+1
1− w
kai afoÔ |w| < 1 : limn→∞ = 11−w
. Sunep¸c
lim xn = lim(Rezn) = Re(lim zn) = Re
(1
1− w
)=
= Re
(1
1− r(cos a + i sin a)
)= Re
(1
(1− r cos a)− ir sin a)
)
= Re
((1− r cos a) + ir sin a
(1− r cos a)2 + r2 sin2 a)
)=
1− r cos a
1− 2r cos a + r2
2. Na brejoÔn ta limn→∞ in
n, limn→∞
(1+i)n
n.
LÔsh
a)∣∣∣ in
n
∣∣∣ = |in|n
= |i|nn
= 1n
< ε, ìtan n > 1ε.
b) un = (1+i)n
n∣∣∣un+1
un
∣∣∣ =∣∣∣ (1+i)n+1
(1+i)nn
n+1
∣∣∣ = nn+1
|1 + i| = n√
2n+1
gia k�je n ≥ 3 èqoumen√
2n+1
> 3√
24
> 1.05 > 1
opìte |un| > (1.03)n−3|u3|,�ra h un de sugklÐnei.
14
1.4 SeirècPrìtash 1.4.1
H gewmetrik seir�∑∞
n=0 zn =
{1
1−z, |z| < 1
apoklÐnei, |z| ≥ 1
Prìtash 1.4.2 (Krit rio SÔgkrishc)
(i) |zk| ≤ |wk| kai∑∞
k=1 |wk|: sugklÐnei ⇒ ∑∞k=1 zk: sugklÐnei apolÔtwc.
(ii) |zk| ≤ |wk| kai∑∞
k=1 |wk|: apoklÐnei ⇒ ∑∞k=1 zk: apoklÐnei all� h
∑wk mporeÐ na
sugklÐnei ìqi.
Prìtash 1.4.3 (Krit rio thc p-seir�c)∑∞
n=11np sugklÐnei an p > 1 kai apoklÐnei sto ∞ an p ≤ 1.
Prìtash 1.4.4 (Krit rio tou Lìgou)
limn→∞∣∣∣zn+1
zn
∣∣∣
< 1> 1= 1
⇒∞∑
n=1
sugklÐnei apolÔtwcapoklÐneiden efarmìzetai to krit rio
Prìtash 1.4.5 (Krit rio thc RÐzac)
limn→∞(|zn|) 1n
< 1> 1= 1
⇒∞∑
n=1
sugklÐnei apolÔtwcapoklÐneiden efarmìzetai to krit rio
Prìtash 1.4.6 (Krit rio Cauchy)
(i) Mia akoloujÐa fn(z) sugklÐnei omoiìmorfa sto sÔnolo A ⇔∀ε > 0∃N : n ≥ N ⇒ |fn(z)− fn+p(z)| < ε ∀z ∈ A, ∀p = 1, 2, . . .
(ii) H seir�∑∞
k=1 gk(z) sugklÐnei omoiìmorfa sto sÔnolo A ⇔
∀ε > 0∃N : n ≥ N ⇒∣∣∣∣
p∑
k=n+1
gk(z)
∣∣∣∣ < ε ∀z ∈ A, ∀p = 1, 2, . . .
Prìtash 1.4.7 (To M-krit rio tou Weierstrass)
'Estw gn akoloujÐa sunart sewn pou orÐzetai sto A ∈ C. 'Estw ìti up�rqei akolou-jÐa pragmatik¸n arijm¸n Mn ≥ 0:
(i) |gn(z)| ≤ Mn, ∀z ∈ A
(ii) h∑∞
n=1 Mn sugklÐnei
Tìte h∑∞
n=1 gn(z) sugklÐnei apolÔtwc kai omoiìmorfa epÐ tou A.
15
1.4.1 ParadeÐgmata � Ask seic1.
∑∞n=1
1nz sugklÐnei apolÔtwc gia Rez > 1 kai omoiìmorfa gia Rez ≥ 1 + ε, ε > 0.
'Estw z = x + iy. Tìte nz = ez log n = e(x+iy) log n.∣∣∣ 1nz
∣∣∣ = 1ex log n = 1
nx kai∑∞
n=1
∣∣∣ 1nz
∣∣∣ =∑∞
n=11
nx sukglÐnei an x > 1, dhl. an Rez > 1.
'Otan Rez ≥ 1 + ε jètw Mn = 1n1+ε kai h omoiìmorfh sÔgklish èpetai apì to M -
krit rio tou Weierstrass.
2.∑∞
n=0e−inz
n2+1sugklÐnei omoiìmorfa sto hmiepÐpedo Imz < −a gia k�je a > 0.
'Estw z = x + iy. Tìte∣∣∣ e−inz
n2+1
∣∣∣ =∣∣∣ e−inxeny
n2+1
∣∣∣ = eny
n2+1
An Imz = y < −a < 0, tìte eny < e−na kai ètsi∣∣∣ e−inz
n2+1
∣∣∣ ≤∣∣∣ e−na
n2+1
∣∣∣ := Mn.
Exet�zw th sÔgklish thc∑∞
n=0 Mn.
Krit rio Lìgou: lim Mn+1
Mn= lim
∣∣∣ e−(n+1)a
(n+1)2+1n2+1e−na
∣∣∣ = e−a lim n2+1n2+2n+2
= e−a < 1
3.∑∞
n=1zn
n
Jètw Aσ = {z : |z| ≤ σ}, 0 ≤ σ < 1.
Jètw gn(z) = zn
n.
Tìte |gn(z)| = |z|nn≤ σn
n≤ σn := Mn kai afoÔ σ < 1, h
∑Mn sugklÐnei.
'Ara h∑∞
n=1zn
nsugklÐnei omoiìmorfa sto Aσ.
H seir� aut sugklÐnei shmeiak� sto A = {z : |z| < 1} afoÔ k�je z ∈ A brÐsketaiarket� kont� se k�poio Aσ gia σ arket� konta sto 1.'Omwc h seir� den sugklÐnei omoiìmorfa epÐ tou A. An sunèkline, h
∑∞n=1
xn
nja
sunèkline omoiìmora epÐ tou [0, 1) pou den isqÔei. (�skhsh).
16
Kef�laio 2
2.1 Kat�taxh Sunìlwn sto Migadikì EpÐpedoOrismìc 2.1.1 .
• D(z0, r) = {z : |z − z0| < r} anoiqtìc dÐskoc, perioq tou z0.
• C(z0, r) = {z : |z − z0| = r} kÔkloc.
• S : anoiktì orc⇔ ∀z ∈ S ∃δ > 0 : D(z0, δ) ⊆ S.
• S = C− S (S = {z ∈ C : z /∈ S}) sumpl rwma tou S.
• S : kleistì orc⇔ S anoiqtì⇔ {zn} ∈ S kai zn → z ⇒ z ∈ S
• ∂S = {z : ∀δ > 0 : D(z, δ) ∩ S = ∅ kai D(z, δ) ∩ S 6= ∅}• S = S ∪ ∂S
• S : fragmèno ⇔ ∃M > 0 : S ⊆ D(0,M)
• S kleistì kai fragmèno orc⇔ S sumpagèc
• S mh sunektikì: up�rqoun dÔo anoiqt�, xèna sÔnola A kai B pou h ènws touc perièqeito S en¸ oÔte to A oÔte to B perièqoun to S.
• S sunektikì an den eÐnai mh sunektikì.
• [z1, z2] : to eujÔgrammo tm ma me �kra z1, z2.
• polugwnik gramm : peperasmènh ènwsh eujugr�mmwn tmhm�twn thc morf c [z0, z1]∪[z1, z2] ∪ . . . ∪ [zn−1, zn].
• An k�je dÔo shmeÐa tou S mporoÔn na enwjoÔn me mia polugwnik gramm pou periè-qetai sto S, tìte to S lègetai polugwnik� sunektikì.
17
• polugwnik� sunektikì ⇒ sunektikì. To antÐstrofo den isqÔei. Oi dÔo ènnoiec eÐnaiisodÔnamec gia ta anoiqt� sÔnola.
• tìpoc orc= anoiqtì + sunektikì.
2.2 SuneqeÐc Sunart seicOrismìc 2.2.1 Mia sun�rthsh migadik¸n tim¸n f(z) orismènh se mia perioq tou z0, eÐnaisuneq c sto z0, an zn → z0 sunep�getai ìti f(zn) → f(z0).
Diaforetik�, h f eÐnai suneq c sto z0 an gia k�je ε > 0 up�rqei δ > 0 ètsi ¸ste an|z − z0| < δ tìte |f(z)− f(z0)| < ε.
H F eÐnai suneq c se ènan tìpo D an gia k�je akoloujÐa {zn} ⊆ D kai z ∈ D tètoia¸ste zn → z, na isqÔei f(zn) → f(z).
An diasp�soume thn f sto pragmatikì kai fantastikì thc mèroc
f(z) = f(x, y) = u(x, y) + iv(x, y)
ìpou h u kai h v paÐrnoun pragmatikèc timèc, eÐnai fanerì ìti h f eÐnai suneq c tìte kaimìno tìte an oi u kai v eÐnai suneqeÐc sunart seic tou (x, y).
ParadeÐgmata.
1. K�je polu¸numo
P (x, y) =m∑
j=1
n∑
k=1
akjxkyj
eÐnai suneq c sun�rthsh se ìlo to epÐpedo.
2. H f(z) = 1z
= xx2+y2 − i y
x2+y2 eÐnai suneq c sto “epÐpedo”{z : z 6= 0}.Idiìthtec
EÐnai profanèc ìti to �jroisma, to ginìmeno kai to phlÐko (me mh mhdenikì paronomast )suneq¸n sunart sewn eÐnai suneq c sun�rthsh.
Lème ìti f ∈ Cn an kai to Ref kai to Imf èqoun suneqeÐc merikèc parag¸gouc n−t�xhc.Mia akoloujÐa sunart sewn {fn} sugklÐnei sthn f omoiìmorfa sto D, an gia k�je ε > 0up�rqei N > 0 tètoio ¸ste n > N sunep�getai ìti |f(zn)− f(z)| < ε gia k�je z ∈ D.Anaferìmenoi p�li sta pragmatik� kai fantastik� mèrh twn {fn}, blèpoume ìti to omoiì-morfo ìrio suneq¸n sunart sewn eÐnai suneq c sun�rthsh.M-test. An fk suneq c sto D, k = 1, 2 . . . kai |fk(z)| ≤ Mk sto D kai an h
∑∞k=1 Mk
sugklÐnei, tìte h∑∞
k=1 fk(z) sugklÐnei se mia sun�rthsh f pou eÐnai suneq c sto D.JumÐzoume ìti mia suneq c sun�rthsh apeikonÐzei sumpag /sunektik� sÔnola se sumpa-g /sunektik� sÔnola, en¸ autì de sumbaÐnei gia kami� �llh kathgorÐa sunìlwn. P.q. h
18
f(z) = 1zapeikonÐzei to fragmèno sÔnolo 0 < |z| < 1 epÐ tou mh fragmènou sunìlou |z| > 1.
Tèloc èqoume to ex c je¸rhma:
Je¸rhma 2.2.1 'Estw ìti h u(x, y) èqei merikèc parag¸gouc ux kai uy pou mhdenÐzontaise k�je shmeÐo enìc tìpou D. Tìte h u eÐnai stajer ston D.
2.2.1 Stereografik Probol EÐnai suqn� qr simo na sumperil�boume sto migadikì epÐpedo to shmeÐo sto �peiro,pou sumbolÐzetai ∞. Gia na antilhfjoÔme “optik�”to shmeÐo sto �peiro, mporoÔme najewr soume ìti to migadikì epÐpedo pern�ei apì ton ishmerinì thc monadiaÐac sfaÐrac mekèntro to z = 0. Se k�je shmeÐo z tou epipèdou antistoiqeÐ akrib¸c èna shmeÐo P thcepif�neic thc sfaÐrac, pou brÐsketai wc tom thc eujeÐac zN (N o bìreioc pìloc) me thnepif�neia aut . Antistrìfwc, se k�je shmeÐo P thc epif�neiac thc sfaÐrac, plhn touN , antistoiqeÐ akrib¸c èna shmeÐo z tou epipèdou. Antistoiq¸ntac sto N to shmeÐo sto�peiro, petuqaÐnoume mia 1-1 antistoiqÐa metaxÔ twn shmeÐwn thc epif�neiac thc sfaÐrac kaitou epektetamènou migadikoÔ epipèdou. Aut h sfaÐra lègetai sfaÐra tou Riemann kaih antistoiqÐa stereografik probol , probol tou PtolemaÐou. To sÔnolo|z| > 1
εlègetai perioq tou ∞.
Orismìc 2.2.2 Lème ìti {zk} → ∞ an |zk| → ∞, dhlad an gia k�je M > 0 up�rqeiN ∈ Z: k > N sunep�getai ìti |zk| > M . OmoÐwc f(z) →∞ an |f(z)| → ∞.
2.3 Sunart seic miac migadik c metablht c z
2.3.1 Analutik� Polu¸numa'Ena polu¸numo P (x, y) lègetai analutikì polu¸numo, an up�rqoun (migadikèc) stajerècak ètsi ¸ste:
P (x, y) = a0 + a1(x + iy) + a2(x + iy)2 + . . . + aN(x + iy)N
Tìte ja lème ìti to P eÐnai polu¸numo wc proc z kai ja to gr�foume wc
P (x, y) = a0 + a1z + a2z2 + . . . + aNzN
P.q. To polu¸numo x2 +y2 +2ixy eÐnai analutikì en¸ eÔkola deÐqnetai ìti to x2 +y2−2ixyden eÐnai analutikì.
Orismìc 2.3.1 'Estw f(x, y) = u(x, y)+ iv(x, y), ìpou u kai v sunart seic pragmatik¸ntim¸n. Me thn proôpìjesh ìti up�rqoun oi ux, uy, vx, vy orÐzoume
fx = ux + ivx, fy = uy + ivy
ApodeiknÔetai ìti èna polu¸numo eÐnai analutikì tìte kai mìno tìte an Py = iPx.
19
Orismìc 2.3.2 Mia sun�rthsh f me migadikèc timèc pou orÐzetai se mia perioq tou z,lègetai diaforÐsimh sto z an up�rqei to
limh→0
f(z + h)− f(z)
h
To ìrio autì sumbolÐzetai me f ′(z). To h den eÐnai aparait twc pragmatikì.
IkanopoioÔntai oi gnwstoÐ tÔpoi gia thn prìsjesh, ton pollaplasiasmì kai th diaÐreshdiaforÐsimwn sunart sewn.
Tèloc apodeiknÔetai ìti an to P eÐnai analutikì, tìte eÐnai diaforÐsimo se k�je z. MiaeurÔterh kl�sh sunart sewn tou z, eÐnai autèc pou dÐnontai apì �peira polu¸numa tou z, alli¸c dunamoseirèc tou z.
2.3.2 DunamoseirècOrismìc 2.3.3 Dunamoseir� eÐnai mia seir� thc morf c
∑∞k=0 ckz
k.
Gia th melèth thc sÔgklishc dunamoseir¸n, mac qrei�zetai h ènnoia tou lim (limsup), miacjetik c pragmatik c akoloujÐac:
limn→∞
an := limn→∞
(supk≥n
ak
).
AfoÔ to supk≥n ak eÐnai mia fjÐnousa sun�rthsh tou n, to ìrio up�rqei p�nta eÐnai ∞.Oi idiìthtec tou lim pou ja qreiastoÔme eÐnai:An limn→∞an = L tìte:
1. gia k�je N kai gia k�je ε > 0, up�rqei κ > N tètoio ¸ste ak ≥ L
2. gia k�je ε > 0, up�rqei N tètoio ¸ste ak ≤ L + ε, gia k�je k > N
Je¸rhma 2.3.1 'Estw ìti lim|ck|1/k = L.
1. An L = 0, h∑
ckzk sugklÐnei gia ìla ta z.
2. An L = +∞, h∑
ckzk sugklÐnei mìno gia z = 0.
3. An 0 < L < +∞, jètoume R = 1L. Tìte h
∑ckz
k sugklÐnei gia |z| < R kai apoklÐneigia |z| > R. To R lègetai aktÐna sÔgklishc thc dunamoseir�c.
Parat rhsh 2.3.1 1. An h∑
ckzk èqei aktÐna sÔgklishc R, tìte sugklÐnei omoiì-
morfa se k�je mikrìtero dÐsko |z| ≤ R− δ ki ètsi eÐnai suneq c sto pedÐo sÔgklis cthc.
20
2. To �jroisma sugklinous¸n dunamoseir¸n eÐnai sugklÐnousa dunamoseir�.
3. To ginìmeno Cauchy sugklinous¸n dunamoseir¸n eÐnai sugklÐnousa dunamoseir� gia|z| < min(Ra, Rb), ìpou Ra, Rb oi aktÐnec sÔgklishc twn dÔo dunamoseir¸n. Upen-jumÐzoume ìti an an, bn eÐnai dÔo akoloujÐec to ginìmeno Cauchy eÐnai h akoloujÐa
cn =n∑
k=0
akbn−k
.
4. An sumbeÐ na up�rqei to lim∣∣∣ ck
ck+1
∣∣∣, tìte
R =1
lim|ck|1/k= lim
∣∣∣∣ck
ck+1
∣∣∣∣ .
Aut h sqèsh èqei meg�lh praktik shmasÐa.
ParadeÐgmata
1.∑∞
n=1 nzn.H dunamoseir� aut sugklÐnei gia |z| < 1 kai apoklÐnei gia |z| > 1 afoÔ n1/n → 1 ∈(0,∞).
2.∑∞
n=1zn
n2 .H dunamoseir� aut èqei, epÐshc, aktÐna sÔgklishc 1. 'Omwc, sugklÐnei kai gia |z| = 1,afoÔ
∣∣ zn
n2
∣∣ = 1n2 → 0.
3.∑∞
n=0zn
n!.
H dunamoseir� sugklÐnei gia k�je z, afoÔ 1(n!)1/n → 0.
4.∑∞
n=0 n!zn.lim
∣∣∣ n!(n+1)!
∣∣∣ = lim 1n+1
= 0 ⇒ R = 0. 'Ara h dunamoseir� sugklÐnei mìno sto z = 0.
5.∑∞
n=0 n2(2z − 1)n =∑∞
n=0 n2(z − 1
2
)n.H aktÐna sÔgklishc thc dunamoseir�c aut c eÐnai R = lim(2nn2)1/n = lim(2n2/n =2 ⇒ R = 1
2kai kèntro eÐnai to 1
2.
Je¸rhma 2.3.2 'Estw f(z) =∑
cnzn kai ìti h seir� sugklÐnei gia |z| < R. Tìte up�rqei
h f ′(z) kai isoÔtai me∑
ncnzn−1 ston |z| < R.
Pìrisma 2.3.1 Oi dunamoseirèc èqoun k�je t�xhc par�gwgo mèsa sto pedÐo sÔgklis ctouc.
Pìrisma 2.3.2 An h f(z) =∑∞
n=0 cnzn èqei mh mhdenik aktÐna sÔgklishc, tìte cn =
f (n)(0)n!
gia k�je n.
21
Je¸rhma 2.3.3 (Monadikìthtac Dunamoseir¸n). 'Estw ìti h∑∞
n=0 cnzn mhdenÐzetai seìla ta shmeÐa miac {zk} : zk → 0 kai zk 6= 0. Tìte h dunamoseir� eÐnai tautotik� Ðsh me tomhdèn.
Pìrisma 2.3.3 An h∑
anzn kai h∑
bnzn sugklÐnoun kai sumpÐptoun se èna sÔnoloshmeÐwn pou èqei shmeÐo suss¸reushc to 0, tìte an = bn gia k�je n.
Pìrisma 2.3.4 'Estw lim|cn|1/n < ∞, jètoume f(z) =∑∞
n=0 cn(z − a)n. Tìte cn =f (n)(a)
n!.
Je¸rhma 2.3.4 (Abel). 'Estw ìti h dunamoseir�∑∞
n=0 cnzn sugklÐnei se k�poio shmeÐo
z1 6= 0. Tìte sugklÐnei apolÔtwc se k�je shmeÐo z: |z| < |z1|. 'Estw r < |z1|. Tìte hdunamoseir� sugklÐnei omoiìmorfa gia |z| ≤ r.
2.3.3 Analutikèc Sunart seicAnalutikìthta kai exis¸seic Cauchy-Riemann
Prìtash 2.3.1 An h f = u + iv eÐnai diaforÐsimh sto z, up�rqoun oi fx kai fy sto z kaiikanopoioÔn tic exis¸seic Cauchy-Riemann.
fy = ifx,
isodÔnama {ux = vy
uy = −vx
To antÐstrofo den isqÔei. Up�rqoun sunart seic pou den diaforÐzontai se èna shmeÐopar‘ìlo pou up�rqoun ekeÐ oi merikèc par�gwgoÐ touc kai ikanopoioÔn tic sunj kec Cauchy-Riemann.
Par�deigma 2.3.1 .
f(z) = f(x, y) =
{xy(x+iy)x2+y2 , z 6= 0
0 , z = 0
H f = 0 kai stouc dÔo �xonec kai sunep¸c fx = fy = 0 sto mhdèn, ìmwc to
limz→0
f(z)− f(0)
z= lim
(x,y)→(0,0)
xy
x2 + y2
den up�rqei. Pr�gmati epÐ thc eujeÐac y = ax : f(z)−f(0)z
= a1+a2 gia z 6= 0 kai sunep¸c to
ìrio exart�tai apì to a!
IsqÔei entoÔtoic to ex c merikì antÐstrofo:
22
Prìtash 2.3.2 'Estw ìti up�rqoun se mia perioq tou z oi fx, fy. Tìte an oi fx, fy
eÐnai suneqeÐc sto z kai isqÔei fy = ifx ekeÐ, h f eÐnai diaforÐsimh sto z.
Orismìc 2.3.4 H f eÐnai analutik (olìmorfh) sto z, an eÐnai diaforÐsimh se mia pe-rioq tou z. H f eÐnai analutik se èna sÔnolo S an eÐnai diaforÐsimh se ìla ta shmeÐa enìcanoiqtoÔ sunìlou pou perièqei to S. Mia analutik sun�rthsh f : C → C (se ìlo to C)lègetai akèraia sun�rthsh.
'Eqoume dh parathr sei ìti to �jroisma, to ginìmeno kai to phlÐko diaforÐsimwn sunar-t sewn eÐnai diaforÐsimh sun�rthsh. OmoÐwc kai h sÔnjesh. 'Opwc kai gia tic pragmatikècsunart seic, h antÐstrofh miac sun�rthshc mporeÐ na mhn eÐnai kan suneq c. O epìmenocorismìc, mac epitrèpei na mil�me gia diaforisimìthta twn antistrìfwn sunart sewn.
Orismìc 2.3.5 'Estw S kai T anoiqt� sÔnola kai èstw ìti h f eÐnai 1-1 sto S me f(S) =T . H g eÐnai h antÐstrofh thc f sto T , an f(g(z)) = z gia z ∈ T . H g eÐnai h antÐstrofhthc f sto z0, an eÐnai h antÐstrofh thc f se k�poia perioq tou z0.
Prìtash 2.3.3 'Estw ìti h g eÐnai h antÐstrofh thc f sto z0 kai ìti h g eÐnai suneq csto z0. An h f eÐnai diaforÐsimh sto g(z0) kai an f ′(g(z0)) 6= 0, tìte h g eÐnai diaforÐsimhsto z0 kai g′(z0) = 1
f ′(g(z0)).
H analutikìthta eÐnai mia exairetik� shmantik idiìthta. Ja asqolhjoÔme idiaÐtera mazÐ thcse epìmena kef�laia. Gia thn ¸ra dÐnoume dÔo �mesec sunèpeièc thc.
Prìtash 2.3.4 An h f = u + iv eÐnai analutik se ènan tìpo D kai h u eÐnai stajer ,tìte h f eÐnai stajer .
Prìtash 2.3.5 An h f = u + iv eÐnai analutik se ènan tìpo D kai h |f | eÐnai stajer ekeÐ, tìte h f eÐnai stajer .
Fusik ermhneÐa thc Diaforisimìthtac
'Eqoume dei ìti gia na eÐnai mia sun�rthsh diaforÐsimh, prèpei na ikanopoieÐtai mia sugkekri-mènh sunj kh, pou “analutik�”ekfr�zetai apì tic sunj kec Cauchy-Riemann. Ja doÔmet¸ra ti shmaÐnei aut h sunj kh “fusik�”. Poia eÐnai dhlad ekeÐnh h xeqwrist idiìth-ta tou dianusmatikoÔ pedÐou pou diakrÐnei mia diaforÐsimh apì mia mh diaforÐsimh migadik sun�rthsh?
H kajarìthta thc ap�nthshc exart�tai polÔ apì ton kalì sumbolismì. 'Estw z ènametablhtì shmeÐo enìc disdi�statou dianusmatikoÔ pedÐou kai s to di�nusma pou antistoiqeÐ
23
sto z. 'Estw x, y oi suntetagmènec tou z kai u, v oi suntetagmènec tou w. Tìte
z = x + iy, w = u + iv opìte w = u− iu.
JewroÔme to w wc sun�rthsh tou z
w = u− iv = f(z) = f(x + iy)
An h f eÐnai diaforÐsimh, ja prèpei na isqÔei (Prìtash 2.3.1) fy = ifx, dhlad
ux − ivx =1
i(uy − vy) ⇔
ux + vy = 0 (2.1)
vx − uy = 0 (2.2)
Autèc oi exis¸seic ekfr�zoun ìti h sun�rthsh pou parist�netai apì to dianusmatikì pedÐoeÐnai diaforÐsimh.
1. JewroÔme to dianusmatikì pedÐo mac wc pedÐo ro c kai to w wc mia taqÔthta, thnèntash tou reÔmatoc sto shmeÐo z. Tìte h èkfrash ux + vy lègetai apìklish toudianÔsmatoc w, sumbolÐzetai me divw kai metr�ei thn exerqìmenh ro an� mon�da ìgkouse mia kleist perioq tou shmeÐou z. An divw > 0 to shmeÐo z dra wc “phg ”(source),en¸ an divw < 0 to z dra wc “katabìjra”(sink). An h apìklish mhdenÐzetai se k�jeshmeÐo, to pedÐo lègetai swlhnoeidèc (sourceless). 'Etsi h (2.1) gr�fetai
divw = 0
kai qarakthrÐzei èna swlhnoeidèc pedÐo.
2. JewroÔme to dianusmatikì pedÐo mac wc pedÐo dun�mewn kai to w wc mia dÔnamh, thnèntash tou pedÐou sto shmeÐo z. H èkfrash vx−uy lègetai strobilismìc (curl) tou wkai metr�ei to èrgo an� mon�da epif�neiac. Pio sugkekrimèna, to èrgo pou par�getaiapì to pedÐo ìtan èna mikrì swmatÐdio diagr�fei mia kleist kampÔlh pou perikleÐeito z diaireÐtai me thn epif�neia pou perikleÐei h kampÔlh. 'Otan oi diast�seic thckampÔlhc teÐnoun sto mhdèn, autì to anhgmèno èrgo teÐnei sto curlw. An to curlmhdenÐzetai se k�je shmeÐo, to pedÐo lègetai astrìbilo. 'Etsi h (2.2) gr�fetai
curlw = 0
kai qarakthrÐzei to pedÐo wc astrìbilo.
SunoyÐzontac, lème ìti mia diaforÐsimh pragmatik sun�rthsh miac migadik c metablht cparist�netai me èna swlhnoeidèc kai astrìbilo pedÐo.
24
H exÐswsh tou Laplace
Mia pragmatik sun�rthsh h dÔo pragmatik¸n metablht¸n x, y lègetai armonik se ènantìpo tou epipèdou xy, an pantoÔ se autìn ton tìpo èqei suneqeÐc pr¸tec kai deÔterec merikècparag¸gouc kai ikanopoieÐ th merik diaforik exÐswsh:
4h = ∇2h = hxx + hyy = 0
pou eÐnai gnwst wc exÐswsh tou Laplace. Apì tic sunj kec Cauchy Riemann pouikanopoioÔn oi suntetagmènec sunart seic miac analutik c sun�rthshc f = u+ iv, paragw-gÐzontac wc proc x, paÐrnoume ìti:
uxx = vyx uyx = −vxx.
Parag¸gish wc proc y, dÐnei antÐstoiqa
uxy = vyy uyy = −vxy.
H sunèqeia twn merik¸n parag¸gwn exasfalÐzei ìti vyx = vxy kai uxy = uyx. Sunep¸c:
uxx + uyy = 0 kai vxx + vyy = 0.
'Eqoume, dhlad , ìti kai to pragmatikì kai to fantastikì mèroc miac analutik c sun�rthshceÐnai armonikèc sunart seic.
Ask seic
1. An oi f, f eÐnai analutikèc ston tìpo D tìte h f eÐnai stajer ston D.LÔsh'Estw f = u + iv. AfoÔ h f eÐnai analutik isqÔoun oi sqèseic Cauchy-Riemann,dhlad ux = vy kai vx = −uy. Ex�llou f = u − iv. AfoÔ ìmwc eÐnai kai h fanalutik , isqÔoun kai gia aut n oi sunj kec Cauchy Riemann, dhlad ux = −vy kai−vx = −uy. Ap’autèc èpetai ìti ux = uy = vx = vy = 0. 'Ara h f eÐnai stajer .
2. (Polikèc Suntetagmènec) ApodeÐxte tic parak�tw sqèseic.
(aþ) Oi exis¸seic Cauchy-Riemann : ∂u∂r
= 1r
∂v∂θ
, ∂v∂r
= −1r
∂u∂θ
.
(bþ) H exÐswsh Laplace : ∂2φ∂r2 + 1
r∂φ∂r
+ 1r2
∂2φ∂θ2 = 0.
(gþ) Par�gwgoc : f ′(z) = e−iθ ∂f∂r
= 1iz
∂f∂θ
.
LÔshJa apodeÐxoume thn pr¸th sqèsh. 'Eqoume ìti z = r(cos θ + i sin θ) = x + iy, r =√
x2 + y2, tan θ = yx. 'Ara
∂r
∂x= cos θ,
∂r
∂y= sin θ,
∂θ
∂x=
1
rsin θ,
∂θ
∂y=
1
rcos θ,
25
∂u
∂x=
∂u
∂r
∂r
∂x+
∂u
∂θ
∂θ
∂x,
∂u
∂y=
∂u
∂r
∂r
∂y+
∂u
∂θ
∂θ
∂y,
∂v
∂x=
∂v
∂r
∂r
∂x+
∂v
∂θ
∂θ
∂x,
∂v
∂y=
∂v
∂r
∂r
∂y+
∂v
∂θ
∂θ
∂y.
Apì tic parap�nw paÐrnoume ìti
∂u
∂x=
∂u
∂rcos θ − 1
r
∂u
∂θsin θ,
∂u
∂y=
∂u
∂rsin θ +
1
r
∂u
∂θcos θ,
∂v
∂x=
∂v
∂rcos θ − 1
r
∂v
∂θsin θ,
∂v
∂y=
∂v
∂rsin θ +
1
r
∂v
∂θcos θ.
Apì tic sunj kec C-R (gia kartesianèc suntetagmènec) paÐrnoume to apotèlesma.
3. 'Estw h f(z) = u(x, y)+iv(x, y), analutik ston tìpo D kai up�rqoun a, b, c ∈ R−{0}¸ste na isqÔei au(x, y) + bv(x, y) = c ston D. DeÐxte ìti h f eÐnai stajer ston D.LÔshParagwgÐzontac thn au(x, y) + bv(x, y) = c ja èqoume
aux + bvx = 0auy + bvy = 0
}C-R⇒ avy + bvx = 0
−avx + bvy = 0
}⇒
abvy + b2vx = 0abvy − a2vx = 0
}⇒ (a2 + b2)vx = 0 ⇒ vx = 0.
'Ara kai vy = 0 kai oi sunj kec C − R dÐnoun ìti ux = uy = 0. Sunep¸c h f eÐnaistajer .
4. H f(z) = z den eÐnai analutik .LÔsh'Estw z = x + yi. Tìte f(z) = u(x, y) + iv(x, y) = x − iy. Epomènwc ux = 1 kaivy = −1. Oi sunj kec C-R den isqÔoun �ra h sun�rthsh den eÐnai analutik .
5. SuzugeÐc suntetagmènec.LÔshz = x + yi, z = x− iy. 'Ara x = 1
2(z + z), y = 1
2i(z − z). 'Eqoume ìti
∂f
∂x=
∂f
∂z
∂z
∂x+
∂f
∂z
∂z
∂x=
∂f
∂z+
∂f
∂zkai
∂f
∂y=
∂f
∂z
∂z
∂y+
∂f
∂z
∂z
∂y= i
(∂f
∂z− ∂f
∂z
)
'Ara∂f
∂z=
1
2
(∂f
∂x− i
∂f
∂y
)
kai∂f
∂z=
1
2
(∂f
∂x+ i
∂f
∂y
)
Oi sunj kec C-R eÐnai ∂f∂y
= i∂f∂x
26
6. 'Estw f(z) = u(x, y) + iv(x, y) analutik . EÐdame, tìte ìti oi u, v eÐnai armonikèc,dhlad
∂2u
∂x2+
∂2u
∂y2= 0 kai
∂2v
∂x2+
∂2v
∂y2= 0.
Oi u, v (pragmatikèc sunart seic) orismènec se ènan tìpo D lègontai suzugeÐc armo-nikèc an h f = u + iv (migadik sun�rthsh) eÐnai analutik ston G isodÔnama an oiu, v eÐnai armonikèc kai ikanopoioÔn tic sunj kec Cauchy-Riemann.
(aþ) H φ(x, y) = x2 − y2 eÐnai armonik . Na brejeÐ h suzug c thc.LÔsh
∂φ∂x
= 2x ⇒ ∂2φ∂x2 = 2
∂φ∂y
= −2y ⇒ ∂2φ∂y2 = −2
}=⇒ ∂2φ
∂x2+
∂2φ
∂y2= 0.
'Estw w h suzug c thc. Ja prèpei oi φ, w na ikanopoioÔn tic sunj kec C-Rwy = φx kai wx = −φy. 'Ara ja prèpei wy = 2x ⇒ w = 2xy + µ(x) ìpouµ(x) stajer wc proc y. 'Ara wx = 2y + µ′(x) ⇒ −φy = 2y + µ′(x) ⇒ 2y =2y + µ′(x) ⇒ µ′(x) = 0 ⇒ µ1(x) = c ⇒ w(x, y) = 2xy + c.
(bþ) Na brejeÐ analutik sun�rthsh f(z) = u(x, y)+iv(x, y), apì tic sqèseic v(x, y) =y
x2+y2 kai f(2) = 0.LÔshvx = − 2xy
(x2+y2)2kai vy = x2−y2
(x2+y2)2. Epeid prèpei na ikanopoioÔntai oi sunj kec C-
R ja èqoume: uy = −vx = 2xy
(x2+y2)2⇒ u(x, y) =
∫2xy
(x2+y2)2dy + µ(x) = − x
x2+y2 +
µ(x). 'Etsi paÐrnoume ìti ux = x2−y2
(x2+y2)+µ′(x) kai afoÔ prèpei ux = vy ⇒ µ(x) =
c. Apì th sqèsh f(2) = 0 paÐrnoume ìti µ(x) = µ(2) = 12, kai ètsi telik�
f(z) =1
2− 1
z
2.4 Oi Stoiqei¸deic Sunart seic
2.4.1 H ekjetik sun�rthshEpijumoÔme na orÐsoume mia ekjetik sun�rthsh miac migadik c metablht c z. Jèloumedhlad na broÔme mia analutik sun�rthsh f tètoia ¸ste
f(z1 + z2) = f(z1)f(z2)
f(x) = ex ∀x ∈ RApì tic exis¸seic autèc paÐrnoume ìti f(z) = f(x + iy) = f(x)f(iy) = exf(iy). Jètontacf(iy) = A(y) + iB(y) paÐrnoume
f(z) = exA(y) + iexB(y).
27
Gia na eÐnai h f analutik , ja prèpei na ikanopoioÔntai oi sunj kec C-R sunep¸c ja prèpeiA(y) = B′(y) kai A′(y) = −B(y). Dhlad A′′(y) = −A(y). 'Etsi èqoume ìti
A(y) = a cos y + b sin y
B(y) = −A′(y) = −b cos y + a sin y.
'Omwc f(x) = ex ⇒ A(0) = 1, B(0) = 0. Ex�llou A(0) = a, B(0) = −b. Katal goume,loipìn, sthn
f(z) = ex cos y + iex sin y
EÔkola epalhjeÔetai ìti h f eÐnai akèraia sun�rthsh pou ikanopoieÐ tic sqèseic pou jèlame.H f eÐnai sunep¸c akèraia “epèktash”thc pragmatik c ekjetik c sun�rthshc. Gr�foumef(z) = ez.
Idiìthtec thc ez
1. |ez| = ex
2. ez = ez
3. eiy = cosy + isiny
4. H exÐswsh ez = a èqei apeÐrou pl jouc lÔseic gia k�je a 6= 0 (a ∈ C)5. To pedÐo tim¸n thc ez eÐnai to C− {0}.6. ez+2πi = ez, ∀z ∈ C7. Sth lwrÐda −π < Im(z) ≤ π h ez eÐnai 1-1.
8. ez1ez2 = ez1+z2
9. (ez)′ = ez
10. H ekjetik sun�rthsh mporeÐ na oristeÐ wc ez =∑∞
n=0zn
n!
2.4.2 Oi trigwnometrikèc sunart seicGia na orÐsoume ta sin z kai cos z parathroÔme ìti gia y ∈ R isqÔei
eiy = cos y + i sin y, e−iy = cos y − i sin y
opìtesin y =
1
2i
(eiy − e−iy
)kai cos y =
1
2
(eiy + e−iy
).
28
'Etsi mporoÔme na orÐsoume akèraiec epekt�seic twn sin x kai cos x jètontac
sin z =1
2i
(eiz − e−iz
)
cos z =1
2
(eiz + eiz
)
Idiìthtec1. cos′ z = − sin z
2. sin′ z = cos z
3. cos (−z) = cos z, cos (z + 2π) = cos z,cos (z + π) = − cos z
4. sin(−z) = − sin z, sin(z + 2π) = sin z, sin(z + π) = − sin z
5. cos2 z + sinz = 1
6. cos z + i sin z = eiz (tÔpoc tou Euler)
7. cos(z1 ± z2) = cos z1 cos z2 ∓ sin z1 sin z2
8. sin(z1 ± z2) = sin z1 cos z2 ± cos z1 sin z2
9. AntÐjeta me to sin x to sin z den fr�ssetai kat’apìluth tim apì to 1.p.q. | sin(10i)| = 1
2(e10 − e−10) > 10000 (!)
10. sin z = 0 ⇔ z = κπ, κ ∈ Z11. cos z = 0 ⇔ z =
(κ + 1
2
)π, κ ∈ Z
12. oi sunart seic sunhmÐtono kai hmÐtono orÐzontai kai wc ex c:
cos z =∞∑
n=0
(−1)n z2n
(2n)!
sin z =∞∑
n=0
(−1)n z2n+1
(2n + 1)!
Oi upìloipec trigwnometrikèc sunart seic orÐzontai wc ex c:
tan z =sin z
cos z, cot z =
cos z
sin z, sec z =
1
cos z, csc z =
1
sin z
en¸ oi uperbolikèc trigwnometrikèc sunart seic wc ex c:
sinh z =1
2
(ez − e−z
), cosh z =
1
2
(ez + e−z
), tanh z =
sinh z
cosh zk.o.k.
H �lgebra kai o logismìc aut¸n twn sunart sewn gÐnetai me b�sh touc prohgoÔme-nouc orismoÔc. Oi tÔpoi pou apodeiknÔontai eÐnai Ðdioi me autoÔc pou gnwrÐzoume gia ticantÐstoiqec sunart seic miac pragmatik c metablht c.
29
2.4.3 H Logarijmik Sun�rthsh'Estw Log r o gnwstìc fusikìc log�rijmoc enìc r ∈ R+, ìpwc orÐzetai ston apeirostikìlogismì. An z ∈ C− {0} kai r = |z|, θ = arg z, orÐzoume
log z = Log r + iθ.
Aut eÐnai mia pleiìtimh sun�rthsh. An θ0 sumbolÐzei thn kÔria tÐmh tou arg z (−π < θ0 ≤π), tìte θ = θ0+2nπ, n ∈ Z kai ètsi h arqik exÐswsh gr�fetai: log z = Log r+i (θ0 + 2nπ).An t¸ra jèsoume n = 0 sthn prohgoÔmenh sqèsh, paÐrnoume thn kÔria tim tou logarÐjmouLog z = Log r + iθ0, r > 0, −π < θ0 ≤ π. H apeikìnish w = log z eÐnai monìtimh me pedÐoorismoÔ to C − {0} kai pedÐo tim¸n to −π < Im(w) ≤ π. Profan¸c an to pedÐo orismoÔperioristeÐ sto R+, o Log z an�getai sto sun jh fusikì log�rijmo.
Parat rhsh 2.4.1 w = Log z ⇔ z = ew
Melet¸ntac tic sunist¸sec sunart seic Log r kai θ0 tou Log z, parathroÔme ìti eÐnai su-neq c sto {(r, θ) : r > 0, pi < θ < π} kai ìti autì eÐnai to mègisto dunatì sÔnolo, ìpou hLog z eÐnai suneq c. EpÐshc parathroÔme ìti h sun�rthsh Log z eÐnai analutik ston para-p�nw tìpo. (Autì èpetai apì tic sunj kec C-R kai pio sugkekrimèna apì thn polik morf touc: ur(r0, θ0) = 1
r0vθ(r0, θ0), 1
r0uθ(r0, θ0) = −vr(r0, θ0)). 'Amesa prokÔptei h idiìthta
d
dzLog z =
1
z, (|z| > 0, −π < Argz < π).
an perioristoÔme sto sÔnolo {(r, θ) : r > 0, a < θ < a+2π, a : aujaÐretoc stajerìc arijmìc }h sun�rthsh log z = Log r + iθ eÐnai monìtimh kai suneq c. ApodeiknÔetai, ìpwc parap�nw,ìti eÐnai analutik kai ìti
d
dzlog z =
1
z(|z| > 0, a < argz < a + 2π)
'Enac kl�doc miac pleiìtimhc sun�rthshc f eÐnai opoiad pote monìtimh sun�rthsh Fpou eÐnai analutik se k�poion tìpo, se k�je shmeÐo z tou opoÐou h tim F (z) eÐnai miaapì tic timèc f(z). Wc proc autìn ton orismì, h sun�rthsh Log z orismènh ston tìpo{(r, θ) : r > 0,−π < θ < π} sunist� ènan kl�do thc log z. Autìc o kl�doc lègetai kÔriockl�doc. H sun�rthsh log z eÐnai ènac �lloc kl�doc thc Ðdiac pleiìtimhc sun�rthshc.
Idiìthtec thc log z
1. elog z = z
2. log ez = z + 2nπi, n ∈ Z3. log(z1z2) = log z1 + log z2
4. log(z
1n
)= 1
nlog z, n ∈ N
5. z1n = exp
(1n
log z), n ∈ N
6. log zn 6= n log z, n ∈ N
30
2.4.4 Oi sunart seic zλ, λz, λ ∈ COrismìc 2.4.1 zλ = exp(λ log z)
H sun�rthsh zλ eÐnai monìtimh kai analutik ston tìpo {(r, θ : r > 0, a < θ < a + 2π}.H par�gwgoc autoÔ tou kl�dou thc zλ dÐnetai apì th sqèsh
d
dzzλ = λzλ−1 (|z| > 0, a < arg z < a + 2π)
'Otan a = −π, dhlad −π < arg z < π, h sun�rthsh zλ lègetai kÔrioc kl�doc thc pleiìtimhcsun�rthshc zλ. Kat’antistoiqÐa me ton orismì thc λz èqoume
Orismìc 2.4.2 λz = exp(z log λ), λ ∈ C− {0}.'Otan kajoristeÐ mia tim thc log λ, h λz eÐnai akèraia sun�rthsh tou z. EÔkola faÐnetai
ìtid
dzλz = λz log λ, λ 6= 0
IsqÔoun oi gnwstoÐ kanìnec �lgebrac kai logismoÔ gi autèc tic sunart seic. Tèloc, isqÔei
(1 + z)λ =∞∑
n=0
(λn
)zn, |z| < 1,
ìpou(
λn
)= λ(λ−1)···(λ−n+1)
n!.
2.4.5 Oi antÐstrofec trigwnometrikèc sunart seic'Estw z = sin w. Tìte w = arcsin z. 'Eqoume
z =eiw − e−iw
2i⇒ e2iw − 2izeiw − 1 = 0 ⇒ eiw = iz + (1− z2)1/2
ìpou wc gnwstìn, h (1 − z2)1/2 eÐnai dÐtimh sun�rthsh tou z. PaÐrnontac logarÐjmouc,èqoume
w = arcsin z = −i log[iz + (1− z2)1/2].
H arcsin z eÐnai pleiìtimh sun�rthsh me �peirou pl jouc timèc se k�je z. 'Otan prosdiori-stoÔn sugkekrimènoi kl�doi thc tetragwnik c rÐzac kai tou logarÐjmou, h sun�rthsh aut gÐnetai monìtimh kai analutik (wc sÔnjesh analutik¸n sunart sewn). An�loga orÐzontaioi sunart seic
arccos z = −i log[z + i(1− z2)1/2]
arctan z =i
2log
i + z
i− z
31
Oi paragwgoÐ aut¸n twn tri¸n sunart sewn, mporoÔn na brejoÔn apì tic parap�nwsqèseic. Oi par�gwgoi twn dÔo pr¸twn exart¸ntai apì tic timèc pou èqoun epilegeÐ gia thntetragwnik rÐza:
d
dzarcsin z =
1
(1− z2)1/2,
d
dzarccos z =
−1
(1− z2)1/2.
AntÐjeta, h par�gwgoc thc trÐthc
d
dzarctan z =
1
1 + z2
den exart�tai apì ton trìpo me ton opoÐo gÐnetai monìtimh h sun�rthsh. Me ton antÐstoiqo,tèloc, trìpo orÐzontai oi antÐstrofec uperbolikèc sunart seic. ProkÔptei ìti:
arcsinh z = log[z + (1 + z2)1/2]
arccosh z = log[z + (z2 − 1)1/2]
arctanh z =1
2log
1 + z
1− z
Ask seic
1. Na epilujoÔn sto C oi exis¸seic ez = 1− i, ez = −1 + i.LÔshWc gnwstìn ew = z ⇒ w = log |z|+ i arg z. Sunep¸c
(aþ) ez = 1− i ⇒ z = log√
2 + i(2kπ − π
4
), k ∈ Z
(bþ) ez = −1 + i ⇒ z = log√
2 + i(2kπ + 3π
4
), k ∈ Z
2. En¸ | sin x| ≤ 1, x ∈ R, isqÔei ìti h sin z, z ∈ C den eÐnai fragmènh.LÔshPr�gmati, èstw z = x + iy. Tìte
| sin z| =∣∣∣∣eiz − e−iz
2i
∣∣∣∣ ≥|e−iz| − |eiz|
2=
ey − e−y
2.
'Estw z = iy, y ∈ R+. Tìte an z →∞⇒ y →∞⇒ | sin z| → ∞.
3. Na epilujeÐ sto C h exÐswsh cos z = 2.LÔshcos z = 2 ⇔ eiz+e−iz
2= 2 ⇔ e2iz − 4eiz + 1 = 0 ⇒ eiz = 4±√16−4
2= 2 ± √
3.'Estw z = x + iy. Tìte eiz = e−y(cos x + i sin x). 'Ara e−y cos x = 2 ± √
3 kaie−y sin x = 0 ⇒ sin x = 0 ⇒ x = kπ, k ∈ Z. Apì thn e−y cos x = 2±√3 èpetai ìti ok prèpei na eÐnai �rtioc, dhlad k = 2m, afoÔ e−y > 0 kai 2±√3 > 0. 'Etsi
e−y = 2±√
3 ⇒ y = − log(2±√
3)
32
kai sunep¸c (afoÔ 2−√3 = 12+√
3) èqoume
z = 2mπ − i log(2±√
3) = 2mπ ± i log(2 +√
3), m ∈ Z.
4. Na brejoÔn oi log�rijmoi twn (−1− i)(1− i), −1− i, 1− i.LÔsh
(aþ) log[(−1− i)(1− i)] = log(−2) = log 2 + πi + 2nπi, n ∈ Z(bþ) log(−1− i) = log
√2 + 5π
4i + 2mπi, m ∈ Z
(gþ) log(1− i) = log√
2 + 7π4
i + 2kπi, k ∈ ZParathroÔme ìti log[(−1− i)(1− i)] = log(−1− i) + log(1− i) mod 2πi.
5. Na upologistoÔn oi timèc twn 31/2, i1/2, ii, (−1)√
2.
(aþ) 31/2 = e(1/2) log 3 = e(1/2)(log 3+2kπi) = e(1/2) log 3ekπi = ±√3.
(bþ) i1/2 = e(1/2) log i = e(1/2)i(π2+2kπ) = ±e
πi4 = ±
√2
2(1 + i).
(gþ) ii = ei log i = ei(log 1+i(π2+2kπ)) = e−(π
2+2kπ), k ∈ Z. (Dhl. to ii ∈ R).
(dþ) (−1)√
2 = e√
2 log(−1) = e√
2(log 1+i(2kπ+π)) = e√
2i(2kπ+π) = cos(π√
2 + 2kπ√
2) +i sin(π
√2 + 2kπ
√2)
6. log zn 6= n log z. Blèpoume merik� paradeÐgmata.
(aþ) z = i, n = 2 :log i2 = log(−1) = (2k + 1)πi, k ∈ Z.En¸ 2 log i = (4k + 1)πi, k ∈ Z.
(bþ) Log ((1 + i)2) = 2 Log(1 + i).(gþ) Log ((−1 + i)2) 6= 2 Log(−1 + i).(dþ) An log z = Log r + iθ
(r > 0, π
4< θ < 9π
4
) ⇒ log i2 = 2 log i.(eþ) An log z = Log r + iθ
(r > 0, 3π
4< θ < 11π
4
) ⇒ log i2 6= 2 log i.
7. 'Estw Aa0 = {z : a0 ≤ Im(z) < a0 + 2π}, a0 ∈ R. Tìte h ez apeikonÐzei to Aa0 1-1kai epÐ tou C− {0}.LÔshAn ez1 = ez2 , tìte ez1−z2 = 1 ⇒ z1−z2 = 2πik, k ∈ Z. Ef’ ìson z1, z2 ∈ Aa0 , isqÔei0 ≤ Im(z1 − z2) < 2π kai afoÔ Re(z1 − z2) = 0 ja èqoume z1 = z2, dhlad h ez eÐnai
33
1-1. 'Estw t¸ra w ∈ C − {0}. JewroÔme thn ez = w sto Aa0 . Tìte an z = x + iykai w = u + iv èqoume ez = w ⇔ ex+iy = u + iv ⇔ ex(cos y + i sin y) = u + iv ⇔
ex cos y = uex sin y = v
}⇔ e2x(cos2 y + sin2 y) = u2 + v2
cos y + i sin y = u+ivex
}⇔ ex =
√u2 + v2
eiy = u+ivex
}⇔
ex = |w|eiy = w
|w|
}⇔
x = log |w| (sun jhc log�rijmoc): mia akrib¸c lÔsh.y = arg w sto [a0, a0 + 2π] èqei akrib¸c mia lÔsh
Dhlad h ez eÐnai epÐ tou C− {0}.8. H sun�rthsh Argz eÐnai suneq c sto C−R−0 kai asuneq c sto R−. OmoÐwc (fusik�)
kai gia thn Log z
2.5 GewmetrÐa twn stoiqeiwd¸n sunart sewn
2.5.1 f1(z) = z2
O z2 èqei m koc Ðso me |z|2 kai ìrisma 2 arg z. Dhlad h f1 uy¸nei sto tetr�gwno to mètrokai diplasi�zei to ìrisma.
2.5.2 f2(z) =√
z
'Estw ìti èqoume dialèxei ènan kl�do, qrhsimopoi¸ntac to 0 ≤ θ < 2π. Tìte z = reiθ ⇒√z =
√rei θ
2 me 0 ≤ θ2
< π, opìte h√
z brÐsketai p�nta sto �nw hmiepÐpedo kai oi gwnÐecupodiplasi�zontai. H f2(z) =
√z eÐnai h antÐstrofh thc f1(z) = z2 ìtan h teleutaÐa
perioristeÐ se perioq pou eÐnai 1-1.
2.5.3 f(z) = sin z
ApeikonÐzei eujeÐec par�llhlec proc ton pragmatikì �xona se elleÐyeic kai eujeÐec par�l-lhlec proc ton fantastikì �xona se uperbolèc. Pr�gmati,
sin z = sin x + iy = sin x cos (iy) + sin (iy) cos x =
= sin x cosh y + i sinh y cos x,
ìpou cosh y = ey+e−y
2, sinh y = ey−e−y
2, afoÔ isqÔei sin iy = eiiy−e−iiy
2i= i ey−e−y
2= i sinh y
kai antistoÐqwc cos (iy) = cosh y. 'Estw, loipìn, pr¸ta y = y0 (eujeÐa par�llhlh me ton�xona twn pragmatik¸n). Tìte an sin z = u + iv, èqoume lìgw twn prohgoÔmenwn sqèsewn
u = sin x cosh y0 ⇒ sin x = ucosh y0
v = cos x sinh y0 ⇒ cos y = vsinh y0
}⇒ u2
cosh2 y0
+v2
sinh2 y0
= 1
34
pou eÐnai èlleiyh sto uv epÐpedo.OmoÐwc, an x = x0 (eujeÐa par�llhlh me ton �xona twn fantastik¸n) brÐskoume (qrhsi-
mopoi¸ntac thn tautìthta cosh2 y− sinh2 y = 1) ìti u2
sin2 x0− v2
cos2 x0= 1, pou eÐnai uperbol .
2.6 Olokl rwshTo olokl rwma eÐnai exairetik� shmantikì sth melèth sunart sewn miac migadik c metablh-t c. H jewrÐa olokl rwshc diakrÐnetai gia th majhmatik thc komyìthta. Ta jewr mataeÐnai isqurìtata kai oi pio pollèc apodeÐxeic eÐnai aplèc. H jewrÐa olokl rwshc eÐnai epilèonidiaÐtera shmantik gia th meg�lh qrhsimìtht� thc sta efarmosmèna majhmatik�.
2.6.1 Orismèno olokl rwmaProkeimènou na eis�goume to olokl rwma thc f(z) me ènan sqetik� aplì trìpo, orÐzoumearqik� to orismèno olokl rwma miac migadik c sun�rthshc F miac pragmatik c metablht ct. 'Estw F (t) = U(t) + iV (t), t ∈ [a, b], ìpou oi sunart seic U kai V eÐnai pragmatikèc kaikat� tm mata suneqeÐc sunart seic tou t orismènec se èna kleistì kai fragmèno di�sthma[a, b]. Lème tìte ìti h F eÐnai kat� tm mata suneq c kai orÐzoume to orismèno olokl rwmathc F sto [a, b] wc ex c:
∫ b
a
F (t)dt =
∫ b
a
U(t)dt + i
∫ b
a
V (t)dt.
oi sunj kec pou upojèsame gia tic U kai V eÐnai ikanèc gia na exasfalÐsoun thn Ôparxh twnoloklhrwm�twn touc. To genikeumèno olokl rwma thc F epÐ enìc mh fragmènou diast matocorÐzetai an�loga kai up�rqei ìtan sugklÐnoun kai ta dÔo genikeumèna oloklhr¸mata thc Ukai thc V . H �lgebra kai o logismìc twn orismènwn oloklhrwm�twn isqÔoun akrib¸c ìpwckai gia tic pragmatikèc sunart seic tou t.
Ja deÐxoume mia basik idiìthta:∣∣∣∣∫ b
a
F (t)dt
∣∣∣∣ =
∫ b
a
|F (t)|dt.
(h Ðdia idiìthta isqÔei bèbaia kai ta genikeumèna oloklhr¸mata∫ +∞
a) 'Estw
∫ b
aF (t)dt = z ∈
C kai èstw z = reiθ. Tìte
r =
∫ b
a
e−iθF (t) ⇒ r =
∫ b
a
Re[e−iθF (t)
]dt
afoÔ Re[∫ b
aG(t)dt
]=
∫ b
aRe[G(t)]dt kai afoÔ h pr¸th sqèsh eÐnai sqèsh pragmatik¸n
arijm¸n. 'OmwcRe
[e−iθF (t)
] ≤∣∣e−iθF (t)
∣∣ = |e−iθ||F (t)| = |F |,
35
opìter <
∫ b
a
|f(t)dt
pou apodeiknÔei to zhtoÔmeno.
2.6.2 KampÔlecJa asqolhjoÔme me kl�seic kampul¸n pou mac qrei�zontai sth melèth oloklhrwm�twnsunart sewn miac migadik c metablht c. KampÔlh C eÐnai èna sÔnolo shmeÐwn z = (x, y)tou C, tètoio ¸ste x = x(t), y = y(t), t ∈ [a, b] ìpou oi x kai y eÐnai suneqeÐc sunart seicthc pragmatik c paramètrou t. Perigr�foume ta shmeÐa thc C me thn exÐswsh z = z(t) =x(t)+ iy(t), t ∈ [a, b] kai afoÔ oi x, y eÐnai suneqeÐc, eÐnai kai h z. H kampÔlh C lègetai apl kampÔlh kampÔlh Jordan an den tèmnei ton eautì thc (dhlad z(t1) 6= z(t2) gia t1 6 t2).Mia kampÔlh pou eÐnai apl , ektìc apì ta �kra thc ìpou z(a) = z(b), lègetai apl kleist kampÔlh kampÔlh Jordan. An oi x, y eÐnai diaforÐsimec sunart seic tou t, h z eÐnai epÐshcdiaforÐsimh sun�rthsh tou t kai èqoume
z′(t) =dz(t)
dt:= x′(t) + iy′(t).
Mia kampÔlh lègetai leÐa, an up�rqei, eÐnai suneq c kai den mhdenÐzetai sto [a, b] h z′(t).To m koc miac leÐac kampÔlhc ekfr�zetai apì ton tÔpo
L =
∫ b
a
|z′(t)|dt, (|z′(t)| =√
[x′(t)]2 + [y′(t)]2)
kai eÐnai analloÐwto apì metabolèc thc parametrik c anapar�stashc thc C thc morf c
t = φ(s),
ìpou φ : [c, d] → [a, b] eÐnai epÐ, suneq c èqei suneq par�gwgo kai φ′(s) > 0 gia k�jes. Kat� tm mata leÐa kampÔlh eÐnai mia kampÔlh pou apoteleÐtai apì peperasmèno pl jocleÐwn kampul¸n pou en¸nontai ta �kra touc. An h z(t) = x(t) + iy(t) parist�nei mia kat�tm mata leÐa kampÔlh, oi x kai y eÐnai eÐnai suneqeÐc, en¸ oi pr¸tec par�gwgoÐ touc eÐnaikat� tm mata suneqeÐc. To m koc miac kat� tm mata leÐac kampÔlhc eÐnai to �jroisma twnmhk¸n twn leÐwn kampul¸n pou thn apoteloÔn.
Se k�je apl kleist kampÔlh apl kleist kat� tm mata kampÔlh C antistoiqoÔn dÔosÔnola pou k�je èna èqei wc sÔnoro mìno thn C. To èna apì aut� pou lègetai eswterikìthc C eÐnai fragmèno, en¸ to �llo (exwterikì) eÐnai mh fragmèno. H apìdeixh den eÐnai apl kai h prìtash aut lègetai je¸rhma kampÔlhc Jordan.
2.6.3 Olokl rwma migadik¸n sunart sewn miac migadik cmetablht c
Orismìc 2.6.1 'Estw C kat� tm mata leÐa kampÔlh pou dÐnetai apì thn z(t), t ∈ [a, b].'Estw ìti h f eÐnai mia kat� tm mata suneq c sun�rthsh sto C. To olokl rwma thc f kat�
36
m koc thc C orÐzetai wc ∫
C
f(z)dz =
∫ b
a
f(z(t))z′(t)dt.
ParathroÔme ìti to olokl rwma kat� m koc thc C den exart�tai mìno apì ta shmeÐa thc C,all� kai apì th dieÔjunsh. 'Omwc den exart�tai apì th sugkekrimènh parametrikopoÐhsh.
Orismìc 2.6.2 Oi dÔo kampÔlec C1 : z(t), t ∈ [a, b], c2 : w(t), t ∈ [c, d] eÐnai omal�isodÔnamec an up�rqei mia 1-1, C1 apeikìnish Ψ(t) : [c, d] → [a, b] tètoia ¸ste Ψ(c) = a,Ψ(d) = b kai w(t) = z(Ψ(t)).
Prìtash 2.6.1 An oi C1 kai C2 eÐnai omal� isodÔnamec, tìte∫
C1f =
∫C2
f .
Orismìc 2.6.3 'Estw ìti h kampÔlh C dÐnetai apì thn z(t), t ∈ [a, b]. Tìte h −C orÐzetaiapì thn z(b + a− t), t ∈ [a, b].
Prìtash 2.6.2∫−C
f = − ∫C
f
Prìtash 2.6.3 'Estw C leÐa kampÔlh, f kai g suneqeÐc sthn C kai a ∈ C. Tìte
• ∫C[f(z) + g(z)]dz =
∫C
f(z)dz +∫
Cg(z)dz
• ∫C
af(z)dz = a∫
Cf(z)dz
Prìtash 2.6.4 'Estw C leÐa kampÔlh m kouc L, f suneq c sthn C kai |f | ≤ M sthnC. Tìte ∣∣∣∣
∫
C
f(z)dz
∣∣∣∣ ≤ ML.
Prìtash 2.6.5 'Estw {fn} akoloujÐa suneq¸n sunart sewn kai èstw ìti fn → f omoiì-morfa epÐ thc leÐac kampÔlhc C. Tìte
∫
C
f(z)dz = limn→+∞
∫
C
fn(z)dz.
H akìloujh genÐkeush tou Jemeli¸douc jewr matoc tou ApeirostikoÔ LogismoÔ eÐnaipolÔ shmantik .
Prìtash 2.6.6 'Estw F analutik epÐ thc leÐac kampÔlhc C kai f h par�gwgoc thc F .Tìte ∫
C
f(z) = F (z(b))− F (z(a))
ParadeÐgmata
37
1. Na brejeÐ to I1 =∫
C1z2dz, ìpou C1 to eujÔgrammo tm ma OB apì to z = 0 sto
z = 2 + i.LÔshParathroÔme ìti ta shmeÐa thc C1 brÐskontai p�nw sthn eujeÐa x = 2y. An, loipìn, hsuntetagmènh y jewrhjeÐ wc par�metroc, mia parametrik exÐswsh thc C1 eÐnai z(y) =2y + iy.EpÐ thc C1, to z2 gÐnetai: z2 = x2 − y2 + i2xy = 3y2 + i4y2. Opìte I1 =
∫ 1
o(3y2 +
i4y2)(2 + i)dy = (3 + 4i)(z + i)∫ 1
0y2dy = 2
3+ 11
3i.
2. Na brejeÐ to I2 =∫
C2z2dz, ìpou C2 to OAB tou sq matoc.
LÔshI2 =
∫C2
z2dz =∫
OAz2dz +
∫AB
z2dz. Mia parametrik anapar�stash tou OA eÐnai
z(x) = x, x ∈ [0, 2]
en¸ gia to AB mporoÔme na gr�youme
z(y) = 2 + iy, y ∈ [0, 1].
Tìte I2 =∫ 2
0x2dx+
∫ 1
0(2+ iy)2idy = 8
3+ i
[∫ 1
0(4− y2)dy + 4i
∫ 1
0ydy
]= 2
3+ 11
3i. Mia
parametrik anapar�stash tou OAB eÐnai h
z(t) =
{t, t ∈ [0, 2]
2 + i(t− 2), t ∈ [2, 3]
ParathroÔme ìti I2 = I1 kai sunep¸c∫
Cz2dz = 0, ìpou C = OABO, pr�gma pou
den eÐnai tuqaÐo all� ofeÐletai sto ìti h z2 eÐnai analutik sto eswterikì kai epÐ thckampÔlhc ìpwc ja doÔme argìtera.
3. Na brejoÔn ta I3 =∫
C3zdz kai I4 =
∫C4
zdz, ìpou C3 : to �nw hmikÔklio tou |z| = 1(apì z = −1 e¸c z = 1) kai C4 : to k�tw hmikÔklio.LÔshMia parametrik exÐswsh tou −C3 eÐnai h
z(θ) = cos θ + i sin θ = eiθ, θ ∈ [0, π].
Sunep¸c I3 =∫
C3zdz = − ∫
−C3zdz = − ∫ π
oe−iθieiθdθ = −πi. AntÐjeta mia parame-
trik exÐswsh tou C4 eÐnai h
z(θ) = eiθ, π ≤ θ ≤ 2,
opìte I4 =∫
C4zdz =
∫ 2π
πe−iθieiθdθ = πi. ParathroÔme ìti I3 6= I4 kai akìma ìti to
olokl rwma IC =∫
Czdz, ìpou C olìklhroc o kÔkloc, den eÐnai 0: IC = I4−I3 = 2πi.
Tèloc epÐ tou C isqÔei ìti |z| = 1 opìte 1z
= z|z|2 = z kai ètsi
∫C
dzz
= 2πi.
38
4. 'Estw C5 to eujÔgrammo tm ma apì to z = i sto z = 1. QwrÐc na upologisteÐ toolokl rwma I5 =
∫C5
dzz4 na brejeÐ èna �nw fr�gma thc apìluthc tim c tou.
LÔshTo C5 brÐsketai p�nw sthn eujeÐa y = 1− x. An z ∈ C5, èqoume |z4| = (x2 + y2)2 =
(x2 + (1 − x)2)2 = (2x2 − 2x + 1)2, opìte |z4| =(2(x− 1
2)2 + 1
2
))2 ≥ 14, afoÔ(
x− 12
)2 ≥ 0. Sunep¸c, gia k�je z ∈ C5:∣∣ 1z4
∣∣ ≤ 4. Jètoume loipìn M = 4 sthnprìtash 2.6.4. Ex�llou to m koc L tou C5 eÐnai profan¸c L =
√2. Telik� paÐrnoume
|I5| ≤ 4√
2.
2.7 To je¸rhma Cauchy-Goursat
2.7.1 To je¸rhma Cauchy
'Estw ìti oi pragmatikèc sunart seic P (x, y) kai Q(x, y) kaj¸c kai oi pr¸tec merikèc par�-gwgoi touc eÐnai suneqeÐc se ènan tìpo R pou apoteleÐtai apì ta shmeÐa pou perib�llontaiapì mia apl kleist kampÔlh C kai epÐ thc C. JewroÔme ìti h kampÔlh èqei jetik dieÔ-junsh. Apì to je¸rhma Green èqoume ìti∫
C
(Pdx + Qdy) =
∫∫(Qx − Py)dxdy.
JewroÔme mia sun�rthsh f(z) = u(x, y) + iv(x, y) pou eÐnai analutik ston R. JewroÔmeepiplèon ìti h f ′(z) eÐnai suneq c ston R. Tìte oi u kai v, kaj¸c kai oi pr¸tecmerikèc par�gwgoÐ touc eÐnai epÐshc suneqeÐc ston R. 'Etsi∫
C
udx− vdy = −∫∫
R
(vx + uy)dxdy
∫
C
vdx + udy =
∫∫
R
(ux − vy)dxdy.
Lìgw twn sunjhk¸n Cauchy-Riemann kai ta dÔo dipl� oloklhr¸mata mhdenÐzontai, opìte∫C
vdx + udy =∫∫
R(ux − vy)dxdy. 'Omwc, an f = u + iv, z = x + iy,
∫
C
f(z)dz =
∫
C
f(z(t))z′(t)dt =
∫ b
a
{u[x(t), y(t)] + iv[x(t), y(t)]}{x′(t) + iy′(t)}dt =
=
∫ b
a
(ux′ − vy′)dt + i
∫ b
a
(vx′ + uy′))dt =
∫
C
udx− vdy + i
∫
C
vdx + udy,
pr�gma pou shmaÐnei ìti∫
Cf(z)dz = 0. Autì to apotèlesma onom�zetai je¸rhma Cauchy.
2.7.2 To je¸rhma Cauchy-Goursat
O Goursat apèdeixe ìti h upìjesh thc sunèqeiac thc f ′(z) mporeÐ na paralhfjeÐ. H nèamorf tou prohgoumènou jewr matoc pou prokÔptei ètsi eÐnai
Je¸rhma 2.7.1 An h f eÐnai analutik ston R kai epÐ thc C, tìte∫
Cf(z)dz = 0.
39
2.8 Apl� kai Pollapl� Sunektik� SÔnolaOrismìc 2.8.1 'Enac tìpoc D lègetai apl� sunektikìc an den perièqei trÔpec, andhlad eÐnai p�nta dunatì dÔo kampÔlec tou D me koin� �kra na metasqhmatistoÔn suneq¸ch mÐa sthn �llh, qwrÐc kat� ton metasqhmatismì h kampÔlh na bgaÐnei èxw apì ton D. 'Enacmh apl� sunektikìc tìpoc lègetai pollapl� sunektikìc.
ParadeÐgmata
• To epÐpedo meÐon ton pragmatikì �xona den eÐnai apl� sunektikì, giatÐ den eÐnai tìpoc(mh sunektikì).
• O daktÔlioc A = {z : 1 < |z| < 3} den eÐnai apl� sunektikìc.
• O monadiaÐoc dÐskoc meÐon ton jetikì pragmatikì �xona eÐnai apl� sunektikìc.
• H �peirh lwrÐda S = {z : −1 < Im z < 1} eÐnai apl� sunektik .
• K�je anoiqtì kurtì sÔnolo eÐnai apl� sunektikì.
To je¸rhma Cauchy-Goursat mporeÐ na diatupwjeÐ wc ex c:
Je¸rhma 2.8.1 An h f eÐnai analutik se ènan apl� sunektikì tìpo D, tìte gia k�jeapl kleist kampÔlh C entìc tou D isqÔei
∫
C
f(z)dz = 0.
Profan¸c to “apl kleist kampÔlh” mporeÐ na antikatastajeÐ apì to “tuqoÔsa kleist kampÔlh”. Ex�llou to je¸rhma Cauchy-Goursat mporeÐ na p�rei kai thn akìloujh morf .
Je¸rhma 2.8.2 'Estw C apl kleist kampÔlh kai Cj (j = 1, 2, . . . , n) aplèc kleistèckampÔlec mèsa sthn C, ètsi ¸ste ta eswterik� sunola twn Cj na mhn èqoun koin� shmeÐa.'Estw R to kleistì sÔnolo pou apoteleÐtai apì ìla ta shmeÐa mèsa kai epÐ thc C, ektìc apìta shmeÐa mèsa se k�je Cj. 'Estw B to prosanatolismèno sÔnoro tou R, pou apoteleÐtaiapì thn C kai ìlec tic Cj (dieÔjunsh tètoia ¸ste ta shmeÐa tou R na brÐskontai arister�tou B). Tìte, an h f eÐnai analutik sto R,
∫
B
f(z)dz = 0
Par�deigmaAn B apoteleÐtai apì ton kÔklo |z| = 2 me jetik dieÔjunsh kai ton kÔklo |z| = 1 me
arnhtik dieÔjunsh tìte ∫
B
dz
z2(z2 + 9)= 0.
H sun�rthsh 1z2(z2+9)
eÐnai analutik , ektìc apì ta shmeÐa z = 0, z = ±3i pou ìmwc denan koun sto daktÔlio me sÔnoro B.
40
2.9 O oloklhrwtikìc tÔpoc tou Cauchy
Ja anafèroume t¸ra èna �llo jemeli¸dec apotèlesma.
Je¸rhma 2.9.1 'Estw ìti h f eÐnai analutik pantoÔ epÐ kai entìc miac apl c kleist ckampÔlhc C (me jetikì prosanatolismì). An z0 eÐnai tuqìn shmeÐo sto eswterikì thc C,tìte
f(z0) =1
2πi
∫
C
f(z)
z − z0
.
O parap�nw lègetai oloklhrwtikìc tÔpoc tou Cauchy kai lèei pwc an mia sun�rthsh f eÐnaianalutik epÐ kai entìc miac apl c kleist c kampÔlhc C, tìte oi kÔriec timèc thc f entìcthc C prosdiorÐzontai pl rwc apì tic timèc thc f epÐ thc C.
Je¸rhma 2.9.2 Me tic upojèseic tou prohgoÔmenou jewr matoc. Gia k�je n ∈ N isqÔei
f (n)(z0) =n!
2πi
∫
C
f(z)dz
(z − z0)n+1
Pìrisma 2.9.1 An mia sun�rthsh f eÐnai analutik se èna shmeÐo, tìte up�rqoun oipar�gwgoi k�je t�xhc thc f kai eÐnai analutikèc sto Ðdio shmeÐo.
Pìrisma 2.9.2 H sqèsh tou jewr matoc 2.9.1 mporeÐ na epektajeÐ sthn perÐptwsh pouh apl kleist kampÔlh C antikatastajeÐ apì to “to prosanatolismèno sÔnoro enìc pol-lapl¸c sunektikoÔ tìpou”.
Pìrisma 2.9.3 (Ektim seic Cauchy). Me tic upojèseic tou jewr matoc 2.9.1. An |f(z) ≤M(r) ston kÔklo |z − z0| = r kai an := f (n)(z0)
n!, tìte |an| ≤ M(r)
rn .
Ask seic
1. 'Estw C mia apl kleist kampÔlh. ApodeÐxe ìti:∫
C
dz
z − z0
=
{0, z0 ektìc thc C
2πi, z0 entìc thc C
LÔsh.An to z0 eÐnai ektìc thc C tìte h 1
z−z0eÐnai analutik epÐ kai entìc thc C. 'Ara apì
to je¸rhma Cauchy ja isquei∫
Cdz
z−z0= 0. An to z0 eÐnai entìc thc C tìte up�rqei
dÐskoc kèntrou z0 kai aktÐnac ε (sunìrou G) ef‘ìson to z0 eÐnai eswterikì shmeÐo.GnwrÐzoume ìti
∫C
dzz−z0
=∫
Γdz
z−z0. EpÐ thc G èqoume |z − z0| = ε ⇒ z − z0 = εeiθ ⇒
z = z0 + εeiθ, θ ∈ [0, 2π]. 'Etsi dz = iεeiθdθ, opìte∫
Γ
dz
z − z0
=
∫ 2π
0
iεeiθdθ
εeiθ= i
∫ 2π
0
dθ = 2πi.
41
2. An C eÐnai to tm ma thc y = x3 − 3x2 + 4x − 1 pou sundèei ta shmeÐa (1,1) kai (2,3)na upologisjeÐ to olokl rwma
∫
C
(12z2 − 4iz)dz.
LÔsh.To olokl rwma eÐnai anex�rthto thc diadrom c. 'Ara:
∫
C
(12z2 − 4iz)dz =
∫ 2+3i
1+i
(12z2 − 4iz)dz =[4z3 − 2iz2
]2+3i
1+i= −156 + 38i.
42
Kef�laio 3
3.1 Memonwmènec AnwmalÐec Analutik¸n Sunar-t sewn
3.1.1 1. Kat�taxh Memonwmènwn Anwmali¸n � Arq touRiemann � Je¸rhma Casorati – Weierstrass
En¸ wc t¸ra eÐqame sugkentr¸sei thn prosoq mac stic genikèc idiìthtec twn analutik¸nsunart sewn, ja asqolhjoÔme sto ex c me thn eidik sumperifor� mia analutik c sun�rth-shc sthn perioq miac {memonwmènhc anwmalÐac}.
Orismìc 3.1.1 H f èqei memonwmènh anwmalÐa sto z0 an h f eÐnai analutik se mia perioq D thc morf c {z : 0 < |z − z0| < d} tou z0, all� den eÐnai analutik sto z0.
Parat rhsh 3.1.1 Lìgw tou Jewr matoc ??, parap�nw, h f ja eÐnai asuneq c se miamemonwmènh anwmalÐa.
ParadeÐgmata
1. f(z) =
{sin z, z 6= 20, z = 2
. 'Eqei memonwmènh anwmalÐa sto z = 2.
2. g(z) = 1z−3
. 'Eqei memonwmènh anwmalÐa sto z = 3.
3. h(z) = exp 1z. 'Eqei memonwmènh anwmalÐa sto z = 0.
Ta paradeÐgmata aut� antiporswpeÔoun touc diaforetikoÔc tÔpouc memonwmènwn anw-mali¸n, pou katat�ssontai wc ex c:
Orismìc 3.1.2 'Estw ìti h f èqei memonwmènh anwmalÐa sto z0.
43
1. An up�rqei analutik sto z0 sun�rthsh g, tètoia ¸ste f(z) = g(z) gia k�je z semia perioq san thn D tou orismoÔ (3.1.1), lème ìti h f èqei epousi¸dh (airìmenh)anwmalÐa sto z0, (dhlad an h tim thc f {diorjwjeÐ} sto z0, gÐnetai analutik kaiekeÐ).
2. An, gia z 6= z0, h f mporeÐ na grafeÐ sth morf f(z) = A(z)B(z)
, ìpou oi A kai B eÐnaianalutikèc sto z0, me A(z0) 6= 0 kai B(z0) = 0, lème ìti h f èqei pìlo sto z0.An h B èqei to z0 wc mhdenikì shmeÐo pollaplìthtac k, lème ìti h f èqei pìlo t�xhck sto z0.
3. An h f den èqei oÔte epousi¸dh anwmalÐa, oÔte pìlo sto z0, lème ìti h f èqei ousi¸dhanwmalÐa sto z0.
Ta akìlouja jewr mata deÐqnoun p¸c h fÔsh thc anwmalÐac pou èqei mia sun�rthsh,mporeÐ na prosdioristeÐ apì th sumperifor� thc se mia perioq , san thn D, thc anwmalÐac.
H Arq tou Riemann gia Epousi¸deic AnwmalÐec An h
f èqei memonwmènh anwmalÐa sto z0 kai an
limz→z0
(z − z0)f(z) = 0
tìte h anwmalÐa eÐnai epousi¸dhc.
Pìrisma 3.1.1 An h f eÐnai fragmènh se mia perioq , ìpwc h D parap�nw, miac memonw-mènhc anwmalÐac, h anwmalÐa eÐnai epousi¸dhc.
Je¸rhma 3.1.1 An h f eÐnai analutik se mia mia perioq , ìpwc h D parap�nw, tou z0
kai an up�rqei jetikìc akèraioc k tètoioc ¸ste
limz→z0
(z − z0)kf(z) 6= 0 all� lim
z→z0
(z − z0)k+1f(z) = 0
tìte h f èqei pìlo t�xhc k sto z0.
Parathr seic
1. Den up�rqei analutik sun�rthsh pou na teÐnei sto ∞, ìpwc mia klasmatik dÔnamhthc 1
z−z0sthn perioq mia memonwmènhc anwmalÐac tou z0 P.q. An h f tan analutik
se mia perioq tou 0 kai ikanopoioÔse thn |f(z) ≤ 1√|z| , epeid h anwmalÐa ja tan
epousi¸dhc h f j� èprepe na eÐnai fragmènh.
2. Se mia perioq miac ousi¸douc anwmalÐac, mia sun�rthsh f den èinai mìno mh fragmènh,all� tètoia ¸ste gia k�je akèraio N h (z − z0)
Nf(z) den teÐnei sto 0 ìtan z → z0.EntoÔtoic, de shmaÐnei ìti f(z) → ∞, ìtan z → z0. To epìmeno je¸rhma deÐqnei ìtito sÔnolo tim¸n pou paÐrnei mia sun�rthsh sthn perioq miac ousi¸douc anwmalÐaceÐnai {puknì} sto migadikì epÐpedo, m' �lla lìgia to pedÐo tim¸n thc f tèmnei k�jedÐsko sto C.
44
Je¸rhma 3.1.2 (Casorati – Weierstrass) An h f èqei ousi¸dh anwmalÐa sto z0 kaian h D eÐnai mia perioq tou z0 tìte to R = {f(z) : z ∈ D} eÐnai puknì to C.
3.1.2 An�ptugma Laurent
GnwrÐzoume ìti oi analutikèc sunart seic s' èna dÐsko, mporoÔn na anaparastajoÔn ekeÐ medunamoseirèc. Mia an�logh anapar�stash, thc morf c
∞∑
k=−∞ak(z − z0)
k
mporeÐ na gÐnei gia analutikèc sunart seic sto daktÔlio R1 < |z − z0| < R2. Oi tètoiecanaparast�seic eÐnai gnwstèc wc an�ptugmata Laurent kai eÐnai polÔ qr sima ergaleÐa sthmelèth menomwmènwn anwmali¸n.
Orismìc 3.1.3 Lème ìti∑∞
k=−∞ µk = L an sugklÐnoun kai h∑∞
k=0 µk = L kai h∑∞
k=1 µ−k =L kai to �jroisma twn orÐwn touc eÐnai L.
Je¸rhma 3.1.3 H f(z) =∑∞
k=−∞ akzk eÐnai sugklÐnousa sto D = {z : R1 < |z| kai |z| <
R2}, ìpouR2 =
1
lim supk→∞ |ak| 1k, R1 = lim sup
k→∞|a−k| 1k .
An R1 < R2, to D eÐnai daktÔlioc kai h f eÐnai analutik sto D.
Je¸rhma 3.1.4 An h f eÐnai analutik sto daktÔlio A : R1 < |z| < R2, tìte h f èqeian�ptugma Laurent, f(z) =
∑∞k=−∞ akz
k ston A.
Parat rhsh: To an�ptugma Laurent eÐnai monadikì.
Pìrisma 3.1.2 An h f eÐnai analutik sto daktÔlio R1 < |z − z0| < R2, tìte èqeimonadikì an�ptugma
f(z) =∞∑
k=−∞ak(z − z0)
k me ak =1
2πi
∫
C
f(z)
(z − z0)k+1dz
kai C = C(z0; R) me R1 < R < R2.
Pìrisma 3.1.3 An h f èqei menomwmènh anwmalÐa sto z0, tìte gia δ > 0 kai 0 < |z−z0| <δ, èqoume
f(z) =∞∑
k=−∞ak(z − z0)
k, ak =1
2πi
∫
C
f(z)
(z − z0)k+1dz
ìpou C = C(z0; R) me 0 < R < δ.
45
Orismìc 3.1.4 An f(z) =∑∞
k=−∞ ak(z−z0)k eÐnai to an�ptugma Laurent thc f , wc proc
mia memonwmènh anwmalÐa z0, to f(z) =∑−1
k=−∞ ak(z − z0)k lègetai kÔrio mèroc thc f sto
z0 en¸ to f(z) =∑∞
k=0 ak(z − z0)k lègetai analutikì mèroc.
Apì th monadikìthta tou anaptÔgmatoc Laurent, mporoÔn na brejoÔn oi qarakthrismoÐtwn kÔriwn mer¸n wc proc ta di�fora eÐdh anwmali¸n.
Prìtash 3.1.1
(i) An h f èqei epousi¸dh anwmalÐa sto z0, tìte c−k = 0, ∀k > 0.
(ii) An h f èqei pìlo t�xhc k sto z0, tìte c−k 6= 0, &c−N = 0 ∀k > 0.
(iii) An h f èqei ousi¸dh anwmalÐa sto z0, to kÔrio mèroc thc èqei apeÐrou pl jouc mhmhdenikoÔc ìrouc.
Oloklhr¸noume thn par�grafo me to akìloujo je¸rhma:
Je¸rhma 3.1.5 (An�lush Rht¸n Sunart sewn se Apl� Kl�smata) K�je gn -sia rht sun�rthsh R(z) = P (z)
Q(z)= P (z)
(z−z1)k1(z−z2)k2 ...(z−zn)kn, ìpou P, Q polu¸numa me
degP < degQ, mporeÐ na analujeÐ wc �jroisma poluwnÔmwn wc proc 1z−zk
, k = 1, 2, . . . , n,dhlad
R(z) =n∑
k=1
Pk
(1
z − zk
).
46
Kef�laio 4
Oloklhrwtik� Upìloipa
4.1 DeÐkthc Strof c kai to Je¸rhma Oloklhrw-tik¸n UpoloÐpwn tou Cauchy
Stìqoc mac eÐnai h genÐkeush tou Jewr matoc Cauchy – Goursat gia sunart seic pou èqounmemonwmènec anwmalÐec.
ParathroÔme ìti an γ eÐnai ènac kÔkloc pou perikleÐei mia memonwmènh anwmalÐa z0 kaian f(z) =
∑∞k=−∞ ck(z− z0)
k se mia perioq D = {z : 0 < |z− z0| < d} tou z0 pou perièqeith γ tìte ∫
γ
f = 2πic−1
'Etsi, blèpoume pwc o sunelest c c−1 èqei idiaÐterh shmasÐa.
Orismìc 4.1.1 An f(z) =∑∞
k=−∞ ck(z − z0)k, se mia perioq ìpwc h D tou z0, to
c−1 lègetai oloklhrwtikì upìloipo thc f sto z0. QrhsimopoioÔme to sumbolismì c−1 =Res(f ; z0).
Prìtash 4.1.1 (Upologismìc Oloklhrwtik¸n UpoloÐpwn)
1. An h f èqei aplì pìlo sto z0, tìte
c−1 = Res(f ; z0) = limz→z0
(z − z0)f(z) =A(z0)
B′(z0)
2. An h f èqei pìlo t�xhc k sto z0, tìte
c−1 = Res(f ; z0) =−1
(k − 1)!· dk−1
dzk−1
[(z − z0)
kf(z)]∣∣∣
z=z0
.
Parat rhsh: Stic perissìterec peript¸seic pìlwn megalÔterhc t�xhc apì 1, kaj¸ckai gia tic ousi¸deic anwmalÐec, o bolikìteroc trìpoc prosdiorismoÔ tou oloklhrwtikoÔupoloÐpou eÐnai kateujeÐan apì to an�ptugma Laurent.
47
ParadeÐgmata:
1. Res(cosec z; 0) = 1cos 0
= 1
2. Res( 1z4−1
; i) = 14i3
= i4
3. Res( 1z3 ; 0) = 0
4. Res(sin 1z−1
; 1) = 1, afoÔ sin 1z−1
= 1z−1
− 13!(z−1)3
− 15!(z−1)5
− · · ·
DeÐkthc Strof cOrismìc 4.1.2 'Estw γ kleist kampÔlh kai ζ /∈ γ. Tìte
Indγ(ζ) =1
2πi
∫
γ
dz
z − ζ
Parathr seic:
1. An γ eÐnai ènac kÔkloc (jetik� prosanatolismènoc), tìte
Indγ(ζ) =
{1, ζ sto eswterikì tou dÐskou0, ζ sto exwterikì tou dÐskou .
2. An γ peristrèfetai perÐ to ζ k forèc, dhlad an γ(θ) = ζ + reiθ, 0 ≤ θ ≤ 2kπ tìteIndγ(ζ) = 1
2πi
∫ 2kπ
0idθ = k, gegonìc pou dikaiologeÐ ton ìro {deÐkthc strof c}.
Je¸rhma 4.1.1 Gia k�je kleist kampÔlh γ kai ζ /∈ γ, o Indγ(ζ) eÐnai akèraioc.
Parathr seic:
1. Apì ton Orismì (4.1.2) èpetai ìti an stajeropoi soume th γ kai af soume to ζ na me-tab�lletai, o Indγ(ζ) eÐnai suneq c sun�rthsh tou ζ (efìson ζ /∈ γ). AfoÔ to Indγ(ζ)eÐnai p�ntote akèraioc, sumperaÐnoume ìti eÐnai stajerìc stic sunektikèc sunist¸sectou sumplhr¸matoc thc γ. Epiplèon Indγ(ζ) → 0 ìtan ζ → ∞. Sunep¸c sth mhfragmènh sunist¸sa tou sumplhr¸matoc thc γ (dhlad sto sÔnolo twn shmeÐwn poumporoÔn na enwjoÔn me to ∞ qwrÐc na tmhjeÐ h γ) Indγ(ζ) = 0.
2. Genik¸c, ìtan asqoloÔmaste me kleistèc kampÔlec, ja eÐmaste se jèsh na diapist¸-noume ap' eujeÐac ìti
Indγ(ζ) = 0 ± 1, ∀ζ /∈ γ.
Orismìc 4.1.3 H γ lègetai kanonik kleist kampÔlh, an h γ eÐnai apl kleist kampÔlhme Indγ(ζ) = 0 1 ∀ζ /∈ γ. S' aut thn perÐptwsh onom�zoume eswterikì thc γ to sÔnolo{ζ : Indγ(ζ) = 1}. Exwterikì thc γ, lègetai to sÔnolo {ζ : Indγ(ζ) = 0}.
48
Je¸rhma 4.1.2 (Oloklhrwtik¸n UpoloÐpwn tou Cauchy) 'Estw f analutik ,s' ènan apl� sunektikì tìpo D, ektìc apì tic memonwmènec anwmalÐec sta shmeÐa z1, . . . , zm.'Estw γ kleist kampÔlh pou den perièqei kamia apì tic anwmalÐec. Tìte
∫
γ
f = 2πi
m∑
k=1
Indγ(ζk)Res(f ; zk)
Pìrisma 4.1.1 An h f eÐnai ìpwc sto Je¸rhma kai h γ eÐnai kanonik kleist kampÔlhsto pedÐo analutikìthtac thc f , tìte
∫
γ
f = 2πi∑
k
Res(f ; zk)
ìpou to �jroisma lamb�netai epÐ ìlwn twn anwmali¸n thc f mèsa sth γ.
4.2 Efarmogèc tou jewr matoc Oloklhrwtik¸n Upo-loÐpwn tou Cauchy ston upologismì oloklh-rwm�twn kai seir¸n
4.2.1 Upologismìc Oloklhrwm�twn
(A)∫∞−∞
P (x)Q(x)
dx, P, Q polu¸numa
EÐnai gnwstì pwc èna tètoio olokl rwma sugklÐnei an Q(x) 6= 0 kai degQ−degP ≥ 2.K�tw ap' autèc tic proôpojèseic
∫ ∞
−∞
P (x)
Q(x)dx = lim
R→∞
∫ R
−R
P (x)
Q(x)dx
kai jèloume na upologÐsoume to deÔtero olokl rwma gia meg�la R. JewroÔme thnkleist kampÔlh CR tou sq matoc, me aktÐna R tìso meg�lh ¸ste na perikleÐei ìlectic rÐzec tou Q pou brÐskontai sto �nw hmiepÐpedo.
'Eqoume ìti:∫
CR
P (z)Q(z)
dz = 2πi∑
k Res
(PQ
; zk
), zk rÐzec tou Q sto �nw hmiepÐpedo,
opìte∫ R
−RP (z)Q(z)
dz +∫ΓR
P (z)Q(z)
dz = 2πi∑
k Res(
PQ
; zk
)
AfoÔ degQ− degP ≥ 2, èqoume∣∣∣∣∫
ΓR
P
Q
∣∣∣∣ ≤ πRA
R2⇒ lim
R→∞
∫
ΓR
P (z)
Q(z)dz = 0.
Telik� ∫ ∞
−∞
P (x)
Q(x)dx = 2πi
∑
k
Res
(P
Q; zk
)
49
Par�deigma∫∞∞
dx1+x4
Oi pìloi thc dz1+z4 sto �nw hmiepÐpedo eÐnai oi z1 = e
iπ4 kai z2 = e
3iπ4 .
Sunep¸c∫∞−∞
dx1+x4 = 2πi
∑2k=1 Res
(1
1+z4 ; zk
).
Epeid kai oi dÔo pìloi eÐnai aploÐ, ta oloklhrwtik� upìloipa dÐnontai apì tic timècthc 1
4z3 stouc pìlouc:
Res(
11+z4 ; z1
)= 1
4z31
= − z1
4= −
√2
8(1 + i)
Res(
11+z4 ; z1
)= 1
4z3 =√
28
(1− i)
'Etsi:∫∞−∞
dx1+x4 =
√2
2π
(B)∫∞−∞ R(x) cos xdx,
∫∞−∞ R(x) sin xdx, R = P
Q, P, Q polu¸numa.
An Q(x) 6= 0 kai an degQ > degP ta oloklhr¸mata sugklÐnoun.Ed¸ den mporoÔme na oloklhr¸soume thn R(z) cos z kat� m koc thc Ðdiac kleist ckampÔlhc me thn perÐptwsh (A), afoÔ limM→∞
∫ΓM
R(z) cos zdz 6= 0.
An jewr soume to∫
CMR(z)eizdz ja deÐxoume ìti
∫ΓM
R(z)eizdz → 0,
opìte∫
CMR(z)eizdz → ∫∞
∞ R(x)eixdx.
Ja èqoume sunep¸c telik� ìti
∫ ∞
∞R(x) cos xdx = Re
{2πi
∑
k
Res(R(z)eiz; zk)
}
∫ ∞
∞R(x) sin xdx = Im
{2πi
∑
k
Res(R(z)eiz; zk)
},
ìpou zk oi pìloi thc R(z) sto �nw hmiepÐpedo.Gia na deÐxoume ìti
∫ΓM
R(z)eizdz → 0, sp�me to ΓM se dÔo uposÔnola:
A ={
z ∈ ΓM : Imz ≥ h}
, B ={
z ∈ ΓM : Imz < h}
.
'Omwc ∣∣∣∣∫
A
R(z)eizdz
∣∣∣∣ ≤K
Me−hπM = c1e
−h
kai ∣∣∣∣∫
B
R(z)eizdz
∣∣∣∣ ≤K
M4h = c2
h
M,
50
opìte∣∣∣∫ΓM
R(z)eizdz∣∣∣ ≤ c1e
−h + c2hM
.
An eklèxoume to h =√
M , paÐrnoume
∣∣∣∣∫
ΓM
R(z)eizdz
∣∣∣∣ ≤ c1e−√M +
c2√M
⇒ limM→∞
∫
ΓM
R(z)eizdz = 0.
ParadeÐgmata
(1)∫∞−∞
sin xx
dx∫∞−∞
sin xx
dx = Im{ ∫∞
−∞eix
xdx
}. H eix
xèqei pìlo sto 0 ki ètsi prèpei na tropo-
poi soume thn teqnik mac: 'Eqoume ìti∫∞−∞
sin xx
dx = Im{ ∫∞
−∞eix−1
xdx
}.
ParathroÔme ìti∫
cM
eiz−1z
dz =∫ M
−Meix−1
xdx +
∫ΓM
eiz−1z
dz
'Omwc∫
cM
eiz−1z
dz = 0 afoÔ h eiz−1z
den èqei pìlouc! 'Etsi,∫ M
−Meix−1
xdx =
∫ΓM
eiz−1z
dz =∫ΓM
dzz− ∫
ΓM
eiz
zdz = πi− ∫
ΓM
eiz
zdz
'Omwc∫ΓM
eiz
zdz → 0, ìtan M →∞⇒ ∫∞
−∞eix−1
xdx = πi ⇒ ∫∞
−∞sin x
xdx = π
(2)∫∞−∞
cos x(x2+1)2
dx∫∞−∞
cos x(x2+1)2
dx = Re{ ∫∞
−∞eix
(x2+1)2dx
}.
H eiz
(z2+1)2èqei pìlouc t�xhc 2 sta shmeÐa z = ±1. Ap' aut� mìno to z = 1
brÐsketai sto �nw hmiepÐpedo. JewroÔme M > 1.
Res(
eiz
(z2+1)2; i
)= d
dz
((z − i)2 eiz
(z2+1)2
)∣∣∣z=i
= ddz
(eiz
(z2+1)2
)∣∣∣z=i
= − i2e.
Sunep¸c∫∞−∞
cos x(x2+1)2
dx = Re{
2πiRes(
eiz
(z2+1)2; i
)}= π
e
(G)∫∞0
P (x)Q(x)
dx, P, Q polu¸numa me Q(x) 6= 0 gia x ≥ 0 kai degQ− degP ≥ 2.
(An � bèbaia � h PQ
eÐnai �rtia, tìte∫ −∞0
PQ
= 12
∫∞∞
PQ).
Jètoume R(z) = P (z)Q(z)
kai jewroÔme to olok rwma thc log z·R(z) sthn kleist kampÔlhKε, M pou apoteleÐtai apì
(a) to orizìntio eujÔgrammo tm ma I1 apì to iε wc to iε +√
M2 − ε2
(b) to tìxo CM kÔklou aktÐnac M pou diatrèqetai me jetikì prosanatolismì apì to√M2 − ε2 + iε wc to
√M2 − ε2 − iε.
(g) to orizìntio eujÔgramma tm ma I2 apì to√
M2 − ε2 − iε wc to −iε.(d) to hmikÔklio Cε aktÐnac ε, kat� arnhtikì prosanatolismì apì −iε wc iε
51
To eswterikì thc Kε, M eÐnai apl� sunektikìc tìpoc pou den perièqei to 0 kai sunep¸co log z orÐzetai ekeÐ wc analutik sun�rthsh (gia lìgouc aplìthtac jewroÔme 0 <Argz < 2π).Apì to Je¸rhma Oloklhrwtik¸n UpoloÐpwn èqoume:
limε→0, M→∞
∫
Kε, M
R(z) log zdz = 2πi∑
k
Res(R(z) log z; zk) (4.1)
ìpou to ε èqei epilegeÐ tìso mikrì kai to M tìso meg�lo, ¸ste oi rÐzec tou Q nabrÐskontai ìlec sto eswterikì thc Kε, M .Tìte
(a)∣∣∣∣∫
CεR(z) log zdz
∣∣∣∣ ≤ πε ·maxCε | log z||R(z)| ≤ Aε| log ε|afoÔ h R eÐnai suneq c sto 0 kai | log z| < log |z|+ 2π.Sunep¸c limε→0
∫Cε
R(z) log zdz = 0
(b)∣∣∣∣∫
CMR(z) log zdz
∣∣∣∣ ≤ 2πM ·maxCM| log z||R(z)| ≤ AM log M
M2
afoÔ |R(z)| ≤ B|z|2 kai ètsi limM→∞
∫CM
R(z) log zdz = 0
(g) limε→0, M→∞∫
I1R(z) log zdz =
∫∞0
R(x) log xdx
(d) limε→0, M→∞∫
I2R(z) log zdz = − ∫∞
0R(x)(log x + 2πi)dx
Apì ta (a), (b), (g) kai (d) paÐrnoume
limε→0, M→∞
∫
I2
R(z) log zdz = −2πi
∫ ∞
0
R(x)dx
kai lìgw thc (4.1):∫ ∞
0
R(x)dx = −∑
k
Res(R(z) log z; zk)
ìpou to k diatrèqei to pl joc twn pìlwn thc R.
Par�deigma∫∞0
dx1+x3
Oi (aploÐ) pìloi thc R(z) = dz1+z3 eÐnai z1 = e
iπ3 , z2 = −1 = eiπ, z3 = ei 5π
3
'Eqoume
Res(
log z1+z3 ; z1
)= log z
3z2
∣∣∣z=z1
= − iπ9
(12
+ i√
32
)
52
Res(
log z1+z3 ; z2
)= iπ
3
Res(
log z1+z3 ; z3
)= −5iπ
9
(12− i
√3
2
)
opìte ∑
k
Res( log z
1 + z3; zk
)= −2
√3
9π
kai ètsi ∫ ∞
0
dx
1 + x3=
2√
3
9π
Parat rhsh 1Oloklhr¸mata thc morf c ∫ ∞
a
P (x)
Q(x)dx
mporoÔn, omoÐwc, na upologistoÔn me to na jewrhjeÐ to∫
CMlog(z − 1)P (z)
Q(z)dz
Parat rhsh 2An�loga gÐnetai kai o upologismìc oloklhrwm�twn thc morf c
∫ ∞
0
xa−1
P (x)dx
me 0 < a < 1 kai P polu¸numo me degP ≥ 1.Sto eswterikì thc kampÔlhc Kε, M èqoume za−1 = exp{(a − 1) log z} kai (me 0 <argz < 2π, p.q.) h sun�rthsh aut mporeÐ na oristeÐ wc analutik .'Opwc prÐn, sto dÔo kuklik� tm mata, ta oloklhr¸mata teÐnoun sto mhdèn kai ètsiarkeÐ na upologistoÔn ta oloklhr¸mata epÐ twn I1, I2.EpÐ tou I1 : za−1 = e(a−1) log x = xa−1,
en¸ epÐ tou I1 : za−1 = e(a−1)(log x+2πi) = xa−1e2πi(a−1)
'Etsi {1− e2πi(a−1)} ∫∞0
xa−1
P (x)dx = 2πi
∑k Res
(za−1
P (z); zk
)
Par�deigma∫∞0
dx(1+x)
√x
Pìloc thc P (z) = 1 + z to −1.
Res(
1(1+z)
√z; −1
)= 1√
z1
(1+z)′
∣∣∣z=−1
= 1i
= −i
Opìte
53
(1− e−πi)
∫ ∞
0
dx
(1 + x)√
x= 2π,
dhlad ∫ ∞
0
dx
(1 + x)√
x= π.
(D) Oloklhr¸mata thc morf c∫ 2π
0R(cos θ, sin θ)dθ, ìpou R rht sun�rthsh.
S' aut thn perÐptwsh, jewroÔme to pragmatikì olokl rwma wc parametrik anapa-r�stash enìc epikampÔliou oloklhr¸matoc epÐ tou monadiaÐou kÔklou.Ac jumhjoÔme, ìti, jètontac z = eiθ, 0 ≤ θ ≤ 2π, èqoume
∫
|z|=1
f(z)dz =
∫ 2π
0
f(eiθ)ieiθdθ
dθ =dz
iz,
cos θ = eiθ+e−iθ
2= 1
2
(z + 1
z
)
sin θ = eiθ−e−iθ
2i= 1
2i
(z − 1
z
)
Sunep¸c ∫ 2π
0
R(cos θ, sin θ)dθ =
∫
|z|=1
R
[z + 1
z
2,
z − 1z
2i
]dz
iz
To deÔtero olokl rwma, mporeÐ � ìpwc gnwrÐzoume � na upologisteÐ me th bo jeiatou Jewr matoc Oloklhrwtik¸n UpoloÐpwn.
Par�deigma 1∫ 2π
0
dθ
2 + cos θ=
2
i
∫
|z|=1
dz
z2 + 4z + 1=
= 4πRes
(1
z2 + 4z + 1;√
3− 2
)=
= 4π1
z2 + 4z + 1
∣∣∣z=√
3−2=
=2√
3
3π.
Par�deigma 2
∫ 2π
0
dθ
5 + 3 sin θ
54
z = eiθ ⇒ sin θ = eiθ−e−iθ
2i=
z− 1z
2i
dz = 3eiθdθ = izdθ∫ 2π
0dθ
5+3 sin θ=
∫C
dziz
5+3(z− 1z) 12i
=∫
C2dz
3z2+10iz−3
ìpou C = C(0; 1), h perifèreia tou kÔklou me kèntro 0 kai aktÐna 1.H 2
3z2+10iz−3èqei pìlouc tic rÐzec thc 3z2 + 10iz − 3 dhlad ta −1
3i, −3i.
Mìno to −13i brÐsketai entìc thc C.
To upìloipo sto z1 = −13i eÐnai:
limz→z1
(z − z1)2
3z2 + 10iz − 3= · · · = 1
4i.
Sunep¸c2dz
3z2 + 10iz − 3= 2πi
1
4i=
π
2.
(E) Oi prohgoÔmenec teqnikèc mporoÔn na epektajoÔn gia ton upologismì enìc oloklhr¸-matoc kat� m koc opoiasd pote kleist c kampÔlhc, lamb�nontac up' ìyin touc pìloucthc upì olokl rwsh sun�rthshc.
Par�deigma
∫
I
ezdz
(z + 2)3, I : z(t) = 1 + it, −∞ < t < ∞
'Estw CR to aristerì hmikÔklio kèntrou z = 1 kai aktÐnac R > 3. Tìte∫ 1+iR
1−iR
ezdz
(z + 2)3+
∫
CR
ezdz
(z + 2)3= 2πiRes
(ezdz
(z + 2)3; −2
).
Sto hmiepÐpedo x ≤ 1, |ez| ≤ e kai ètsi ìtan R →∞∫
CR
ezdz
(z + 2)3→ 0,
opìte ∫
I
ezdz
(z + 2)3= 2πiRes
(ezdz
(z + 2)3; −2
).
Gia ton upologismì tou oloklhrwtikoÔ upoloÐpou parathroÔme ìti
ez = e−2ez+2 = e−2
(1 + (z + 2) +
(z + 2)2
2+ · · ·
),
55
opìteRes
(ezdz
(z + 2)3; −2
)= ”c−1” =
1
2e2
kai ètsi ∫
I
ezdz
(z + 2)3=
πi
e2
56
BibliografÐa
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[2] N. Artemi�dhc. Migadik An�lush. Aj na, 1992.
[3] S. Tersènob. Analutikèc Sunart seic kai Merikèc Efarmogèc touc,. DÐauloc, 1995.
[4] A. Angot. Complements de Mathematiques. Masson, 1972.
[5] J. Back, D. J. Newman. Migadik An�lush. Leader Books, Aj na, 2004.
[6] J. E. Marsden, M. J. Hoffman. Basik Migadik An�lush. Ekd. SummetrÐa, 1994.
[7] M. J. Ablowitz, A. S. Fokas. Complex Variables:Introduction and Applications. Cam-bridge University Press, 1997.
[8] M. R. Spiegel. Migadikèc metablhtèc. Schaum’s Outline Series, ESPI, 1980.
[9] R. V. Churchill, J. W. Brown. Migadikèc Sunart seic kai Efarmogèc. PanepisthmiakècEkdìseic Kr thc, 2001.
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