ΜΑΘΗΜΑΤΙΚΑ ΤΗΛΕΠΙΚΟΙΝΩΝΙΩΝ

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1 ΕΘΝΙΚΟ ΚΑΙ ΚΑΠΟΔΙΣΤΡΙΑΚΟ ΠΑΝΕΠΙΣΤΗΜΙΟ ΑΘΗΝΩΝ ΤΜΗΜΑ ΠΛΗΡΟΦΟΡΙΚΗΣ & ΤΗΛΕΠΙΚΟΙΝΩΝΙΩΝ ΜΑΘΗΜΑΤΙΚΑ ΤΗΛΕΠΙΚΟΙΝΩΝΙΩΝ Εισαγωγή στη Μιγαδική Ανάλυση Ι. Γ. Στρατής Καθηγητής Τμήμα Μαθηματικών Αθήνα, 2006

2 Εισαγωγή Οι σημειώσεις αυτές αποτελούν βοήθημα μελέτης των φοιτητών του Τμήματος Πληροφορικής και Τηλεπικοινωνιών του Εθνικού και Καποδιστριακού Πανεπιστημίου Αθηνών για το μάθημα «Μαθηματικά Τηλεπικοινωνιών». Περιέχουν και ενότητες που ο χρόνος δεν επιτρέπει να διδαχθούν στο μάθημα, αλλά εμφανίζονται εδώ για την καλύτερη επαφή των φοιτητών με το γνωστικό αντικείμενο αυτού του μέρους του μαθήματος (το άλλο μέρος αναφέρεται στις Συνήθεις Διαφορικές Εξισώσεις). Στις σημειώσεις αυτές δεν περιέχονται ορισμένες ιδιαίτερα σημαντικές εφαρμογές της θεωρίας των Μιγαδικών Συναρτήσεων, όπως ο ασυμπτωτικός υπολογισμός ολοκληρωμάτων και τα προβλήματα Riemann Hilbert. Όλα τα αποτελέσματα παρουσιάζονται χωρίς απόδειξη, αφ ενός εφ όσον (λόγω της φιλοσοφίας του Τμήματος Πληροφορικής και Τηλεπικοινωνιών για το συγκεκριμένο μάθημα) ο στόχος των παραδόσεων είναι η χρηστικότητα των μαθηματικών εννοιών και εργαλείων, και αφ ετέρου επειδή ο χρόνος που διατίθεται δεν επαρκεί για αυστηρή θεώρηση των εννοιών. Οι αποδείξεις, καθώς και πληρέστερη ανάπτυξη της θεωρίας, μπορούν να βρεθούν στα συγγράμματα της βιβλιογραφίας. Ευχαριστώ τους συναδέλφους και φίλους καθηγητές Σ. Θεοδωρίδη, Η. Κουτσουπιά, Θ. Σφηκόπουλο (του Τμήματος Πληροφορικής και Τηλεπικοινωνιών), Ν. Αλικάκο, Σ. Μερκουράκη, Τ. Χατζηαφράτη και τον ομότιμο καθηγητή Σ. Τερσένοβ (του Τμήματος Μαθηματικών) για χρήσιμες συζητήσεις που είχαμε κατά τη διάρκεια της συγγραφής αυτών των σημειώσεων και για τις εύστοχες παρατηρήσεις τους. Επίσης ευχαριστώ τον Δρ. Π. Μπουμπούλη και τη μεταπτυχιακή φοιτήτρια Μ. Κοτέ για τη δακτυλογράφηση των σημειώσεων. Αθήνα, Ιανουάριος 2006 Ιωάννης Γ. Στρατής Καθηγητής Τμήμα Μαθηματικών, ΕΚΠΑ

3 Perieqìmena Eisagwg 5 1 Oi MigadikoÐ ArijmoÐ To S ma twn Migadik n Arijm n To Migadikì EpÐpedo Polikèc Suntetagmènec - TÔpoi De Moivre kai Euler ParadeÐgmata - Ask seic AkoloujÐec kai Seirèc Migadik n Arijm n Ask seic Seirèc ParadeÐgmata Ask seic Sunart seic Migadik n Arijm n Kat taxh Sunìlwn sto Migadikì EpÐpedo SuneqeÐc Sunart seic Stereografik Probol Sunart seic miac migadik c metablht c z Analutik Polu numa Dunamoseirèc Analutikèc Sunart seic Oi Stoiqei deic Sunart seic H Ekjetik Sun rthsh Oi trigwnometrikèc sunart seic H Logarijmik Sun rthsh Oi sunart seic z λ, λ z, λ C Oi antðstrofec trigwnometrikèc sunart seic Olokl rwsh Orismèno olokl rwma KampÔlec Olokl rwma migadik n sunart sewn miac migadik c metablht c To je rhma Cauchy-Goursat

4 4.4.1 Diaisjhtik -Fusik apìdeixh To je rhma Cauchy To je rhma Cauchy-Goursat Apl kai Pollapl Sunektik SÔnola To Aìristo Olokl rwma O oloklhrwtikìc tôpoc tou Cauchy To je rhma tou Green Ask seic 'Allec idiìthtec twn analutik n sunart sewn 54 6 Analutik Sunèqish 56 7 Memonwmènec AnwmalÐec Analutik n Sunart sewn Kat taxh Memonwmènwn Anwmali n Arq tou Riemann Je rhma Casorati Weierstrass An ptugma Laurent Oloklhrwtik Upìloipa DeÐkthc Strof c kai to Je rhma Oloklhrwtik n UpoloÐpwn tou Cauchy Efarmogèc tou Jewr matoc Oloklhrwtik n UpoloÐpwn Efarmogèc tou jewr matoc Oloklhrwtik n UpoloÐpwn tou Cauchy ston upologismì oloklhrwm twn kai seir n Upologismìc Oloklhrwm twn Upologismìc OrÐwn Seir n SÔmmorfh Apeikìnish Je rhma SÔmmorfhc Apeikìnishc tou Riemann Efarmogèc thc SÔmmorfhc Apeikìnishc Probl mata Dirichlet kai Neumann EpÐlush problhm twn Dirichlet kai Neumann me sômmorfh apeikìnish Efarmogèc Hlektrik kukl mata - Hmitonoeid ReÔmata Grafik Anapar stash Hmitonoeid n Sunart sewn Migadik Anapar stash Kukl mata RLC Oi kanìnec tou Kirchhoff Migadikèc Emped seic se Seiriak Par llhlh SÔndesh GenÐkeush thc ènnoiac thc migadik c empèdhshc Migadikì Di nusma Efarmogèc sto Statikì Hlektrismì O nìmoc tou Coulomb 'Entash HlektrikoÔ PedÐou. Hlektrostatikì Dunamikì

5 To je rhma tou Gauss To migadikì hlektrostatikì dunamikì Grammik fortða AgwgoÐ Qwrhtikìthta Ask seic BibliografÐa

6 Kef laio 1 Oi MigadikoÐ ArijmoÐ 'Hdh apì ton 16 o ai na emfanðzontai arijmoð thc morf a + b 1, (a, b R). O Cardan touc qrhsimopoðhse gia th lôsh exis sewn 2 oυ kai 3 oυ bajmoô. To 18 o ai na o Euler èkane qr 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 c kurðwc q rh sth gewmetrik touc anapar stash kai ston Gauss. O pr toc pl rhc kai austhrìc 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 n To s ma twn migadik n arijm n C eðnai to sônolo twn diatetagmènwn zeug n pragmatik n arijm n (a, b) me prìsjesh kai pollaplasiasmì pou orðzontai wc ex c: Idiìthtec (a, b)+(c, d) =(a + c, b + d) (a, b)(c, d) =(ac bd, ad + bc) 'Estw z k =(a k,b k ) tuqìntec migadikoð arijmoð. Tìte z 1 + z 2 = z 2 + z 1 antimetajetikìthta thc prìsjeshc z 1 +(z 2 + z 3 )=(z 1 + z 2 )+z 3 prosetairistikìthta thc prìsjeshc (0, 0) oudètero stoiqeðo thc prìsjeshc antðjetoc tou z =(a, b), eðnai o z =( a, b) 'Estw λ, µ R, z,w C. Tìte 6

7 λ(µz) =(λµ)z (λ + µ)z =(λz + µz) λ(z + w) =λz + λw z 1 z 2 = z 2 z 1 antimetajetikìthta tou pollaplasiasmoô z 1 (z 2 z 3 )=(z 1 z 2 )z 3 prosetairistikìthta tou pollaplasiasmoô z 1 (z 2 + z 3 )=z 1 z 2 + z 1 z 3 epimeristikìthta tou pollaplasiasmoô wc proc thn prìsjesh (1, 0) oudètero stoiqeðo tou pollaplasiasmoô antðstrofoc tou z =(a, b) (0, 0), eðnai o 1 z = ( a, a 2 +b 2 ) b a 2 +b 2 Sunep c to C eðnai èna s ma (antimetajetikìc daktôlioc me antðstrofo pollaplasiasmoô.) Mia idiìthta tou R pou den metafèretai sto C eðnai ekeðnh thc di taxhc. MporeÐ eôkola na deiqjeð ìti to i =(0, 1) den mporoôme na to qarakthrðsoume wc arnhtikì jetikì, qwrðc na 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 tic arijmhtikèc 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 tetragwnikèc rðzec tou 1: to i kai to i. Epiplèon, up rqoun dôo migadikèc tetragwnikèc rðzec k je mh mhdenikoô migadikoô arijmoô a + bi. Pr gmati: opìte (x + iy) 2 =(a + bi) { x 2 y 2 = a 2xy = b a + a x = ± 2 + b 2 2 { 4x 4 4ax b 2 =0 y = b 2x 7

8 kai ètsi y = b 2x = ± a + { a 2 + b 2 1, b 0 sgn(b), ìpou sgn(b) = 2 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 suntelestèc dèqetai lôsh sto C. Pr gmati: ( az 2 + bz + c =0 a, b, c C, a 0 z + b ) 2 = b2 4ac z = b ± b2 4ac 2a 4a 2 2a AntÐjeta, ìpwc gnwrðzoume, to x 2 +1 = 0, p.q., den èqei rðza sto R. Dhl. to C eðnai algebrik c kleistì (afoô to anwtèrw isqôei gia k je poluwnumik exðswsh). 1.2 To Migadikì EpÐpedo (a, b) a + ib xonac twn x pragmatikìc xonac xonac twn y fantastikìc xonac Fti qnoume èna trðgwno me dôo pleurèc to 1 kai to z 1. Met fti qnoume èna ìmoio trðgwno, me ton Ðdio prosanatolismì, kai to z 2 na antistoiqeð sto 1. Tìte to di nusma pou antistoiqeð sto z 1 eðnai to z 1 z 2 (sq ma 1.2(a) ). ParathroÔme ìti pollaplasiasmìc epð i eðnai gewmetrik isodônamoc me strof 90 antðjeta me th for thc kðnhshc twn deikt n tou rologioô (sq ma 1.2(b) ). T ra, an z = x + iy, èqoume touc ex c ìrouc: Re z := x: to pragmatikì mèroc tou z. Im z := y: to fantastikì mèroc tou z, Imz R. z := x iy: o suzug c tou z. z := z z = x 2 + y 2 : h apìluth tim, mètro, nìrma tou z. arg z := θ: to ìrisma tou z, opìte sin θ = Imz, cos θ = Rez z z ParadeÐgmata (i) Re z>0: to dexð hmiepðpedo (blèpe sq ma 1.3). (ii) {z : z = z}: h pragmatik eujeða. 8

9 Sq ma 1.1: Prìsjesh. (iii) {z : θ <argz<θ}: blèpe sq ma 1.4. (iv) {z : z +1 < 1}: blèpe sq ma 1.5. { } (v) z : argz π 2 < π = {z : Imz > 0}: to nw hmiepðpedo. 2 Prìtash 'Estw z, w C, tìte: 1. z + w = z + w 2. zw = z w ) 3. = ( z w 4. z = z z w,w 0 5. Rez = 1 (z + z) 2 6. Imz = 1 (z z) 2i 7. z z =(Rez) 2 +(Imz) 2 8. z Rez z, z Imz z 9. z z = z 2, zw = z w 10. z + w z + w : trigwnik anisìthta 9

10 (a) (b) Sq ma 1.2: Pollaplasiasmìc. 11. z w z w 12. z Rez + Imz 13. z 1 w 1 + z k w k 2 ( z z k 2)( ) w w k 2 anisìthta Cauchy - Schwarz to fl = fl isqôei 14. arg z = argz λ, µ C :(λ, µ) (0, 0) kai λz j = µ w j,j=1,..., k 15. arg(zw) = argz + argw (mod2π) 16. arg z w = argz argw (mod2π) 10

11 Sq ma 1.3: Re z>0. Sq ma 1.4: {z : θ <argz<θ}. Sq ma 1.5: {z : z +1 < 1}. 11

12 1.3 Polikèc Suntetagmènec - TÔpoi De Moivre kai Euler 'Enac mh mhdenikìc migadikìc arijmìc prosdiorðzetai pl rwc apì thn apìluth tim tou kai to 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 tìte z 1 = r 1 (cos θ 1 + i sin θ 1 ) z 2 = r 2 (cos θ 2 + i sin θ 2 ) z 1 z 2 = r 1 r 2 [cos(θ 1 + θ 2 )+isin(θ 1 + θ 2 )] z 1 = r 1 [cos(θ 1 θ 2 )+isin(θ 1 θ 2 )] z 2 r 2 z n = r n (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 z n = z 0. Par deigma: H eôresh twn kubik n riz n thc mon dac. z 3 =1 r 3 (cos 3θ + i sin 3θ) =1(cos0+isin 0) r =1, 3θ =0(mod2π) z 1 =cos0+isin 0 = 1 z 2 =cos 2π 3 + i sin 2π 3 3 = i 2 z 3 =cos 4π 3 + i sin 4π 3 = i 2 H polik morf twn tri n aut n riz n deðqnei ìti eðnai oi korufèc enìc isopleôrou trig nou eggegrammènou sto monadiaðo kôklo. OmoÐwc, oi n-ostèc rðzec enìc z C eðnai oi korufèc kanonikoô polug nou me n pleurèc pou eðnai eggegrammèno ston kôklo kèntrou 0 kai aktðnac r 1 n. Suqn qrhsimoipoieðtai oi tôpoc tou Euler: e iθ =cosθ + i sin θ Parat rhsh 'Estw z = x + iy = r(cos θ + i sin θ) =r(cos(θ +2kπ)+i sin θ +2kπ)). 12

13 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 arg λ z ekeðnh thn tim tou argz gia thn opoða isqôei λ< arg λ z λ +2π. EÔresh tou Argz z = x + iy, x 2 + y 2 0 JewroÔme 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 ParadeÐgmata - Ask seic 1. Na brejeð to arg 3π ( 1 i) 2 LÔsh z := 1 i z =1+i Argz = arctan( 1)= π 1 4 Argz = π π = 3π 4 4 kai argz =2kπ 3π,k Z. 4 T ra 3π 2 < 2kπ 3π 4 3π 2 +2k k =2, dhlad arg 3π ( 1 i) =4π 3π 2 4 = 13π 4 2. Na brejeð h anagkaða kai ikan sunj kh ste oi z 1,z 2,z 3 C na eðnai suneujeiakoð. LÔsh a) 'Estw ìti ta z 1, z 2, z 3 eðnai suneujeiak. H eujeða pou dièrqetai apì ta z 1, z 2 èqei exðswsh z = z 1 + τ(z 1 z 2 ),τ R, h opoða epalhjeôetai kai apì to z 3. 'Etsi z 3 z 1 z 2 z 1 R. b) 'Estw ìti up rqei τ R : z 3 z 1 = τ(z 1 z 2 ). Tìte z 3 = z 1 + τ(z 1 z 2 ) z 2 = z 1 (z 1 z 2 ) z 1 = z 1 +0(z 1 z 2 ) 13

14 ap' ìpou èpetai ìti ta z 1,z 2,z 3 brðskontai epð thc eujeðac z = z 1 + τ(z 1 z 2 ),τ R 3. Poiìc eðnai o gewmetrikìc tìpoc twn shmeðwn z = x + iy tou migadikoô epipèdou pou ikanopoioôn thn exðswsh LÔsh 'Estw z j = a j + b j i, j =1, 2. Tìte z z 1 z z 2 = k, k : staj., z 1,z 2 C. x + yi (a 1 + b 1 i) x + yi (a 2 + b 2 i) = k (x a 1) 2 +(y b 1 ) 2 = k 2 (x a 2 ) 2 + k 2 (y b 2 ) 2 (1 k 2 )x 2 +2(k 2 (a 2 a 1 )x +(1 k 2 )y 2 +2(k 2 b 2 b 1 )y = k 2 (a b 2 2) (a b 2 1). a) k =1. Tìte 2(b 1 b 2 )y =2(a 2 a 1 )x + a b 2 1 a 2 2 b 2 2, pou parist nei eujeða (kai m lista th mesok jeto tou eujugr mmou tm matoc pou en nei ta z 1,z 2 ). b) k 1. Tìte ( ) 2 ( ) 2 x + k2 a 2 a 1 1 k + y + k2 b 2 b k = A, ìpou 2 A := k2 (a b 2 2) (a b 2 1) 1 k 2 + pou parist nei kôklo kèntrou kai aktðnac R = A. ( ) k 2 2 ( ) a 2 a 1 k 2 2 b 2 b 1 +, 1 k 2 1 k 2 z 0 = a 1 a 2 k 2 + i b 1 b 2 k 2 1 k 2 1 k 2 4. Na apodeiqjeð ìti, an a, b R kai ζ C, h exðswsh az z + ζz + ζz + b =0 parist nei eujeða ìtan {a =0&ζ ζ > 0} kai kôklo peperasmènhc, mh mhdenik c aktðnac ìtan {a 0&ζ ζ >ab}. LÔsh 14

15 '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 ètsi az z + ζz + ζz + b = a(x 2 + y 2 )+2γx 2δy + b =0 (i) An a =0 2γx 2δy + b =0, pou parist nei eujeða ìtan γ 2 + δ 2 > 0 ζ ζ >0. (ii) An a 0 x 2 + y 2 +2 γ x 2 δ y + b =0 a a a ( ) 2 ( ) 2 x + γ a + y + δ a = r 2, ìpou r 2 = γ2 +δ 2 ab, a 2 pou parist nei kôklo ìtan r 2 > 0, dhlad ìtan ζ ζ = γ 2 + δ 2 >ab. 5. a) Na lujeð h exðswsh z 8 =1 b) Na lujeð h exðswsh z 5 = 32 UpenjumÐzoume ton tôpo tou De Moivre: An z = r(cos θ + i sin θ), tìte z n = r n (cos nθ + i sin nθ). Gia na lôsoume thn exðswsh z n = w upojètoume ìti z = r(cos θ + i sin θ), w= ϱ(cos φ + i sin φ) kai èqoume LÔsh r = n ρ kai nθ = φ +2kπ { ( ) ( )} z = n φ +2kπ φ +2kπ ρ cos + i sin n n a) AfoÔ 1=cos2kπ + i sin 2kπ, èqoume z = cos 2kπ 2kπ + i sin,k=0, 1, 2, 3, 4, 5, 6, = 1, + i,i, 1 + i, 1, 1 i, i, i 2 15

16 b) 'Eqoume 32 = 32{cos(π +2kπ)+isin(π +2kπ)}, k Z = 2 5 {cos(π +2kπ)+isin(π +2kπ)} { ( ) ( )} π +2kπ π +2kπ z =2 cos + i sin,k=0, 1, 2, 3, An z =1, n.d.o. w 1z+w 2 w 2 z+w 1 =1, gia opoiousd pote w1,w 2 C. LÔsh 'Eqoume opìte w 1 z + w 2 z z= z 2 =1 = w 2 z + w 1 w 1z + w 2 w 1 z + w 2 = w 2 z + w 1 w 2 + w 1 z z z = z = w 1 z + w 2 (w 2 + w 1 z)z w 1 z + w 2 w 1 z + w 2 z = w 1z + w 2 w 1 z + w 2 z =1 7. Migadikìc Sumbolismìc ExÐswsh EujeÐac: z = w 1 + w 2 τ, w 1,w 2 C,τ R ExÐswsh KÔklou: z w = r, w C, r>0 ExÐswsh 'Elleiyhc: z w + z + w =2a, w C, a > 0, estðec: ± w, hmi xonac: a. meg loc 8. z = re iθ ze ia = re iθ e ia = re i(θ+a) Dhlad, z epð e ia shmaðnei strof tou z kat th jetik for kat gwnða a. 9. i) 'Estw p + qi rðza thc a 0 z n + a 1 z n a n =0 me a 0 0, a 1,...,a n R, p, q R. Tìte h p qi eðnai epðshc rðza (jèse p + qi = re iθ kai met gr ye th suzug exðswsh). ii) z 2 +(2i 3)z +5 i =0 z = (2i 3)± (2i 3) (5 i) z =2 3i, 2 1 z =1+i. OQI suzugeðc rðzec. (Ed oi suntelestèc eðnai migadikoð arijmoð.) 16

17 Kef laio 2 AkoloujÐec kai Seirèc Migadik n Arijm n Orismìc z n z orc z n z 0 (sugklðnei sto R) ParathroÔme ìti z n z Rez n Rez Imz n Imz Orismìc {z n } akoloujða Cauchy ε>0 N Z : n, m > N z n z m <ε Prìtash H {z n } eðnai sugklðnousa h {z n } eðnai Cauchy. Orismìc Mia seir k=1 z k sugklðnei an h akoloujða twn merik n ajroism twn {s n } sugklðnei, ìpou s n = z 1 +z 2 + +z n. To ìrio, tìte, thc {s n } 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 z k eðnai: z n 0 ìtan n. (3) Ikan sunj kh gia th sôgklish thc k=1 z k eðnai h sôgklish thc k=1 z k (opìte h k=1 z k lègetai apolôtwc sugklðnousa). ParadeÐgmata (1) z n 0 an z < 1, afoô z n 0 = z n 0 n (2) n+i 1, afoô n 1 n+i = i n+i = 1 0 n 2 +1 (3) H i k k=1 k 2 +i sugklðnei, afoô i k k 2 +i = 1 k 4 +1 kai afoô h k=1 1 k 4 +1 sugklðnei. 17

18 1 1 apoklðnei, afoô k+i Pr gmati, ( ) k=1 Re 1 (4) H k=1 k+i Jèse a k = 1 k+1,b k = k k Tìte b k a k 0 gia k 1. Ex llou h k=1 2.1 Ask seic k+i = k i = k=1 k 2 +1 kai afoô h ( k=1 Re k k k+i ) apoklðnei. 1 k+1 apoklðnei. Apì to Krit rio SÔgklishc apoklðnei kai h k=1 k k 'Estw x n =1+r cos a+r 2 cos 2a+ +r n cos na, r (0, 1). Na brejeð to lim n x n. LÔsh Jètw y n = r sin a + r 2 sin 2a + + r n sin na kai z n := x n + iy n =1+r(cos a + i sin a)+ + r n (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: z n =1+w + w w n = 1 wn+1 1 w kai afoô w < 1 : lim n = 1. 1 w Sunep c ( ) 1 lim x n = lim(rez n )=Re(lim z n )=Re = 1 w ( ) ( ) 1 1 = Re = Re 1 r(cos a + i sin a) (1 r cos a) ir sin a) ( ) (1 r cos a)+ir sin a 1 r cos a = Re (1 r cos a) 2 + r 2 sin 2 = a) 1 2r cos a + r 2 i 2. Na brejoôn ta lim n n, lim n n (1+i)n. n LÔsh a) in n = in n b) u n = (1+i)n n u n+1 u n = (1+i) n+1 (1+i) n = i n n = 1 n <ε, ìtan n> 1 ε, ra lim n in n =0. n n+1 = gia k je n 3 èqoume n 2 > 3 2 > 1.05 > 1 n+1 4 opìte u n > (1.03) n 3 u 3, ra h u n den sugklðnei. n 1+i = n 2 n+1 n+1 18

19 2.2 Seirèc Prìtash H gewmetrik seir n=0 zn = { 1 Prìtash (Krit rio SÔgkrishc), z < 1 1 z apoklðnei, z 1 (i) z k w k kai k=1 w k : sugklðnei k=1 z k: sugklðnei apolôtwc. (ii) z k w k kai k=1 z k : apoklðnei k=1 w k : apoklðnei all h w k mporeð na sugklðnei ìqi. Prìtash (Krit rio thc p-seir c) n=1 1 n p sugklðnei an p>1 kai apoklðnei sto an p 1. Prìtash (Krit rio tou Lìgou) z < 1 n+1 lim n > 1 z n z n =1 n=1 Prìtash (Krit rio thc RÐzac) < 1 lim n ( z n ) 1 n > 1 =1 n=1 Prìtash (Krit rio Cauchy) z n sugklðnei apolôtwc apoklðnei den efarmìzetai to krit rio sugklðnei apolôtwc apoklðnei den efarmìzetai to krit rio (i) Mia akoloujða f n (z) sugklðnei omoiìmorfa sto sônolo A ε >0 N : n N f n (z) f n+p (z) <ε z A, p =1, 2,... (ii) H seir k=1 g k(z) sugklðnei omoiìmorfa sto sônolo A p ε >0 N : n N g k (z) <ε z A, p =1, 2,... k=n+1 Prìtash (To M-krit rio tou Weierstrass) 'Estw g n akoloujða sunart sewn pou orðzetai sto A C. 'Estw ìti up rqei akoloujða pragmatik n arijm n M n 0: (i) g n (z) M n, z A (ii) h n=1 M n sugklðnei Tìte h n=1 g n(z) sugklðnei apolôtwc kai omoiìmorfa epð tou A. 19

20 2.2.1 ParadeÐgmata Ask seic 1. 1 n=1 sugklðnei apolôtwc gia Rez > 1 kai omoiìmorfa gia Rez 1+ε, ε > 0. n z 'Estw z = x + iy. Tìte n z = e z log n = e (x+iy)logn. 1 n = 1 kai z n x n=1 1 n = 1 z n=1 sugklðnei an x>1, dhl. an Rez > 1. n x e x log n = 1 'Otan Rez 1+ε jètw M n = 1 n 1+ε kai h omoiìmorfh sôgklish èpetai apì to M- krit rio tou Weierstrass. 2. e inz n=0 sugklðnei omoiìmorfa sto hmiepðpedo Imz < a gia k je a>0. n 2 +1 'Estw z = x + iy. Tìte e inz n 2 +1 = e inx e ny n 2 +1 = eny n 2 +1 An Imz = y< a <0, tìte e ny <e na kai ètsi e inz := M n. Exet zw th sôgklish thc n=0 M n. Krit rio Lìgou: lim M n+1 M n = lim 3. z n n=1 n e (n+1)a (n+1) 2 +1 Jètw A σ = {z : z σ}, 0 σ<1. Jètw g n (z) = zn n. Tìte g n (z) = z n 'Ara h n=1 n σn n n 2 +1 e na n 2 +1 n 2 +1 e na = e a lim n2 +1 n 2 +2n+2 = e a < 1 σn := M n kai afoô σ<1, h M n sugklðnei. z n n sugklðnei omoiìmorfa sto A σ. H seir aut sugklðnei shmeiak sto A = {z : z < 1} afoô k je z A brðsketai arket kont se k poio A σ gia σ arket konta sto 1. 'Omwc h seir den sugklðnei omoiìmorfa epð tou A. An sunèkline, h x n n=1 n sunèkline omoiìmorfa epð tou [0, 1), pou den isqôei. ( skhsh). ja 20

21 Kef laio 3 Sunart seic Migadik n Arijm n 3.1 Kat taxh Sunìlwn sto Migadikì EpÐpedo Orismìc D(z 0,r)={z : z z 0 <r} anoiqtìc dðskoc, perioq tou z 0. C(z 0,r)={z : z z 0 = r} kôkloc. S : anoiktì orc z S δ >0:D(z 0,δ) S. S = C S ( S = {z C : z/ S}) sumpl rwma tou S. S : kleistì orc S anoiqtì {z n } S kai z n z z S S = {z : δ >0: D(z,δ) S = kai D(z,δ) S } sônoro tou S. 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 perièqei to S en oôte to A oôte to B perièqoun to S. S sunektikì an den eðnai mh sunektikì. [z 1,z 2 ]:to eujôgrammo tm ma me kra z 1, z 2. polugwnik gramm : peperasmènh ènwsh eujugr mmwn tmhm twn thc morf c [z 0,z 1 ] [z 1,z 2 ]... [z n 1,z n ]. 21

22 An k je dôo shmeða tou S mporoôn na enwjoôn me mia polugwnik gramm pou perièqetai sto S, tìte to S lègetai polugwnik sunektikì. polugwnik sunektikì sunektikì. To antðstrofo den isqôei. Oi dôo ènnoiec eðnai isodônamec gia ta anoiqt sônola. tìpoc orc = anoiqtì + sunektikì. Sq ma 3.1: Polugwnik sunektikì SÔnolo. 3.2 SuneqeÐc Sunart seic Orismìc Mia sun rthsh migadik n tim n f(z) orismènh se mia perioq tou z 0, eðnai suneq c sto z 0, an z n z 0 sunep getai ìti f(z n ) f(z 0 ). Diaforetik, h f eðnai suneq c sto z 0 an gia k je ɛ>0 up rqei δ>0 ètsi ste an z z 0 <δtìte f(z) f(z 0 ) <ɛ. H f eðnai suneq c se ènan tìpo D an gia k je akoloujða {z n } D kai z D tètoia ste z n z, na isqôei f(z n ) 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 kai mìno tìte an oi u kai v eðnai suneqeðc sunart seic tou (x, y). 22

23 ParadeÐgmata. 1. K je polu numo m n P (x, y) = a kj x k y j eðnai suneq c sun rthsh se ìlo to epðpedo. j=1 k=1 2. H f(z) = 1 = x i z x 2 +y 2 y x 2 +y 2 eðnai suneq c sto epðpedo {z : z 0}. Orismènec 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 C n an kai to Ref kai to Imf èqoun suneqeðc merikèc parag gouc n t xhc. Mia akoloujða sunart sewn {f n } sugklðnei sthn f omoiìmorfa sto D, an gia k je ɛ>0 up rqei N>0 tètoio ste n>n sunep getai ìti f(z n ) f(z) <ɛgia k je z D. Anaferìmenoi p li sta pragmatik kai fantastik mèrh twn {f n }, blèpoume ìti to omoiìmorfo ìrio suneq n sunart sewn eðnai suneq c sun rthsh. M-test. An f k suneq c sto D, k =1, 2...kai f k (z) M k sto D kai an h k=1 M k sugklðnei, tìte h k=1 f k(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 sumpag /sunektik sônola, en autì de sumbaðnei gia kami llh kathgorða sunìlwn. P.q. h f(z) = 1 apeikonðzei to fragmèno sônolo 0 < z < 1 epð tou mh fragmènou z sunìlou z > 1. 'Estw ìti h u(x, y) èqei merikèc parag gouc u x kai u y pou mhdenðzontai se k je shmeðo enìc tìpou D. Tìte h u eðnai stajer ston D 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 na jewr soume ìti to migadikì epðpedo pern ei apì ton ishmerinì thc monadiaðac sfaðrac me kèntro to z =0. Se k je shmeðo z tou epipèdou antistoiqeð akrib c èna shmeðo P thc epif neic thc sfaðrac, pou brðsketai wc tom thc eujeðac zn (N o bìreioc pìloc) me thn epif neia aut. Antistrìfwc, se k je shmeðo P thc epif neiac thc sfaðrac, plhn tou N, antistoiqeð akrib c èna shmeðo z tou epipèdou. Antistoiq ntac sto N to shmeðo sto 23

24 peiro, petuqaðnoume mia 1-1 antistoiqða metaxô twn shmeðwn thc epif neiac thc sfaðrac kai tou epektetamènou migadikoô epipèdou. Aut h sfaðra lègetai sfaðra tou Riemann kai h antistoiqða stereografik probol, probol tou PtolemaÐou. To sônolo z > 1 lègetai perioq tou. ɛ Sq ma 3.2: Stereografik probol. Orismìc Lème ìti {z k } an z k, dhlad an gia k je M>0 up rqei N Z: k>n sunep getai ìti z k >M. OmoÐwc f(z) an f(z). 3.3 Sunart seic miac migadik c metablht c z Analutik Polu numa 'Ena polu numo P (x, y) lègetai analutikì polu numo, an up rqoun (migadikèc) stajerèc a k ètsi ste: P (x, y) =a 0 + a 1 (x + iy)+a 2 (x + iy) a N (x + iy) N Tìte ja lème ìti to P eðnai polu numo wc proc z kai ja to gr foume wc P (x, y) =a 0 + a 1 z + a 2 z a N z N P.q. To polu numo x 2 +y 2 +2ixy eðnai analutikì en eôkola deðqnetai ìti to x 2 +y 2 2ixy den eðnai analutikì. Orismìc 'Estw f(x, y) =u(x, y)+iv(x, y), ìpou u kai v sunart seic pragmatik n tim n. Me thn proôpìjesh ìti up rqoun oi u x, u y, v x, v y orðzoume f x = u x + iv x, f y = u y + iv y 24

25 ApodeiknÔetai ìti èna polu numo eðnai analutikì tìte kai mìno tìte an P y = ip x. Orismìc Mia sun rthsh f me migadikèc timèc pou orðzetai se mia perioq tou z, lègetai diaforðsimh sto z an up rqei to f(z + h) f(z) lim h 0 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 diaðresh diaforðsimwn sunart sewn. Tèloc apodeiknôetai ìti an to P eðnai analutikì, tìte eðnai diaforðsimo se k je z. Mia eurôterh kl sh sunart sewn tou z, eðnai autèc pou dðnontai apì peira polu numa tou z, alli c dunamoseirèc tou z Dunamoseirèc Orismìc Dunamoseir eðnai mia seir thc morf c k=0 c kz k. Gia th melèth thc sôgklishc dunamoseir n, mac qrei zetai h ènnoia tou lim (limsup), miac jetik c pragmatik c akoloujðac: ( lim a n := lim n n ) sup a k. k n AfoÔ to sup k n a k eðnai mia fjðnousa sun rthsh tou n, to ìrio up rqei p nta eðnai. Oi idiìthtec tou lim pou ja qreiastoôme eðnai: An lim n a n = L tìte: 1. gia k je N kai gia k je ɛ>0, up rqei κ>n tètoio ste a k L ɛ 2. gia k je ɛ>0, up rqei N tètoio ste a k L + ɛ, gia k je k>n Je rhma 'Estw ìti lim c k 1/k = L. 1. An L =0, h c k z k sugklðnei gia ìla ta z. 2. An L =+, h c k z k sugklðnei mìno gia z =0. 3. An 0 <L<+, jètoume R = 1 L. Tìte h c k z k sugklðnei gia z <Rkai apoklðnei gia z > R. To R lègetai aktðna sôgklishc thc dunamoseir c. Parat rhsh An h c k z k èqei aktðna sôgklishc R, tìte sugklðnei omoiìmorfa se k je mikrìtero dðsko z R δ ki ètsi eðnai suneq c sto pedðo sôgklis c thc. 25

26 2. To jroisma sugklinous n dunamoseir n eðnai sugklðnousa dunamoseir. 3. To ginìmeno Cauchy sugklinous n dunamoseir n eðnai sugklðnousa dunamoseir gia z < min(r a,r b ), ìpou R a, R b oi aktðnec sôgklishc twn dôo dunamoseir n. UpenjumÐzoume ìti an a n, b n eðnai dôo akoloujðec to ginìmeno Cauchy eðnai h akoloujða c n =. 4. An sumbeð na up rqei to lim c k, c tìte k+1 R = n a k b n k k=0 1 = lim lim c k 1/k 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 n 1/n 1 (0, ). 2. z n n=1. n 2 H dunamoseir aut èqei, epðshc, aktðna sôgklishc 1. 'Omwc, sugklðnei kai gia z =1, afoô z n = 1 0. n 2 n 2 3. z n n=0 n!. H dunamoseir sugklðnei gia k je z, afoô 4. n=0 n!zn. n! lim = lim 1 (n+1)! n+1 c k c k+1 1 (n!) 1/n 0. =0 R =0. 'Ara h dunamoseir sugklðnei mìno sto z =0. 5. n=0 n2 (2z 1) n = n=0 n2 2 n ( z 1 2) n. H aktðna sôgklishc thc dunamoseir c aut c eðnai R = lim(2 n n 2 ) 1/n = lim(2n 2/n )= 2 R = 1 2 kai kèntro eðnai to 1 2. Je rhma 'Estw f(z) = c n z n kai ìti h seir sugklðnei gia z <R. Tìte up rqei h f (z) kai isoôtai me nc n z n 1 ston z <R. Pìrisma Oi dunamoseirèc èqoun k je t xhc par gwgo mèsa sto pedðo sôgklis c touc. Pìrisma An h f(z) = n=0 c nz n èqei mh mhdenik aktðna sôgklishc, tìte c n = f (n) (0) n! gia k je n.. 26

27 Je rhma (Monadikìthtac Dunamoseir n). 'Estw ìti h n=0 c nz n mhdenðzetai se ìla ta shmeða miac {z k } : z k 0 kai z k 0. Tìte h dunamoseir eðnai tautotik Ðsh me to mhdèn. Pìrisma An h a n z n kai h b n z n sugklðnoun kai sumpðptoun se èna sônolo shmeðwn pou èqei shmeðo suss reushc to 0, tìte a n = b n gia k je n. Pìrisma 'Estw lim c n 1/n f (n) (a) n!. <, jètoume f(z) = n=0 c n(z a) n. Tìte c n = Je rhma (Abel). 'Estw ìti h dunamoseir n=0 c nz n sugklðnei se k poio shmeðo z 1 0. Tìte sugklðnei apolôtwc se k je shmeðo z: z < z 1. 'Estw r< z 1. Tìte h dunamoseir sugklðnei omoiìmorfa gia z r Analutikèc Sunart seic Analutikìthta kai exis seic Cauchy-Riemann Prìtash An h f = u + iv eðnai diaforðsimh sto z, up rqoun oi f x kai f y sto z kai ikanopoioôn tic exis seic Cauchy-Riemann. isodônama f y = if x, { ux = v y u y = v x To antðstrofo den isqôei. Up rqoun sunart seic pou den diaforðzontai se èna shmeðo par ìlo pou up rqoun ekeð oi merikèc par gwgoð touc kai ikanopoioôn tic sunj kec Cauchy- Riemann. Par deigma f(z) =f(x, y) = { xy(x+iy) x 2 +y 2,z 0 0,z =0 H f =0kai stouc dôo xonec kai sunep c f x = f y =0sto mhdèn, ìmwc to f(z) f(0) lim z 0 z xy = lim (x,y) (0,0) x 2 + y 2 den up rqei. Pr gmati epð thc eujeðac y = ax : f(z) f(0) ìrio exart tai apì to a! IsqÔei entoôtoic to ex c merikì antðstrofo: z = a 1+a 2 gia z 0kai sunep c to 27

28 Prìtash 'Estw ìti up rqoun se mia perioq tou z oi f x, f y. Tìte an oi f x, f y eðnai suneqeðc sto z kai isqôei f y = if x ekeð, h f eðnai diaforðsimh sto z. Orismìc H f eðnai analutik (olìmorfh) sto z, an eðnai diaforðsimh se mia perioq tou z. H f eðnai analutik se èna sônolo S an eðnai diaforðsimh se ìla ta shmeða enìc anoiqtoô 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 sunart sewn eðnai diaforðsimh sun rthsh. OmoÐwc kai h sônjesh. 'Opwc kai gia tic pragmatikèc sunart seic, h antðstrofh miac sun rthshc mporeð na mhn eðnai kan suneq c. O epìmenoc orismìc, mac epitrèpei na mil me gia diaforisimìthta twn antistrìfwn sunart sewn. Orismìc '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 antðstrofh thc f sto z 0, an eðnai h antðstrofh thc f se k poia perioq tou z 0. Prìtash 'Estw ìti h g eðnai h antðstrofh thc f sto z 0 kai ìti h g eðnai suneq c sto z 0. An h f eðnai diaforðsimh sto g(z 0 ) kai an f (g(z 0 )) 0, tìte h g eðnai diaforðsimh sto z 0 kai g (z 0 )= 1 f (g(z 0 )). H analutikìthta eðnai mia exairetik shmantik idiìthta. Ja asqolhjoôme idiaðtera mazð thc se epìmena kef laia. Gia thn ra dðnoume dôo mesec sunèpeièc thc. Prìtash An h f = u + iv eðnai analutik se ènan tìpo D kai h u eðnai stajer, tìte h f eðnai stajer. Prìtash 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 sugkekrimènh sunj kh, pou analutik ekfr zetai apì tic sunj kec Cauchy-Riemann. Ja doôme t ra ti shmaðnei aut h sunj kh fusik. Poi eðnai dhlad ekeðnh h xeqwrist idiìthta 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 èna metablhtì shmeðo enìc disdi statou dianusmatikoô pedðou kai w to di nusma pou antistoiqeð 28

29 sto z, mèsw thc f. '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 3.3.1) f y = if x, dhlad u x iv x = 1 i (u y iv y ) kai u x + v y =0 (3.1) v x u y =0 (3.2) Autèc oi exis seic ekfr zoun ìti h sun rthsh pou parist netai apì to dianusmatikì pedðo eðnai diaforðsimh. 1. JewroÔme to dianusmatikì pedðo mac wc pedðo ro c kai to w wc mia taqôthta, thn èntash thc ro c sto shmeðo z. Tìte h èkfrash u x + v y lègetai apìklish tou dianôsmatoc w, sumbolðzetai me div w kai metr ei thn exerqìmenh ro an mon da ìgkou se mia kleist perioq tou shmeðou z. An div w > 0 to shmeðo z dra wc phg (source), en an div w <0 to z dra wc katabìjra (sink). An h apìklish mhdenðzetai se k je shmeðo, to pedðo lègetai swlhnoeidèc (sourceless). 'Etsi h (3.1) gr fetai kai qarakthrðzei èna swlhnoeidèc pedðo. div w =0 2. JewroÔme to dianusmatikì pedðo mac wc pedðo dun mewn kai to w wc mia dônamh, thn èntash tou pedðou sto shmeðo z. H èkfrash v x u y lègetai strobilismìc (curl) tou w kai metr ei to èrgo an mon da epif neiac. Pio sugkekrimèna, to èrgo pou par getai apì to pedðo ìtan èna mikrì swmatðdio diagr fei mia kleist kampôlh pou perikleðei to z diaireðtai me thn epif neia pou perikleðei h kampôlh. 'Otan oi diast seic thc kampôlhc teðnoun sto mhdèn, autì to anhgmèno èrgo teðnei sto curl w. An to curl mhdenðzetai se k je shmeðo, to pedðo lègetai astrìbilo. 'Etsi h (3.2) gr fetai kai qarakthrðzei to pedðo wc astrìbilo. curl w =0 SunoyÐzontac, lème ìti mia diaforðsimh pragmatik sun rthsh miac migadik c metablht c parist netai me èna swlhnoeidèc kai astrìbilo pedðo. 29

30 H exðswsh tou Laplace Mia pragmatik sun rthsh h dôo pragmatik n metablht n x, y lègetai armonik se ènan tìpo tou epipèdou xy, an pantoô se autìn ton tìpo èqei suneqeðc pr tec kai deôterec merikèc parag gouc kai ikanopoieð th merik diaforik exðswsh: h = 2 h = h xx + h yy =0 pou eðnai gnwst wc exðswsh tou Laplace. Apì tic sunj kec Cauchy Riemann pou ikanopoioôn oi suntetagmènec sunart seic miac analutik c sun rthshc f = u + iv, paragwgðzontac wc proc x, paðrnoume ìti: Parag gish wc proc y, dðnei antðstoiqa u xx = v yx u yx = v xx. u xy = v yy u yy = v xy. H sunèqeia twn merik n parag gwn exasfalðzei ìti v yx = v xy kai u xy = u yx. Sunep c: u xx + u yy =0 kai v xx + v yy =0. 'Eqoume, dhlad, ìti kai to pragmatikì kai to fantastikì mèroc miac analutik c sun rthshc eðnai 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 u x = v y kai v x = u y. Ex llou f = u iv. AfoÔ ìmwc eðnai kai h f analutik, isqôoun kai gia aut n oi sunj kec Cauchy Riemann, dhlad u x = v y kai v x = u y. Ap autèc èpetai ìti u x = u y = v x = v y =0. 'Ara h f eðnai stajer. 2. (Polikèc Suntetagmènec) ApodeÐxte tic parak tw sqèseic. (a ) Oi exis seic Cauchy-Riemann : u = 1 v r r θ, v = 1 u r r θ. (b ) H exðswsh Laplace : 2 φ r r (g ) Par gwgoc : f iθ f (z) =e = 1 r iz LÔsh φ r + 1 r 2 2 φ θ 2 =0. f θ. Ja apodeðxoume thn pr th sqèsh. 'Eqoume ìti z = r(cos θ + i sin θ) =x + iy, r = x2 + y 2, tan θ = y x. 'Ara r x =cosθ, r y =sinθ, θ x = 1 θ sin θ, r y = 1 cos θ, r 30

31 u x = u r r x + u θ θ x, v x = v r r x + v θ θ x, Apì tic parap nw paðrnoume ìti u y = u r r y + u θ θ y, v y = v r r y + v θ θ y. u x = u r cos θ 1 u u sin θ, r θ v x = v r cos θ 1 v v sin θ, r θ y = u r sin θ + 1 u cos θ, r θ y = v r sin θ + 1 v cos θ. r 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. LÔsh ParagwgÐzontac thn au(x, y)+bv(x, y) =c ja èqoume } } au x + bv x =0 C-R av y + bv x =0 au y + bv y =0 av x + bv y =0 } abv y + b 2 v x =0 abv y a 2 (a 2 + b 2 )v v x =0 x =0 v x =0. 'Ara kai v y =0kai oi sunj kec C R dðnoun ìti u x = u y =0. Sunep c h f eðnai stajer. 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 u x =1kai v y = 1. Oi sunj kec C-R den isqôoun ra h sun rthsh den eðnai analutik. 5. SuzugeÐc suntetagmènec. LÔsh z = x + yi, z = x iy. 'Ara x = 1(z 2 + z), y = 1 (z z). 'Eqoume ìti 2i f x = f z z x + f z z x = f z + f z kai f y = f z z y + f z z y = i ( f z f ) z 'Ara kai f z = 1 ( f 2 x i f y f z = 1 2 ( f x + i f y ) ) 31

32 Oi sunj kec C-R eðnai f = i f. Sunep c oi sunj kec C-R se suzugeðc suntetagmènec y x eðnai f z =0. 6. 'Estw f(z) =u(x, y) +iv(x, y) analutik. EÐdame, tìte ìti oi u, v eðnai armonikèc, dhlad 2 u x + 2 u 2 y 2 =0kai 2 v x + 2 v 2 y =0. 2 Oi u, v (pragmatikèc sunart seic) orismènec se ènan tìpo D lègontai suzugeðc armonikèc an h f = u + iv (migadik sun rthsh) eðnai analutik ston D isodônama an oi u, v eðnai armonikèc kai ikanopoioôn tic sunj kec Cauchy-Riemann. (a ) H φ(x, y) =x 2 y 2 eðnai armonik. Na brejeð h suzug c thc. LÔsh φ =2x 2 φ } =2 x x 2 φ = 2y 2 φ = 2 φ = 2 x + 2 φ 2 y =0. 2 y y 2 'Estw w h suzug c thc. Ja prèpei oi φ, w na ikanopoioôn tic sunj kec C-R w y = φ x kai w x = φ y. 'Ara ja prèpei w y =2x w =2xy + µ(x) ìpou µ(x) stajer wc proc y. 'Ara w x =2y + µ (x) φ y =2y + µ (x) 2y = 2y + µ (x) µ (x) =0 µ 1 (x) =c w(x, y) =2xy + c. (b ) Na brejeð analutik sun rthsh f(z) =u(x, y)+iv(x, y), apì tic sqèseic v(x, y) = y x 2 +y 2 kai f(2) = 0. LÔsh v x = 2xy kai v (x 2 +y 2 ) 2 y = x2 y 2. Epeid prèpei na ikanopoioôntai oi sunj kec C- (x 2 +y 2 ) 2 R ja èqoume: u y = v x = 2xy u(x, y) = 2xy dy + µ(x) = x + (x 2 +y 2 ) 2 (x 2 +y 2 ) 2 x 2 +y 2 µ(x). 'Etsi paðrnoume ìti u x = x2 y 2 + (x 2 +y 2 ) µ (x) kai afoô prèpei u x = v y µ(x) = c. Apì th sqèsh f(2) = 0 paðrnoume ìti c = 1, kai ètsi telik 2 f(z) = z Orjog niec Oikogèneiec 'Estw dôo oikogèneiec kampul n u(x, y) =c 1 kai v(x, y) =c 2 kai èstw ìti h f = u + iv eðnai analutik. Tìte oi oikogèneiec kampul n ( u kai v eðnai orjog niec. dy ArkeÐ na deiqjeð ìti to ginìmeno thc klðshc dx )u thc u(x, y) ( ) =c 1, epð thn klðsh dy dx v thc v(x, y) =c 2 eðnai Ðso me -1. IsodÔnama dhlad arkeð na deiqjeð ìti u v =0. Pr gmati, ( ) dy u ( ) x dy v x = kai =. dx u u dx v y v y 32

33 AntÐstoiqa: u v = ( u x, u ) ( v y x, v ) y Apì tic sunj kec C-R paðrnoume ìti ( ) ( ) dy dy dx dx u v = u v x x + u v y y. = 1 u v =0. Par deigma ProsdiorÐste tic orjog niec troqièc miac oikogèneiac kampul n sto xy epðpedo me exðswsh e x (x sin y y cos y) =c, c R stajer. 'Estw u(x, y) =e x (x sin y y cos y). AnazhtoÔme v tètoia ste h f = u + iv na eðnai analutik. Apì tic sunj kec C-R èqoume oti v y = u x = e x sin y xe x sin y + ye x cos y v x = u y = e x cos y xe x cos y ye x sin y. 'Ara oloklhr nontac wc proc y èqoume v = ye x sin y + xe x cos y + F (x). Sth sunèqeia paragwgðzoume wc proc x kai antikajistoôme opìte telik paðrnoume F (x) =c. 3.4 Oi Stoiqei deic Sunart seic H Ekjetik Sun rthsh EpijumoÔme na orðsoume mia ekjetik sun rthsh miac migadik c metablht c z. dhlad na broôme mia analutik sun rthsh f tètoia ste Jèloume f(z 1 + z 2 )=f(z 1 )f(z 2 ) f(x) =e x x R Apì tic exis seic autèc paðrnoume ìti f(z) =f(x + iy) =f(x)f(iy) =e x f(iy). Jètontac f(iy) =A(y) +ib(y) paðrnoume f(z) =e x A(y)+ie x B(y). Gia na eðnai h f analutik, ja prèpei na ikanopoioôntai oi sunj kec C-R sunep c ja prèpei A(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) =e x A(0) = 1, B(0) = 0. Ex llou A(0) = a, B(0) = b. Katal goume, loipìn, sthn f(z) =e x cos y + ie x sin y 33

34 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 foume f(z) =e z. Idiìthtec thc e z 1. e z = e x 2. e z = ē z 3. e iy = cosy + isiny 4. H exðswsh e z = a èqei apeðrou pl jouc lôseic gia k je a 0(a C) 5. To pedðo tim n thc e z eðnai to C {0}. 6. e z+2πi = e z, z C 7. Sth lwrðda π <Im(z) π h e z eðnai e z 1 e z 2 = e z 1+z 2 9. (e z ) = e z 10. H ekjetik sun rthsh mporeð na oristeð wc e z = z n n=0 n! Oi trigwnometrikèc sunart seic Gia na orðsoume ta sin z kai cos z parathroôme ìti gia y R isqôei e iy =cosy + i sin y, e iy =cosy i sin y opìte sin y = 1 2i ( e iy e iy) kai cos y = 1 2 ( e iy + e iy). 'Etsi mporoôme na orðsoume akèraiec epekt seic twn sin x kai cos x jètontac sin z = 1 ( e iz e iz) 2i cos z = 1 ( e iz + e iz) 2 Idiìthtec 1. cos z = sin z 2. sin z =cosz 3. cos ( z) =cosz, cos (z +2π) =cosz, cos (z + π) = cos z 34

35 4. sin( z) = sin z, sin(z +2π) = sin z, sin(z + π) = sin z 5. cos 2 z +sin z =1 6. cos z + i sin z = e iz (tôpoc tou Euler) 7. cos(z 1 ± z 2 )=cosz 1 cos z 2 sin z 1 sin z 2 8. sin(z 1 ± z 2 ) = sin z 1 cos z 2 ± cos z 1 sin z 2 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 ) > (!) 10. sin z =0 z = κπ, κ Z 11. cos z =0 z = ( κ + 2) 1 π, κ Z 12. Oi sunart seic sunhmðtono kai hmðtono orðzontai kai wc ex c: cos z = sin z = ( 1) n z2n (2n)! ( 1) n z 2n+1 (2n +1)! n=0 n=0 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 ( e z e z), cosh z = 1 2 ( e z + e z), tanh z = sinh z cosh z k.o.k. H lgebra kai o logismìc aut n twn sunart sewn gðnetai me b sh touc prohgoômenouc orismoôc. Oi tôpoi pou apodeiknôontai eðnai Ðdioi me autoôc pou gnwrðzoume gia tic antðstoiqec sunart seic miac pragmatik c metablht c H Logarijmik Sun rthsh 'Estw Log r o gnwstìc fusikìc log rijmoc enìc r R +, ìpwc orðzetai ston apeirostikì logismì. An z C {0} kai r = z, θ =argz, orðzoume log z =Logr + iθ. 35

36 Aut eðnai mia pleiìtimh sun rthsh. An θ 0 sumbolðzei thn kôria tðmh tou arg z ( π <θ 0 π), tìte θ = θ 0 +2nπ, n Z kai ètsi h arqik exðswsh gr fetai: log z =Logr+i (θ 0 +2nπ). An t ra jèsoume n =0sthn prohgoômenh sqèsh, paðrnoume thn kôria tim tou logarðjmou Log z =Logr + iθ 0, r>0, π <θ 0 π. H apeikìnish w =logz eðnai monìtimh me pedðo orismoô to C {0} kai pedðo tim n to π <Im(w) π. Profan c an to pedðo orismoô perioristeð sto R +, o Log z an getai sto sun jh fusikì log rijmo. Parat rhsh w =Logz z = e w Melet ntac tic sunist sec sunart seic Log r kai θ 0 tou Log z, parathroôme ìti eðnai suneq c sto {(r, θ) :r>0,π<θ<π} kai ìti autì eðnai to mègisto dunatì sônolo, ìpou h Log z eðnai suneq c. EpÐshc parathroôme ìti h sun rthsh Log z eðnai analutik ston parap nw tìpo. (Autì èpetai apì tic sunj kec C-R kai pio sugkekrimèna apì thn polik morf touc: u r (r 0,θ 0 )= 1 1 r 0 v θ (r 0,θ 0 ), r 0 u θ (r 0,θ 0 )= v r (r 0,θ 0 )). 'Amesa prokôptei h idiìthta d dz Log 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 = Logr + iθ eðnai monìtimh kai suneq c. ApodeiknÔetai, ìpwc parap nw, ìti eðnai analutik kai ìti d dz log z = 1 z ( z > 0, a<argz<a+2π) 'Enac kl doc miac pleiìtimhc sun rthshc f eðnai opoiad pote monìtimh sun rthsh F pou eðnai analutik se k poion tìpo, se k je shmeðo z tou opoðou h tim F (z) eðnai mia apì 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 kôrioc kl doc. H sun rthsh log z eðnai ènac lloc kl doc thc Ðdiac pleiìtimhc sun rthshc. Idiìthtec thc log z 1. e log z = z 2. log e z = z +2nπi, n Z 3. log(z 1 z 2 )=logz 1 +logz 2 ( ) 4. log z 1 n log z, n N = 1 n 5. z 1 n =exp ( 1 n log z), n N 6. log z n n log z, n N 36

37 3.4.4 Oi sunart seic z λ, λ z, λ C Orismìc z λ =exp(λ log z) H sun rthsh z λ eðnai monìtimh kai analutik ston tìpo {(r, θ) :r>0,a<θ<a+2π}. H par gwgoc autoô tou kl dou thc z λ dðnetai apì th sqèsh d dz zλ = λz λ 1 ( z > 0,a<arg z<a+2π) 'Otan a = π, dhlad π <arg z<π, h sun rthsh z λ lègetai kôrioc kl doc thc pleiìtimhc sun rthshc z λ. Orismìc λ z =exp(z log λ), λ C {0}. 'Otan kajoristeð mia tim thc log λ, h λ z eðnai akèraia sun rthsh tou z. EÔkola faðnetai ìti d dz λz = λ z log λ, λ 0 IsqÔoun oi gnwstoð kanìnec lgebrac kai logismoô gi autèc tic sunart seic. Tèloc, isqôei ìpou ( ) λ = λ(λ 1) (λ n+1). n n! (1 + z) λ = n=0 ( ) λ z n, z < 1, n Oi antðstrofec trigwnometrikèc sunart seic 'Estw z =sinw. Tìte w = arcsin z. 'Eqoume z = eiw e iw 2i e 2iw 2ize iw 1=0 e iw = iz +(1 z 2 ) 1/2 ìpou wc gnwstìn, h (1 z 2 ) 1/2 eðnai dðtimh sun rthsh tou z. PaÐrnontac logarðjmouc, èqoume w = arcsin z = i log[iz +(1 z 2 ) 1/2 ]. H arcsin z eðnai pleiìtimh sun rthsh me peirou pl jouc timèc se k je z. 'Otan prosdioristoôn 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 orðzontai oi sunart seic arccos z = i log[z + i(1 z 2 ) 1/2 ] arctan z = i 2 log i + z i z 37

38 Oi par gwgoi aut n twn tri n sunart sewn, mporoôn na brejoôn apì tic parap nw sqèseic. Oi par gwgoi twn dôo pr twn exart ntai apì tic timèc pou èqoun epilegeð gia thn tetragwnik rðza: AntÐjeta, h par gwgoc thc trðthc d dz arcsin z = 1 (1 z 2 ) 1/2, d dz arccos z = d dz arctan z = 1 1+z 2 1 (1 z 2 ) 1/2. den exart tai apì ton trìpo me ton opoðo gðnetai monìtimh h sun rthsh. Me antðstoiqo, tèloc, trìpo orðzontai oi antðstrofec uperbolikèc sunart seic. ProkÔptei ìti: Ask seic arcsinh z = log[z +(1+z 2 ) 1/2 ] arccosh z = log[z +(z 2 1) 1/2 ] arctanh z = 1 1+z log 2 1 z 1. Na epilujoôn sto C oi exis seic e z =1 i, e z = 1+i. LÔsh Wc gnwstìn e w = z w =log z + i arg z. Sunep c (a ) e z =1 i z =log 2+i ( ) 2kπ π 4,k Z (b ) e z = 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. LÔsh Pr gmati, èstw z = x + iy. Tìte sin z = e iz e iz 2i e iz e iz = ey e y. 2 2 'Estw z = iy, y R +. Tìte an z y sin z. 3. Na epilujeð sto C h exðswsh cos z =2. LÔsh cos z = 2 eiz +e iz 2 = 2 e 2iz 4e iz +1 = 0 e iz = 4± = 2 ± 3. 'Estw z = x + iy. Tìte e iz = e y (cos x + i sin x). 'Ara e y cos x =2± 3 kai e y sin x =0 sin x =0 x = kπ, k Z. Apì thn e y cos x =2± 3 èpetai ìti o k prèpei na eðnai rtioc, dhlad k =2m, afoô e y > 0 kai 2 ± 3 > 0. 'Etsi e y =2± 3 y = log(2 ± 3) 38

39 kai sunep c (afoô 2 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 (a ) log[( 1 i)(1 i)] = log( 2) = log 2 + πi +2nπi, n Z (b ) log( 1 i) =log 2+ 5π 4 i +2mπi, m Z (g ) log(1 i) =log 2+ 7π 4 i +2kπi, k Z ParathroÔme ìti log[( 1 i)(1 i)] = log( 1 i)+log(1 i) (mod2πi). 5. Na upologistoôn oi timèc twn 3 1/2, i 1/2, i i, ( 1) 2. (a ) 3 1/2 = e (1/2) log 3 = e (1/2)(log 3+2kπi) = e (1/2) log 3 e kπi = ± 3. (b ) i 1/2 = e (1/2) log i = e (1/2)i( π +2kπ) 2 = ±e πi 4 = ± 2(1 + i). (g ) i i = e i log i = e i(log 1+i( π 2 +2kπ)) = e ( π 2 +2kπ),k Z. (Dhl. to i i R). (d ) ( 1) 2 = e 2log( 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 z n n log z. Blèpoume merik paradeðgmata. (a ) z = i, n =2: log i 2 =log( 1) = (2k +1)πi, k Z. En 2logi =(4k +1)πi, k Z. (b ) Log ((1 + i) 2 )=2Log(1+i). (g ) Log (( 1+i) 2 ) 2Log( 1+i). (d ) An log z =Logr + iθ ( r>0, π 4 <θ< 9π 4 (e ) An log z =Logr + iθ ( r>0, 3π 4 <θ< 11π 4 2 ) log i 2 =2logi. ) log i2 2logi. 7. 'Estw A a0 = {z : a 0 Im(z) <a 0 +2π}, a 0 R. Tìte h e z apeikonðzei to A a0 1-1 kai epð tou C {0}. LÔsh An e z 1 = e z 2, tìte e z 1 z 2 =1 z 1 z 2 =2πik, k Z. Ef ìson z 1,z 2 A a0, isqôei 0 Im(z 1 z 2 ) < 2π kai afoô Re(z 1 z 2 )=0ja èqoume z 1 = z 2, dhlad h e z eðnai 1-1. 'Estw t ra w C {0}. JewroÔme thn e z = w sto A a0. Tìte an z = x + iy kai w = u + iv èqoume e z = w e x+iy = u + iv e x (cos y + i sin y) =u + iv e x cos y = u e x sin y = v } e2x (cos 2 y +sin 2 y)=u 2 + v 2 cos y + i sin y = u+iv e x 39 } ex = u 2 + v 2 e iy = u+iv e x }

40 } e x = w e iy = w w x = log w (sun jhc log rijmoc): mia akrib c lôsh. y = argw sto [a 0,a 0 +2π] èqei akrib c mia lôsh Dhlad h e z 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 9. f 1 (z) =z 2. O z 2 èqei m koc Ðso me z 2 kai ìrisma 2argz. Dhlad h f 1 uy nei sto tetr gwno to mètro kai diplasi zei to ìrisma. 10. f 2 (z) = z. 'Estw ìti èqoume dialèxei ènan kl do, qrhsimopoi ntac to 0 θ<2π. Tìte z = re iθ z = re i θ 2 me 0 θ 2 <π, opìte h z brðsketai p nta sto nw hmiepðpedo kai oi gwnðec upodiplasi zontai. H f 2 (z) = z eðnai h antðstrofh thc f 1 (z) =z 2 ìtan h teleutaða perioristeð se perioq pou eðnai f(z) = sin z. ApeikonÐzei eujeðec par llhlec proc ton pragmatikì xona se elleðyeic kai eujeðec par llhlec proc ton fantastikì xona se uperbolèc. Pr gmati, sin z = sinx + iy =sinxcos (iy)+sin(iy)cosx = = sinx cosh y + i sinh y cos x, ìpou cosh y = ey +e y, sinh y = ey e y, afoô isqôei sin iy = eiiy e iiy = i ey e y = 2 2 2i 2 i sinh y kai antistoðqwc cos (iy) = coshy. 'Estw, loipìn, pr ta y = y 0 (eujeða par llhlh me ton xona twn pragmatik n). Tìte an sin z = u + iv, èqoume lìgw twn prohgoômenwn sqèsewn u =sinxcosh y 0 sin x = v =cosxsinh y 0 cos y = pou eðnai èlleiyh sto uv epðpedo. u cosh v y 0 sinh y 0 } u2 cosh 2 y 0 + v2 sinh 2 y 0 =1 OmoÐwc, an x = x 0 (eujeða par llhlh me ton xona twn fantastik n) brðskoume (qrhsimopoi ntac thn tautìthta cosh 2 y sinh 2 u y =1) ìti 2 sin 2 x 0 v2 cos 2 x 0 =1, pou eðnai uperbol. 40

41 Kef laio 4 Olokl rwsh To olokl rwma eðnai exairetik shmantikì sth melèth sunart sewn miac migadik c metablht c. H jewrða olokl rwshc diakrðnetai gia th majhmatik thc komyìthta. Ta jewr mata eðnai isqurìtata kai oi pio pollèc apodeðxeic eðnai aplèc. H jewrða olokl rwshc eðnai epilèon idiaðtera shmantik gia th meg lh qrhsimìtht thc sta efarmosmèna majhmatik. 4.1 Orismèno olokl rwma Prokeimènou na eis goume to olokl rwma thc f(z) me ènan sqetik aplì trìpo, orðzoume arqik to orismèno olokl rwma miac migadik c sun rthshc F miac pragmatik c metablht c t. 'Estw F (t) =U(t)+iV (t), t [a, b], ìpou oi sunart seic U kai V eðnai pragmatikèc kai kat 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 rwma thc 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 twn oloklhrwm twn touc. To genikeumèno olokl rwma thc F epð enìc mh fragmènou diast matoc orðzetai an loga kai up rqei ìtan sugklðnoun kai ta dôo genikeumèna oloklhr mata thc U kai thc V. H lgebra kai o logismìc twn orismènwn oloklhrwm twn isqôoun akrib c ìpwc kai gia tic pragmatikèc sunart seic tou t. Ja deðxoume mia basik idiìthta: b b F (t)dt = F (t) dt. a + b (h Ðdia idiìthta isqôei bèbaia kai ta genikeumèna oloklhr mata ) 'Estw F (t)dt = z a a C kai èstw z = re iθ. Tìte r = b a e iθ F (t) r = 41 a b a Re [ e iθ F (t) ] dt

42 [ ] b afoô Re G(t)dt a arijm n. 'Omwc opìte pou apodeiknôei to zhtoômeno. 4.2 KampÔlec = b Re[G(t)]dt kai afoô h pr th sqèsh eðnai sqèsh pragmatik n a Re [ e iθ F (t) ] e iθ F (t) = e iθ F (t) = F, r< b a f(t)dt Ja asqolhjoôme me kl seic kampul n pou mac qrei zontai sth melèth oloklhrwm twn sunart 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 seic thc 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(t 1 ) z(t 2 ) gia t 1 t 2 ). 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 epðshc diaforðsimh 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 je s. Kat tm mata leða kampôlh eðnai mia kampôlh pou apoteleðtai apì peperasmèno pl joc leðwn 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 eðnai kat tm mata suneqeðc. To m koc miac kat tm mata leðac kampôlhc eðnai to jroisma twn mhk n twn leðwn kampul n pou thn apoteloôn. Se k je apl kleist kampôlh apl kleist kat tm mata kampôlh C antistoiqoôn dôo sônola pou k je èna èqei wc sônoro mìno thn C. To èna apì aut pou lègetai eswterikì thc C eðnai fragmèno, en to llo (exwterikì) eðnai mh fragmèno. H apìdeixh den eðnai apl kai h prìtash aut lègetai je rhma kampôlhc Jordan. 42

43 4.3 Olokl rwma migadik n sunart sewn miac migadik c metablht c Orismìc '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 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 Oi dôo kampôlec C 1 : z(t), t [a, b], c 2 : w(t), t [c, d] eðnai omal isodônamec an up rqei mia 1-1, C 1 apeikìnish Ψ(t) :[c, d] [a, b] tètoia ste Ψ(c) =a, Ψ(d) =b kai w(t) =z(ψ(t)). Prìtash An oi C 1 kai C 2 eðnai omal isodônamec, tìte C 1 f = C 2 f. Orismìc 'Estw ìti h kampôlh C dðnetai apì thn z(t), t [a, b]. Tìte h C orðzetai apì thn z(b + a t), t [a, b]. Prìtash C f = C f Prìtash 'Estw C leða kampôlh, f kai g suneqeðc sthn C kai a C. Tìte C [f(z)+g(z)]dz = C f(z)dz + C g(z)dz C af(z)dz = a C f(z)dz Prìtash 'Estw C leða kampôlh m kouc L, f suneq c sthn C kai f M sthn C. Tìte f(z)dz ML. C Prìtash 'Estw {f n } akoloujða suneq n sunart sewn kai èstw ìti f n f omoiìmorfa epð thc leðac kampôlhc C. Tìte f(z)dz = lim f n (z)dz. C n + C H akìloujh genðkeush tou Jemeli douc jewr matoc tou ApeirostikoÔ LogismoÔ eðnai polô shmantik. Prìtash 'Estw F analutik epð thc leðac kampôlhc C kai f h par gwgoc thc F. Tìte f(z) =F (z(b)) F (z(a)) C 43

44 ParadeÐgmata 1. Na brejeð to I 1 = C 1 z 2 dz, ìpou C 1 to eujôgrammo tm ma OB apì to z =0sto z =2+i. LÔsh ParathroÔme ìti ta shmeða thc C 1 brðskontai p nw sthn eujeða x =2y. An, loipìn, h suntetagmènh y jewrhjeð wc par metroc, mia parametrik exðswsh thc C 1 eðnai z(y) = 2y + iy. EpÐ thc C 1, to z 2 gðnetai: z 2 = x 2 y 2 + i2xy =3y 2 + i4y 2. Opìte I 1 = 1 o (3y2 + i4y 2 )(2 + i)dy =(3+4i)(z + i) 1 0 y2 dy = i. 2. Na brejeð to I 2 = C 2 z 2 dz, ìpou C 2 : to eujôgrammo tm ma apì thn arq twn axìnwn (O) mèqri to A:(2,0) kai apì ekeð to eujôgrammo tm ma mèqri to B:(2,1), me jetikì prosanatolismì (blèpe sq ma 4.1). Sq ma 4.1: To trðgwno OAB. LÔsh I 2 = C 2 z 2 dz = OA z2 dz + AB z2 dz. Mia parametrik anapar stash tou OA eðnai en gia to AB mporoôme na gr youme z(x) =x, x [0, 2] z(y) =2+iy, y [0, 1]. Tìte I 2 = 2 0 x2 dx + [ 1 (2 + 0 iy)2 idy = 8 + i 1 (4 3 0 y2 )dy +4i ] 1 ydy = i. Mia parametrik anapar stash tou OAB eðnai h { t, t [0, 2] z(t) = 2+i(t 2), t [2, 3] 44

45 ParathroÔme ìti I 2 = I 1 kai sunep c C z2 dz =0, ìpou C = OABO, pr gma pou den eðnai tuqaðo all ofeðletai sto ìti h z 2 eðnai analutik sto eswterikì kai epð thc kampôlhc ìpwc ja doôme argìtera. 3. Na brejoôn ta I 3 = C 3 zdz kai I 4 = C 4 zdz, ìpou C 3 : to nw hmikôklio tou z =1 (apì z = 1 e c z =1) kai C 4 : to k tw hmikôklio. LÔsh Mia parametrik exðswsh tou C 3 eðnai h z(θ) =cosθ + i sin θ = e iθ, θ [0,π]. Sunep c I 3 = C 3 zdz = C 3 zdz = π o e iθ ie iθ dθ = πi. AntÐjeta mia parametrik exðswsh tou C 4 eðnai h z(θ) =e iθ, π θ 2, opìte I 4 = C 4 zdz = 2π π e iθ ie iθ dθ = πi. ParathroÔme ìti I 3 I 4 kai akìma ìti to olokl rwma I C = C zdz, ìpou C olìklhroc o kôkloc, den eðnai 0: I C = I 4 I 3 =2πi. Tèloc epð tou C isqôei ìti z =1opìte 1 z = z z 2 = z kai ètsi C dz z =2πi. 4. 'Estw C 5 to eujôgrammo tm ma apì to z = i sto z =1. QwrÐc na upologisteð to olokl rwma I 5 = dz C 5 na brejeð èna nw fr gma thc apìluthc tim c tou. z LÔsh 4 To C 5 brðsketai p nw sthn eujeða y =1 x. An z C 5, èqoume z 4 =(x 2 + y 2 ) 2 = (x 2 +(1 x) 2 ) 2 = (2x 2 2x +1) 2, opìte z 4 = ( 2 ( x 1 2 )2 + 2)) ( 4, afoô x 1 2 2) 0. Sunep c, gia k je z C5 : 1 z 4. Jètoume loipìn M =4sthn 4 prìtash Ex llou to m koc L tou C 5 eðnai profan c L = 2. Telik paðrnoume I To je rhma Cauchy-Goursat Diaisjhtik -Fusik apìdeixh JewroÔme èna di nusma w = u+iv pou antistoiqeð se tuqìn shmeðo (x, y) enìc disdi statou pedðou. JewroÔme mia kampôlh C se autì to pedðo. An jewr soume to w wc dônamh, mporoôme na jewr soume thn C wc diadrom kat m koc thc opoðac kineðtai èna ulikì shmeðo. H dieôjunsh miac tètoiac kðnhshc dðnetai apì to monadiaðo efaptìmeno di nusma sthn kampôlh e iτ ìpou τ h gwnða metaxô thc efaptìmenhc kai tou jetikoô hmi xona twn x. JewroÔme jetik dieôjunsh thn antðjeth me aut thc kðnhshc twn deikt n tou rologioô, gia kleistèc kampôlec. An jewr soume to w wc puknìthta reômatoc, eðnai fusikì na jewr soume thn C wc to sônoro diasqðzontac to opoðo mporeð na kinhjeð èna ulikì shmeðo. Gia mia tètoia kðnhsh to monadiaðo exwterikì k jeto di nusma e iν paðzei shmantikì rìlo. ν eðnai h gwnða metaxô 45

46 Sq ma 4.2: thc (exwterik c) kajètou kai tou jetikoô hmi xona twn x. Jetik dieôjunsh eðnai aut thc exwterik c kajètou gia kleistèc kampôlec. Sunep c e iτ = ie iν cos τ = sin ν, sin τ = cos ν. 'Estw w τ, w ν oi probolèc tou w epð thc efaptìmenhc kai thc kajètou sthn C, antistoðqwc. 'Estw ds to diaforikì tou tìxou thc C sto shmeðo z. To olikì èrgo dðnetai apì th sqèsh w C τds, an jewr soume to w wc dônamh pou epidr epð tou ulikoô shmeðou metafèrontac to apì to èna kro a thc C sto llo b. An jewr soume to w wc puknìthta reômatoc, to sunolikì posì Ôlhc pou pern ei k jeta apì thn C an qronik mon da, h ro (flux), dðnetai apì th sqèsh w C νds. Apì th gewmetrða tou probl matoc, èqoume afenìc w τ = Re( we iτ ) = u cos τ + v sin τ, w ν = Re( we iν ) = u cos v + v sin ν kai af etèrou dx dy =cosτ, =sinτ ds ds, opìte jètontac dx + idy = dz, telik èqoume w C τds + i w C νds = (u iv)(dx + idy) = C C wdz, alloi c wdz =èrgo+i ro. C Sq ma 4.3: An t ra ta a kai b sumpðptoun (C : apl kleist kampôlh) kai upojèsoume ìti h w = f(z) eðnai analutik, tìte (ìpwc èqoume dei) to pedðo tou w eðnai swlhnoeidèc kai astrìbilo, 46

47 sunèpwc mhdenðzontai kai to èrgo kai h ro. 'Eqoume loipìn telik ìti an h w = f(z) eðnai analutik se k je shmeðo tou tìpou R pou fr ssetai apì thn kleist kampôlh C kai epð thc C, tìte: f(z)dz = To je rhma Cauchy C '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 llontai apì mia apl kleist kampôlh C kai epð thc C. JewroÔme ìti h kampôlh èqei jetik dieôjunsh. Apì to je rhma Green èqoume ìti (Pdx+ Qdy) = (Q x P y )dxdy. C JewroÔme mia sun rthsh f(z) =u(x, y) +iv(x, y) pou eðnai analutik ston R. JewroÔme epiplèon ìti h f (z) eðnai suneq c ston R. Tìte oi u kai v, kaj c kai oi pr tec merikèc par gwgoð touc eðnai epðshc suneqeðc ston R. 'Etsi udx vdy = (v x + u y )dxdy C R vdx + udy = (u x v y )dxdy. C R Lìgw twn sunjhk n Cauchy-Riemann kai ta dôo dipl oloklhr mata mhdenðzontai, opìte vdx + udy = C (u R x v y )dxdy. 'Omwc, an f = u + iv, z = x + iy, b f(z)dz = f(z(t))z (t)dt = {u[x(t),y(t)] + iv[x(t),y(t)]}{x (t)+iy (t)}dt = C C a b b = (ux vy )dt + i (v x + uy ))dt = udx vdy + i vdx + udy, a a pr gma pou shmaðnei ìti f(z)dz =0. Autì to apotèlesma onom zetai je rhma Cauchy. C To je rhma Cauchy-Goursat O Goursat apèdeixe ìti h upìjesh thc sunèqeiac thc f (z) mporeð na paralhfjeð. H nèa morf tou prohgoumènou jewr matoc pou prokôptei ètsi eðnai Je rhma An h f eðnai analutik ston R kai epð thc C, tìte f(z)dz =0. C C C 47

48 4.5 Apl kai Pollapl Sunektik SÔnola Orismìc 'Enac tìpoc D lègetai apl sunektikìc an den perièqei trôpec, an dhlad eðnai p nta dunatì dôo kampôlec tou D me koin kra na metasqhmatistoôn suneq c h mða sthn llh, qwrðc kat ton metasqhmatismì h kampôlh na bgaðnei èxw apì ton D. 'Enac mh 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 (blèpe sq ma 4.4). O monadiaðoc dðskoc meðon ton jetikì pragmatikì xona eðnai apl sunektikìc (blèpe sq ma 4.5). H peirh lwrðda S = {z : 1 < Im z<1} eðnai apl sunektik (blèpe sq ma 4.6). K je anoiqtì kurtì sônolo eðnai apl sunektikì. To je rhma Cauchy-Goursat mporeð na diatupwjeð wc ex c: Je rhma An h f eðnai analutik se ènan apl sunektikì tìpo D, tìte gia k je apl kleist kampôlh C entìc tou D isqôei f(z)dz =0. C 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 'Estw C apl kleist kampôlh kai C j (j =1, 2,...,n) aplèc kleistèc kampôlec mèsa sthn C, ètsi ste ta eswterik sunola twn C j na mhn èqoun koin shmeða. 'Estw R to kleistì sônolo pou apoteleðtai apì ìla ta shmeða mèsa kai epð thc C, ektìc apì ta shmeða mèsa se k je C j. 'Estw B to prosanatolismèno sônoro tou R, pou apoteleðtai apì thn C kai ìlec tic C j (dieôjunsh tètoia ste ta shmeða tou R na brðskontai arister tou B). Tìte, an h f eðnai analutik sto R, f(z)dz =0 B Par deigma An B apoteleðtai apì ton kôklo z =2me jetik dieôjunsh kai ton kôklo z =1me arnhtik dieôjunsh tìte 1 B dz z 2 (z 2 +9) =0. H sun rthsh eðnai analutik, ektìc apì ta shmeða z =0, z = ±3i pou ìmwc den z 2 (z 2 +9) an koun sto daktôlio me sônoro B. 48

49 Sq ma 4.4: Sq ma 4.5: Sq ma 4.6: 49

50 4.6 To Aìristo Olokl rwma 'Estw z 0 kai z dôo aujaðreta shmeða enìc apl sunektikoô tìpou D, ìpou h f eðnai analutik. An C 1 kai C 2 eðnai dôo kampôlec pou en noun to z 0 me to z, keðmenec ex' olokl rou ston D, tìte h C 1 mazð me thn C 2 sunistoôn mia kleist kampôlh. Apì to je rhma Cauchy- Goursat, èqoume f(s)ds C 1 f(s)ds =0, C 2 ìpou s sumbolðzei shmeða epð twn C 1, C 2. To olokl rwma apì to z 0 sto z, eðnai sunep c anex rthto apì thn kampôlh, me thn proôpìjesh ìti h kampôlh brðsketai olìklhrh mèsa ston D. Autì to olokl rwma orðzei mia sun rthsh F ston apl sunektikì tìpo D, F (z) = z z 0 f(s)ds. DeÐqnoume ìti up rqei h par gwgoc thc F (z) kai isoôtai me f(z). 'Estw z + z tuqìn shmeðo, di foro tou z, pou brðsketai se mia perioq tou z pou an kei pl rwc ston D. Tìte F (z + z) F (z) = z+ z z 0 f(s)ds z z 0 f(s)ds = z+ z z f(s)ds ìpou o drìmoc olokl rwshc apì to z sto z + z mporeð na eklegeð wc eujôgrammo tm ma. Epeid f(z) = f(z) z+ z ds = 1 z+ z f(z)ds z z z z èpetai ìti F (z + z) F (z) z f(z) = 1 z z+ z z (f(s) f(z))ds. 'Omwc h f eðnai suneq c sto z kai ètsi gia k je ɛ>0 up rqei δ>0 tètoio ste f(s) f(z) <ɛìtan s z <δ. An, loipìn, to z + z eðnai tìso kont sto z ste z <δ, tìte F (z + z) F (z) f(z) z < 1 ɛ z = ɛ. z Dhlad F (z + z) F (z) lim z 0 z = f(z) F (z) =f(z). H sun rthshf (z) eðnai èna aìristo olokl rwma ( antipar gwgoc) thc f kai gr fetai F (z) = f(z)dz. Dhlad, h F (z) eðnai mia analutik sun rthsh thc opoðac h par gwgoc eðnai f(z). 50

51 Parat rhsh An G(z) eðnai mia llh analutik sun rthsh, tètoia ste G (z) = f(z), tìte ta dôo aìrista oloklhr mata F (z) kai G(z) diafèroun to polô kat mia migadik stajer. Pr gmati, èstw H(z) =F (z) G(z). Tìte H (z) =F (z) G (z) =f(z) f(z) =0. An, loipìn, H(z) =u(x, y) +iv(x, y), èpetai ìti u x (x, y) +iv x (x, y) =0kai sunep c kai h u x (x, y) kai h v x (x, y) eðnai mhdèn ston tìpo ìpou kai h F kai h G eðnai analutikèc. Apì tic sunj kec Cauchy-Riemann èpetai ìti kai h u y (x, y) kai h v y (x, y) mhdenðzontai. 'Etsi h u(x, y) kai h v(x, y) eðnai stajerèc, dhlad H(z) = c F (z) =G(z)+ c. Par deigma 1 Ac upologðsoume to 1 z1/2 dz, ìpou z 1/2 = r exp iθ, r>0, 0 <θ<2π kai h kampôlh 2 pou en nei ta dôo kra olokl rwshc keðtai p nw apì ton pragmatikì xona sto epðpedo z. H sun rthsh den eðnai analutik sta shmeða thc aktðnac θ =0, eidikìtera sto z =1. 'Omwc ènac lloc kl doc, f(z) = r exp iθ 2,r>0, π 2 <θ<3π 2 thc pleiìtimhc sun rthshc z 1/2 eðnai analutikìc pantoô ektìc apì ta shmeða thc aktðnac θ = π 2. Oi timèc thc f(z) p nw apì ton pragmatikì xona, sumpðptoun me ekeðnec tou pr tou kl dou r exp iθ, r>0, 0 <θ<2π kai ètsi h oloklhrwtèa sun rthsh mporeð na 2 antikatastajeð apì thn f(z). T ra èna aìristo olokl rwma thc f(z) eðnai h 2 3 z3/2 = 2 3 r3/2 exp i3θ 2,r>0, π 2 <θ<3π 2 ki ètsi 1 z 1/2 dz = 2 ( e 0 e 3iπ/2) = 2 (1 + i) To olokl rwma pou upologðsame, kat m koc miac kampôlhc k tw apì ton pragmatikì xona èqei llh tim. Ed antikajistoôme thn oloklhrwtèa sun rthsh me thn g(z) = r exp iθ, r > 0, π <θ< 5π H analutik sun rthsh 2 3 z3/2 = 2 3 r3/2 exp i3θ, r > 0, π < 2 2, eðnai èna aìristo olokl rwma thc g(z). 'Etsi θ< 5π 2 1 z 1/2 dz = 2 ( ) e i3π e i3π 2 = ( 1+i). 4.7 O oloklhrwtikìc tôpoc tou Cauchy Ja anafèroume t ra èna llo jemeli dec apotèlesma. Je rhma 'Estw ìti h f eðnai analutik pantoô epð kai entìc miac apl c kleist c kampôlhc C (me jetikì prosanatolismì). An z 0 eðnai tuqìn shmeðo sto eswterikì thc C, tìte f(z 0 )= 1 2πi C f(z) z z 0. 51

52 O parap nw lègetai oloklhrwtikìc tôpoc tou Cauchy kai lèei pwc an mia sun rthsh f eðnai analutik epð kai entìc miac apl c kleist c kampôlhc C, tìte oi kôriec timèc thc f entìc thc C prosdiorðzontai pl rwc apì tic timèc thc f epð thc C. Je rhma Me tic upojèseic tou prohgoômenou jewr matoc. Gia k je n N isqôei f (n) (z 0 )= n! f(z)dz 2πi (z z 0 ) n+1 Pìrisma An mia sun rthsh f eðnai analutik se èna shmeðo, tìte up rqoun oi par gwgoi k je t xhc thc f kai eðnai analutikèc sto Ðdio shmeðo. Pìrisma H sqèsh tou jewr matoc mporeð na epektajeð sthn perðptwsh pou h apl kleist kampôlh C antikatastajeð apì to to prosanatolismèno sônoro enìc pollapl c sunektikoô tìpou. Pìrisma (Ektim seic Cauchy). Me tic upojèseic tou jewr matoc An f(z) M(r) ston kôklo z z 0 = r kai a n := f (n) (z 0 ), tìte a n! n M(r). r n 4.8 To je rhma tou Green 'Estw P (x, y), Q(x, y) suneqeðc sunart seic me suneqeðc pr tec merikèc parag gouc se ènan pollapl sunektikì tìpo D kai èstw C to sônorì tou. Tìte C Pdx+ Qdy = D C ( Q x P y ) dxdy Parat rhsh Ikan kai anagkaða sunj kh gia na isqôei Pdx+ Qdy =0eÐnai h C P = Q y x gia C to sônoro enìc pollapl sunektikoô tìpou. Migadik morf tou jewr matoc Green. 'Estw h sun rthsh B(z, z) suneq c me suneqeðc pr tec merikèc parag gouc se ènan tìpo R me sônoro C. Tìte (ìpou z = x + yi, z = x yi). C B B(z, z)dz =2i R z dxdy 4.9 Ask seic 1. 'Estw C mia apl kleist kampôlh. ApodeÐxte ìti: { dz 0, z0 ektìc thc C = z z 0 2πi, z 0 entìc thc C C 52

53 LÔsh. An to z 0 eðnai ektìc thc C tìte h 1 z z 0 dz z z 0 eðnai analutik epð kai entìc thc C. 'Ara apì to je rhma Cauchy ja isquei =0. An to z C 0 eðnai entìc thc C tìte up rqei dðskoc kèntrou z 0 kai aktðnac ɛ (sunìrou G) ef ìson to z 0 eðnai eswterikì shmeðo. dz GnwrÐzoume ìti C z z 0 = dz Γ z z 0. EpÐ thc G èqoume z z 0 = ɛ z z 0 = ɛe iθ z = z 0 + ɛe iθ,θ [0, 2π]. 'Etsi dz = iɛe iθ dθ, opìte Γ dz 2π iɛe iθ dθ = z z 0 0 ɛe iθ 2π = i dθ =2πi An C eðnai to tm ma thc y = x 3 3x 2 +4x 1 pou sundèei ta shmeða (1,1) kai (2,3) na upologisjeð to olokl rwma (12z 2 4iz)dz. LÔsh. To olokl rwma eðnai anex rthto thc diadrom c. 'Ara: C (12z 2 4iz)dz = 2+3i 1+i C (12z 2 4iz)dz = [ 4z 3 2iz 2] 2+3i 1+i = i. 53

54 Kef laio 5 'Allec idiìthtec twn analutik n sunart sewn Je rhma (Je rhma Liouville). An mia akèraia sun rthsh eðnai fragmènh ( f(z) M, z C) tìte eðnai stajer. Je rhma (Jemeli dec je rhma thc 'Algebrac). K je polu numo (me pragmatikoôc suntelestèc) me bajmì megalôtero tou mhdenìc èqei rðza sto C. Parat rhsh K je polu numo bajmoô n èqei akrib c n rðzec (endeqomènwc Ðsec). Je rhma (Je rhma Mèshc Tim c). An h f eðnai analutik ston tìpo D kai z 0 D, tìte to f(z 0 ) isoôtai me th mèsh tim thc f epð tou sunìrou tuqìntoc dðskou me kèntro z 0 pou perièqetai sto D. Dhlad f(z 0 )= 1 2π 2π 0 f(z 0 + re iθ )dθ. Parat rhsh To Je rhma Mèshc Tim c apoteleð anadiatôpwsh tou oloklhrwtikoô tôpou tou Cauchy. Je rhma (Arq tou MegÐstou). Mia mh stajer analutik sun rthsh se ènan tìpo D, den èqei eswterik shmeða megðstou. Dhlad z D, kai δ>0, w S(z,δ) D : f(w) > f(z). Je rhma (Arq tou ElaqÐstou). An f eðnai mia mh stajer analutik sun rthsh se ènan tìpo D, tìte kanèna shmeðo z D den mporeð na eðnai sqetikì el qisto thc f, ektìc kai an f(z) =0. JumÐzoume ìti ston apeirostikì logismì, ta shmeða sqetikoô megðstou anazhtoôntai an - mesa sta krðsima shmeða (ekeðna ìpou f =0) miac diaforðsimhc sun rthshc f. H parak tw prìtash dðnei mia ousiastik antðjesh sth sumperifor miac analutik c sun rthshc se èna shmeðo pou paðrnei mègisto. 54

55 Prìtash (Erdos). 'Estw f analutik sun rthsh se ènan kleistì dðsko pou paðrnei to mègistì thc se èna sunoriakì shmeðo z 0. Tìte f (z 0 ) 0, ektìc kai an h f eðnai stajer. Je rhma (Anoiqt c Apeikìnishc). H eikìna enìc anoiqtoô kai sunektikoô sunìlou mèsw miac mh stajer c analutik c apeikìnishc eðnai anoiqtì sônolo. ProqwroÔme t ra sto antðstrofo tou jewr matoc Cauchy. Je rhma (Je rhma Morera). 'Estw f suneq c se ènan tìpo D, me thn idiìthta f(z)dz =0gia k je apl kleist kampôlh Γ pou me to eswterikì thc perièqetai ston Γ D. Tìte h f eðnai analutik ston D. Pìrisma 'Estw {f n } akoloujða analutik n sunart sewn se èna anoiqtì qwrðo D, tètoia ste f n f omoiìmorfa se k je sumpagèc uposônolo K tou D. Tìte h f eðnai analutik sto D. Je rhma (Arq an klashc tou Schwarz). An h f eðnai analutik sto T = {z C :Imz>0}, h f eðnai suneq c sto T kai an f(x) R, x R tìte h f epekteðnetai se akèraia sun rthsh. 55

56 Kef laio 6 Analutik Sunèqish 'Opwc gnwrðzoume k je analutik sun rthsh sto z = a mporeð na anaptuqjeð se dunamoseir f(z) = n=0 f (n) (a) (z a) n, z a <R n! kai k je tètoia sugklðnousa dunamoseir, anaparist mia analutik sun rthsh kont } sto z = a. P c mporoôme na upologðsoume tic timèc thc f(z) èxw apì ton dðsko z a <R, an h f eðnai analutik se eurôterh perioq? Par deigma f(z) = z n, z < 1 n=0 ( = 1 ) 1 z MporoÔme na upologðsoume to an ptugma Taylor gia thn f(z) perð èna nèo kèntro z = z 0, z 0 < 1. 'Estw z 0 = 1 2. ( f 1 ) 2 ( f 1 ) 2 = = ( 1 ) n = = ( (n +1) 1 n 1 = ( ) 2) 2 = n=0 n=0 ( f (n) 1 ) 2. ( 2 = n! 3 ) n+1 T ra paðrnoume (blèpe sq ma 6.1) ( f(z) = z + 1 ) n ( ) n+1 2, 2 3 n=0 56 z < 3 2

57 Sq ma 6.1: Ex' llou wc anamènoume èqoume: ( z + 1 ) n ( ) n+1 2 = 2 3 n= ( ) 2 z = 3 3 2z 1 = 1 1 z 2 'Eqoume epekteðnei analutik c thn f(z) = n=0 zn, z < 1 kai se shmeða ektìc tou arqikoô dðskou sôgklishc. Aut h diadikasða mporeð na epanalhfjeð kai telik h f(z) epekteðnetai analutik s' ìlo to C ektìc tou z =1. Den eðnai safèc ìti h f pou paðrnoume ètsi eðnai monìtimh kont sto z =1. JewroÔme mia diadrom me shmeða pou perikleðei to z =1, apofeôgont c} to, wc ex c: UpologÐzoume th dunamoseir perð to z 1 epð thc diadrom c aut c, suneqðzoume sto z 2 epð thc diadrom c kai entìc tou dðskou sto z 1, k.o.k., mèqri na epistrèyoume se mia perioq tou z 1 m' èna dðsko kèntrou z n epð thc diadrom c (blèpe sq ma 6.2). H sun rths mac ja eðnai monìtimh tìte kai mìnon tìte an oi timèc sunèqishc tou f(z) tautðzontai me tic dedomènec timèc thc f(z). EÐnai bolikì na jewr soume ìti ìlec oi pijanèc suneqðseic episun ptontai} ston orismì thc f(z). 'Etsi, p.q. h sun rthsh f(z) = n=0 ( 1) n zn+1 n +1, z < 1 anaparist gia z < 1 th monìtimh analutik sun rthsh log(1 + z) pou èqei thn tim 0 sto z =0, dhlad thn kôria tim tou logarðjmou. EntoÔtoic up rqoun peirec diaforetikèc 57

58 Sq ma 6.2: suneqðseic (perð to z = 1), pou k je mia diafèrei apì th dedomènh kat èna akèraio pollapl sio tou 2πi. JewroÔme thn olìthta twn sunarthsiak n tim n na anaparistoôn thn analutik sun rthsh. IsqÔei to akìloujo je rhma: Je rhma 'Estw f(z), g(z) analutikèc sunart seic, ste f(z) = g(z) se èna qwrðo D ìpou eðnai kai oi dôo analutikèc. Tìte isqôei f(z) g(z) pantoô ìpou orðzontai. Pìrisma 'Estw {z n } akoloujða ste lim n z n = z. 'Estw ìti f(z n )=g(z n ) gia k je n N kai ìti oi f(z), g(z) eðnai analutikèc sto z. Tìte f(z) g(z) pantoô. ParadeÐgmata 1. Epanerqìmaste sthn f 1 (z) = n=0 zn, z < 1. GnwrÐzoume ìti f 1 (z) = 1, z < 1 1 z kai h f 1(z) den orðzetai gia z 1. T ra h sun rthsh f 2 (z) = 1, z 1eÐnai analutik pantoô ektìc tou z =1. 1 z Ef' ìson f 2 (z) =f 2 (z), z {z : z < 1}, h f 2 (z) eðnai h mình dunat analutik sunèqish thc f 1 (z). 2. An èqoume tic plhroforðec ìti h dunamoseir n=0 zn sugklðnei kai anaparist mia analutik sun rthsh tou z ìtan z < 1 kai ìti to jroisma thc eðnai ìtan z = x, tìte mporoôme na sumper noume ìti to jroisma thc eðnai Autì èpetai apì to ìti h 1 1 z 1 1 x kai pou paðrnei thn tim {z : z < 1} ton x xona. 1 1 z 1 1 x gia z < 1. eðnai h analutik sun rthsh pou orðzetai sto {z : z < 1} kat m koc tou eujugr mmou tm matoc pou tèmnei o 58

59 3. Na deðxete ìti h f(z) = 0 t 3 e zt dt eðnai analutik sto {z :Rez>0}. Na breðte analutik sunèqish thc f(z) sto {z :Rez<0}. (i) M f(z) = t 3 e zt dt = lim t 3 e zt dt 0 M 0 M ( ) e = lim t 3 zt dt = paragontik olokl rwsh M 0 z M } = lim {t 3 e zt e zt 3t2 +6t e zt 6 e zt M M 0 z z 2 z 3 z 4 0 = lim M = 6 z 4 { 6 z M 3 e Mz 4 z 3M 2 e Mz z 2 6Me Mz z 3 6e Mz z 4 (ii) Gia Re z>0, t 3 e zt dt = 6, gia Re z. 0 z 4 H sun rthsh g(z) = 6 eðnai analutik sto C {0}. z 4 Epeid f(z) =g(z) gia Re z>0, h g(z) eðnai h zhtoômenh sun rthsh. 4. Ω tìpoc sto C, z 0 Ω, f:ω C analutik. Ta ex c eðnai isodônama: (i) H f epekteðnetai se akèraia sun rthsh f (ii) lim (n) 1 (z 0 ) n n n! =0 5. Ω tìpoc sto C, z 0 Ω, F:Ω C kai f :Ω C analutikèc. Tìte F = f F (n+1) (z 0 )=f (n) (z 0 ),n=1, 2, Na brejoôn ìlec oi akèraiec sunart seic me f = f 'Estw f : C C analutik. Tìte f(z) = n=0 a nz n, z C, ìpou a n = f (n) (0) n!. E n f = f, epagwgik èqoume n =0, 1,... } f (n+1) (0) = f (n) (0) (n +1)!a n+1 = n! a n a n+1 = a n n +1 a n = a n 1 n = 1 n(n 1) a n 2 = a 0 n! = f(0) n! 'Ara f(z) = a 0 n=0 n! zn = a z n 0 n=0 n! = a 0 e z = f(0)e z,z C. 59

60 Sunep c ìlec oi akèraiec sunart seic me f = f eðnai thc morf c f(z) =f(0)e z. Je rhma (Arq Analutik c Sunèqishc) 'Estw f suneq c s' èna anoiqtì sônolo D. 'Estw ìti eðnai analutik ekeð me endeqìmenh exaðresh sta shmeða enìc eujugr mmou tm matoc L. Tìte h f eðnai analutik s' olìklhro to D. 60

61 Kef laio 7 Memonwmènec AnwmalÐec Analutik n Sunart sewn Kat taxh Memonwmènwn Anwmali n Arq tou Riemann Je rhma Casorati Weierstrass En wc t ra eðqame sugkentr sei thn prosoq mac stic genikèc idiìthtec twn analutik n sunart sewn, ja asqolhjoôme sto ex c me thn eidik sumperifor mia analutik c sun rthshc sthn perioq miac memonwmènhc anwmalðac}. Orismìc H f èqei memonwmènh anwmalða sto z 0 an h f eðnai analutik se mia perioq D thc morf c {z :0< z z 0 <d} tou z 0, all den eðnai analutik sto z 0. Parat rhsh Lìgw tou Jewr matoc , parap nw, h f ja eðnai asuneq c se mia memonwmènh anwmalða. ParadeÐgmata 1. f(z) = { sin z, z 2 0, z =2. 'Eqei memonwmènh anwmalða sto z =2. 2. g(z) = 1. 'Eqei memonwmènh anwmalða sto z =3. z 3 3. h(z) =exp 1. 'Eqei memonwmènh anwmalða sto z =0. z Ta paradeðgmata aut antiproswpeôoun touc diaforetikoôc tôpouc memonwmènwn anwmali n, pou katat ssontai wc ex c: Orismìc 'Estw ìti h f èqei memonwmènh anwmalða sto z 0. 61

62 1. An up rqei analutik sto z 0 sun rthsh g, tètoia ste f(z) =g(z) gia k je z se mia perioq san thn D tou orismoô (7.1.1), lème ìti h f èqei epousi dh (airìmenh) anwmalða sto z 0, (dhlad an h tim thc f diorjwjeð} sto z 0, gðnetai analutik kai ekeð). 2. An, gia z z 0, h f mporeð na grafeð sth morf f(z) = A(z), ìpou oi A kai B eðnai B(z) analutikèc sto z 0, me A(z 0 ) 0kai B(z 0 )=0, lème ìti h f èqei pìlo sto z 0. An h B èqei to z 0 wc mhdenikì shmeðo pollaplìthtac k, lème ìti h f èqei pìlo t xhc k sto z An h f den èqei oôte epousi dh anwmalða, oôte pìlo sto z 0, lème ìti h f èqei ousi dh anwmalða sto z 0. 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. Je rhma (H Arq tou Riemann gia Epousi deic AnwmalÐec) An h f èqei memonwmènh anwmalða sto z 0 kai an tìte h anwmalða eðnai epousi dhc. lim (z z 0 )f(z) =0 z z 0 Pìrisma An h f eðnai fragmènh se mia perioq, ìpwc h D parap nw, miac memonwmènhc anwmalðac, h anwmalða eðnai epousi dhc. Je rhma An h f eðnai analutik se mia mia perioq, ìpwc h D parap nw, tou z 0 kai an up rqei jetikìc akèraioc k tètoioc ste tìte h f èqei pìlo t xhc k sto z 0. Parathr seic lim (z z 0 ) k f(z) 0 all lim (z z 0 ) k+1 f(z) =0 z z 0 z z0 1. Den up rqei analutik sun rthsh pou na teðnei sto, ìpwc mia klasmatik dônamh 1 thc z z 0 sthn perioq mia memonwmènhc anwmalðac tou z 0 P.q. An h f tan analutik se mia perioq tou 0 kai ikanopoioôse thn f(z) 1, epeid h anwmalða ja tan z 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 z 0 ) N f(z) den teðnei sto 0 ìtan z z 0. EntoÔtoic, de shmaðnei ìti f(z), ìtan z z 0. To epìmeno je rhma deðqnei ìti to sônolo tim n pou paðrnei mia sun rthsh sthn perioq miac ousi douc anwmalðac eðnai puknì} sto migadikì epðpedo, m' lla lìgia to pedðo tim n thc f tèmnei k je dðsko tou C. 62

63 Je rhma (Casorati Weierstrass) An h f èqei ousi dh anwmalða sto z 0 kai an h D eðnai mia perioq tou z 0 tìte to R = {f(z) :z D} eðnai puknì to C An ptugma Laurent GnwrÐzoume ìti oi analutikèc sunart seic s' èna dðsko, mporoôn na anaparastajoôn ekeð me dunamoseirèc. Mia an logh anapar stash, thc morf c k= a k (z z 0 ) k mporeð na gðnei gia analutikèc sunart seic sto daktôlio R 1 < z z 0 <R 2. Oi tètoiec anaparast seic eðnai gnwstèc wc anaptôgmata Laurent kai eðnai polô qr sima ergaleða sth melèth memonwmènwn anwmali n. Orismìc Lème ìti k= µ k = L an sugklðnoun kai h k=0 µ k kai h k=1 µ k kai to jroisma twn orðwn touc eðnai L. Je rhma H f(z) = k= a kz k eðnai sugklðnousa sto D = {z : R 1 < z kai z < R 2 }, ìpou R 2 = 1 lim sup k a k 1 k,r 1 = lim sup a k 1 k. k An R 1 <R 2, to D eðnai daktôlioc kai h f eðnai analutik sto D. Je rhma An h f eðnai analutik sto daktôlio A : R 1 < z <R 2, tìte h f èqei an ptugma Laurent, f(z) = k= a kz k ston A. Parat rhsh: To an ptugma Laurent eðnai monadikì. Pìrisma An h f eðnai analutik sto daktôlio R 1 monadikì an ptugma f(z) = k= kai C = C(z 0 ; R) me R 1 <R<R 2. a k (z z 0 ) k me a k = 1 2πi C < z z 0 < R 2, tìte èqei f(z) dz (z z 0 ) k+1 Pìrisma An h f èqei menomwmènh anwmalða sto z 0, tìte gia δ>0 kai 0 < z z 0 < δ, èqoume f(z) = k= ìpou C = C(z 0 ; R) me 0 <R<δ. a k (z z 0 ) k,a k = 1 2πi C f(z) dz (z z 0 ) k+1 63

64 Orismìc An f(z) = k= a k(z z 0 ) k eðnai to an ptugma Laurent thc f, wc proc mia memonwmènh anwmalða z 0, to f(z) = 1 k= a k(z z 0 ) k lègetai kôrio mèroc thc f sto z 0 en to f(z) = k=0 a k(z z 0 ) k lègetai analutikì mèroc. Apì th monadikìthta tou anaptôgmatoc Laurent, mporoôn na brejoôn oi qarakthrismoð twn kôriwn mer n wc proc ta di fora eðdh anwmali n. Prìtash (i) An h f èqei epousi dh anwmalða sto z 0, tìte a k =0, k >0. (ii) An h f èqei pìlo t xhc k sto z 0, tìte a k 0, & a N =0 k >0. (iii) An h f èqei ousi dh anwmalða sto z 0, to kôrio mèroc thc èqei apeðrou pl jouc mh mhdenikoôc ìrouc. Oloklhr noume thn par grafo me to akìloujo je rhma: Je rhma (An lush Rht n Sunart sewn se Apl Kl smata) K je gn sia rht sun rthsh R(z) = P (z) P (z) = Q(z) (z z 1 ) k 1(z z 2 ) k 2...(z z n) kn, ìpou P, Q polu numa me 1 degp < degq, mporeð na analujeð wc jroisma poluwnômwn wc proc z z k,k =1, 2,..., n, dhlad R(z) = n k=1 ( ) 1 P k. z z k 64

65 Kef laio 8 Oloklhrwtik Upìloipa 8.1 DeÐkthc Strof c kai to Je rhma Oloklhrwtik n UpoloÐpwn tou Cauchy Stìqoc mac eðnai h genðkeush tou Jewr matoc Cauchy Goursat gia sunart seic pou èqoun memonwmènec anwmalðec. ParathroÔme ìti an γ eðnai ènac kôkloc pou perikleðei mia memonwmènh anwmalða z 0 kai an f(z) = k= c k(z z 0 ) k se mia perioq D = {z :0< z z 0 <d} tou z 0 pou perièqei th γ tìte f =2πic 1 'Etsi, blèpoume pwc o sunelest c c 1 èqei idiaðterh shmasða. γ Orismìc An f(z) = k= c k(z z 0 ) k, se mia perioq ìpwc h D tou z 0, to c 1 lègetai oloklhrwtikì upìloipo thc f sto z 0. QrhsimopoioÔme to sumbolismì c 1 = Res(f; z 0 ). Prìtash (Upologismìc Oloklhrwtik n UpoloÐpwn) 1. An h f èqei aplì pìlo sto z 0, tìte 2. An h f èqei pìlo t xhc k sto z 0, tìte c 1 = Res(f; z 0 ) = lim z z0 (z z 0 )f(z) = A(z 0) B (z 0 ) c 1 = Res(f; z 0 )= 1 (k 1)! d k 1 [ ] (z z dz k 1 0 ) k f(z). z=z0 Parat rhsh: Stic perissìterec peript seic pìlwn megalôterhc t xhc apì 1, kaj c kai gia tic ousi deic anwmalðec, o bolikìteroc trìpoc prosdiorismoô tou oloklhrwtikoô upoloðpou eðnai kateujeðan apì to an ptugma Laurent. 65

66 ParadeÐgmata: 1. Res(cosec z; 0)= 1 cos 0 =1 2. Res( 1 z 4 1 ; i) = 1 4i 3 = i 4 3. Res( 1 z 3 ;0)=0 4. Res(sin 1 1 ;1)=1, afoô sin = z 1 z 1 z 1 3!(z 1) 3 5!(z 1) 5 DeÐkthc Strof c Orismìc 'Estw γ kleist kampôlh kai ζ/ γ. Tìte Ind γ (ζ) = 1 dz 2πi z ζ Parathr seic: 1. An o γ eðnai ènac kôkloc (jetik prosanatolismènoc), tìte γ Ind γ (ζ) = { 1, ζ sto eswterikì tou dðskou 0, ζ sto exwterikì tou dðskou. 2. An o γ peristrèfetai perð to ζkforèc, dhlad an γ(θ) =ζ + re iθ, 0 θ 2kπ tìte Ind γ (ζ) = 1 2kπ idθ = k, gegonìc pou dikaiologeð ton ìro deðkthc strof c}. 2πi 0 Je rhma Gia k je kleist kampôlh γ kai ζ/ γ, o Ind γ (ζ) eðnai akèraioc. Parathr seic: 1. Apì ton Orismì (8.1.2) èpetai ìti an stajeropoi soume th γ kai af soume to ζ na metab lletai, o Ind γ (ζ) eðnai suneq c sun rthsh tou ζ (efìson ζ/ γ). AfoÔ to Ind γ (ζ) eðnai p ntote akèraioc, sumperaðnoume ìti eðnai stajerìc stic sunektikèc sunist sec tou sumplhr matoc thc γ. Epiplèon Ind γ (ζ) 0 ìtan ζ. Sunep c sth mh fragmènh sunist sa tou sumplhr matoc thc γ (dhlad sto sônolo twn shmeðwn pou mporoôn na enwjoôn me to qwrðc na tmhjeð h γ) Ind γ (ζ) =0. 2. Genik c, ìtan asqoloômaste me kleistèc kampôlec, ja eðmaste se jèsh na diapist noume ap' eujeðac ìti Ind γ (ζ) =0 ± 1, ζ / γ. Orismìc H γ lègetai kanonik kleist kampôlh, an h γ eðnai apl kleist kampôlh me Ind γ (ζ) =0 1 ζ / γ. S' aut thn perðptwsh onom zoume eswterikì thc γ to sônolo {ζ : Ind γ (ζ) =1}. Exwterikì thc γ, lègetai to sônolo {ζ : Ind γ (ζ) =0}. 66

67 Je rhma (Oloklhrwtik n UpoloÐpwn tou Cauchy) 'Estw f analutik, s' ènan apl sunektikì tìpo D, ektìc apì tic memonwmènec anwmalðec sta shmeða z 1,..., z m. 'Estw γ kleist kampôlh pou den perièqei kamia apì tic anwmalðec. Tìte γ f =2πi m Ind γ (ζ k )Res(f; z k ) k=1 Pìrisma An h f eðnai ìpwc sto Je rhma kai h γ eðnai kanonik kleist kampôlh sto pedðo analutikìthtac thc f, tìte f =2πi Res(f; z k ) γ k ìpou to jroisma lamb netai epð ìlwn twn anwmali n thc f mèsa sth γ Efarmogèc tou Jewr matoc Oloklhrwtik n UpoloÐpwn Orismìc Lème ìti h f eðnai merìmorfh s' èna pedðo D an eðnai analutik ekeð, ektìc apì èna pl joc shmeðwn pou èqei pìlouc. Je rhma 'Estw γ kanonik kleist kampôlh. An f eðnai merìmorfh entìc kai epð thc γ kai den èqei mhdenik shmeða oôte pìlouc epð thc γ kai an jèsoume P = pl joc riz n thc f entìc thc γ (mia rðza t xhc k metriètai k forèc) kai Π=pl joc pìlwn thc f entìc thc γ (epðshc lamb netai up' ìyin h pollaplìthta), tìte P Π= 1 2πi Pìrisma (Arq tou OrÐsmatoc) An f eðnai analutik entìc kai epð miac kanonik c kleist c kampôlhc γ (kai de mhdenðzetai epð thc γ), tìte γ f f (pl joc riz n thc f entìc thc γ :=) P = 1 2πi γ f f Parat rhsh: To parap nw lègetai Arq tou OrÐsmatoc}, giatð an h γ dðnetai apì th z(t), t [0, 1], tìte 1 f =logf(z(1)) log f(z(0)) = 1 2πi f 2π Argf(z) γ ParathroÔme ìti mporoôme na jewr soume to wc to deðkth strof c thc kampôlhc γ f(γ(z)) gôrw apì to z =0. Aut h optik gwnða, mac odhgeð se mia apl apìdeixh tou akìloujou shmantikoô jewr matoc: 67 f f

68 Je rhma (Rouché) 'Estw f kai g analutikèc entìc kai epð miac kanonik c kleist c kampôlhc γ kai èstw akìma ìti f(z) > g(z) z γ. Tìte P (f + g) =P (f) entìc thc γ. ParadeÐgmata 1. AfoÔ 4z 2 > 2z epð tou z =1, kaj' èna apì ta polu numa èqei akrib c dôo rðzec ston z < 1. 2z 10 +4z 2 +1 kai 2z 10 4z O arijmìc twn riz n tou poluwnômou z 8 4z 5 + z 2 1 sthn anoiqt monadiaða sfaðra S(0; 1) eðnai 5. Pr gmati, jètontac f(z) = 1 4 z z2 1 4, èqoume epð tou z =1ìti f(z) < 1= z5. Je rhma (Hurwitz) 'Estw {f n } akoloujða mh mhdenik n analutik n sunart sewn s' ènan tìpo D, kai ìti ìti h f n f omoiìmorfa sta sumpag tou D. Tìte eðte f 0 sto D, f(z) 0 z D. Parat rhsh: EÐnai dunatìn na èqoume f 0 parìlo pou n : f n (z) 0. P.q. f n (z) = 1 n ez. Je rhma 'Estw f n f omoiìmorfa sta sumpag tou D. An h f n eðnai 1 1 ston D gia k je n, tìte eðte h f eðnai stajer, h f eðnai 1 1 ston D. 8.2 Efarmogèc tou jewr matoc Oloklhrwtik n UpoloÐpwn tou Cauchy ston upologismì oloklhrwm twn kai seir n Upologismìc Oloklhrwm twn (A) P (x) dx,p,q polu numa Q(x) EÐnai gnwstì pwc èna tètoio olokl rwma sugklðnei an Q(x) 0kai degq degp 2. K tw ap' autèc tic proôpojèseic P (x) dx = lim Q(x) R R R P (x) Q(x) dx kai jèloume na upologðsoume to deôtero olokl rwma gia meg la R. JewroÔme thn kleist kampôlh C R tou sq matoc, me aktðna R tìso meg lh ste na perikleðei ìlec tic rðzec tou Q pou brðskontai sto nw hmiepðpedo. 68

69 Sq ma 8.1: P (z) 'Eqoume ìti: dz =2πi ( ) C R Q(z) k Res P ; z Q k, z k rðzec tou Q sto nw hmiepðpedo, R P (z) opìte dz + P (z) dz =2πi ( ) R Q(z) Γ R Q(z) k Res P ; z Q k AfoÔ degq degp 2, èqoume P Γ R Q πr A R lim P (z) dz =0. 2 R Γ R Q(z) Telik P (x) Q(x) dx =2πi k ( ) P Res Q ; z k Par deigma dx 1+x 4 Oi pìloi thc Sunep c dz sto nw hmiepðpedo eðnai oi z 1+z 4 1 = e iπ 4 kai z 2 = e 3iπ 4. ( k=1 Res 1 dx =2πi 2 1+x 4 1+z 4 ; z k ). Epeid kai oi dôo pìloi eðnai aploð, ta oloklhrwtik upìloipa dðnontai apì tic timèc thc 1 stouc pìlouc: 4z ( 3 ) 1 Res ; z 1+z 4 1 = 1 = z 4z = 2(1 + i) 8 ( ) 1 Res ; z 1+z 4 1 = 1 = 2(1 i) 4z

70 dx 'Etsi: = 2 π 1+x 4 2 (B) R(x)cosxdx, R(x)sinxdx, R = P,P,Q polu numa. Q An Q(x) 0kai an degq > degp ta oloklhr mata sugklðnoun. Ed den mporoôme na oloklhr soume thn R(z)cosz kat m koc thc Ðdiac kleist c kampôlhc me thn perðptwsh (A), afoô lim M Γ M R(z)coszdz 0 ìpou Γ M to hmikôklio tou sq matoc 8.2. Sq ma 8.2: 'Estw C M to hmikôklio Γ M mazð me to eujôgrammo tm ma [ M,M]. An jewr soume to C M R(z)e iz dz ja deðxoume ìti Γ M R(z)e iz dz 0, opìte C M R(z)e iz dz R(x)eix dx. Ja èqoume sunep c telik ìti { R(x)cosxdx = Re 2πi k { R(x)sinxdx = Im 2πi k } Res(R(z)e iz ; z k ) } Res(R(z)e iz ; z k ), ìpou z k oi pìloi thc R(z) sto nw hmiepðpedo. Gia na deðxoume ìti Γ M R(z)e iz dz 0, sp me to Γ M se dôo uposônola: 70

71 (G) 0 { } { } A = z Γ M : Imz h,b= z Γ M : Imz < h. 'Omwc R(z)e iz dz K A M e h πm = c 1 e h kai R(z)e iz dz K B M 4h = c h 2 M, opìte Γ M R(z)e iz dz c 1 e h h + c 2 An eklèxoume to h = M, paðrnoume ParadeÐgmata (1) M. R(z)e iz dz c 1e M + c 2 lim Γ M M sin x dx x { sin x dx = Im e ix x poi soume thn teqnik mac: 'Eqoume ìti M Γ M R(z)e iz dz =0. } dx. H eix èqei pìlo sto 0 ki ètsi prèpei na tropo- x x { sin x } dx = Im e ix 1dx. x x e ParathroÔme ìti iz 1 c M dz = M e ix 1dx + e iz 1 z M x Γ M dz z e 'Omwc iz 1 c M dz =0afoÔ h eiz 1 den èqei pìlouc! 'Etsi, z z M M (2) e ix 1 x dx = Γ M e iz 1 z dz = Γ M dz z Γ M e iz z dz = πi Γ M 'Omwc Γ M cos x dx (x 2 +1) 2 e iz z dz 0, ìtan M cos x dx = Re (x 2 +1) 2 { } e ix dx. (x 2 +1) 2 e iz dz z e ix 1dx = πi x sin x dx = π x H eiz èqei pìlouc t xhc 2 sta shmeða z = ±1. Ap' aut mìno to z =1 (z 2 +1) 2 brðsketai sto nw hmiepðpedo. JewroÔme M>1. ( ) ( ) ( ) e Res iz ; i = d (z i) 2 eiz z=i = d e iz z=i = i (z 2 +1) 2 dz (z 2 +1) 2 dz (z 2 +1) 2 2e. { ( )} cos x e Sunep c dx = Re 2πiRes iz ; i = π (x 2 +1) 2 (z 2 +1) 2 e P (x) Q(x) dx, P, Q polu numa me Q(x) 0gia x 0 kai degq degp 2. (An bèbaia h P Q eðnai rtia, tìte 0 P Q = 1 2 Jètoume R(z) = P (z) kai jewroôme to olok rwma thc log z R(z) sthn kleist kampôlh Q(z) K ε, M pou apoteleðtai apì 71 P Q ).

72 Sq ma 8.3: (a) to orizìntio eujôgrammo tm ma I 1 apì to iε wc to iε + M 2 ε 2 (b) to tìxo C M kôklou aktðnac M pou diatrèqetai me jetikì prosanatolismì apì to M 2 ε 2 + iε wc to M 2 ε 2 iε. (g) to orizìntio eujôgramma tm ma I 2 apì to M 2 ε 2 iε wc to iε. (d) to hmikôklio C ε aktðnac ε, kat arnhtikì prosanatolismì apì iε wc iε To eswterikì thc K ε, M eðnai apl sunektikìc tìpoc pou den perièqei to 0 kai sunep c o log z orðzetai ekeð wc analutik sun rthsh (gia lìgouc aplìthtac jewroôme 0 < Argz < 2π). Apì to Je rhma Oloklhrwtik n UpoloÐpwn èqoume: lim R(z)logzdz =2πi ε 0, M K ε, M k Res(R(z)logz; z k ) (8.1) ìpou to ε èqei epilegeð tìso mikrì kai to M tìso meg lo, ste oi rðzec tou Q na brðskontai ìlec sto eswterikì thc K ε, M. Tìte (a) C ε R(z)logzdz πε max C ε log z R(z) Aε log ε afoô h R eðnai suneq c sto 0 kai log z < log z +2π. Sunep c lim ε 0 C ε R(z)logzdz =0 72

73 (b) C M R(z)logzdz 2πM max C M log z R(z) AM log M M 2 afoô R(z) B z 2 kai ètsi lim M C M R(z)logzdz =0 (g) lim ε 0, M (d) lim ε 0, M I 1 R(z)logzdz = 0 R(x)logxdx Apì ta (a), (b), (g) kai (d) paðrnoume kai lìgw thc (8.1): I 2 R(z)logzdz = 0 R(x)(log x + 2πi)dx lim ε 0, M I 2 R(z)logzdz = 2πi 0 R(x)dx 0 R(x)dx = k Res(R(z)logz; z k ) ìpou to k diatrèqei to pl joc twn pìlwn thc R. Par deigma 0 dx 1+x 3 Oi (aploð) pìloi thc R(z) = 'Eqoume ( ) log z Res ; z 1+z 3 1 ( ) log z Res ; z 1+z 3 2 = iπ 3 ( ) log z Res ; z 1+z 3 3 opìte kai ètsi Parat rhsh 1 = log z 3z 2 z=z1 = iπ 9 = 5iπ 9 Oloklhr mata thc morf c ( 1 i ) k dz 1+z 3 eðnai z 1 = e iπ 3,z 2 = 1 =e iπ,z 3 = e i 5π 3 ( 1 + i ) ( log z ) Res 1+z ; z 3 k = π 0 dx 1+x = π a P (x) Q(x) dx mporoôn, omoðwc, na upologistoôn me to na jewrhjeð to C M log(z 1) P (z) Q(z) dz 73

74 Sq ma 8.4: Parat rhsh 2 An loga gðnetai kai o upologismìc oloklhrwm twn thc morf c me 0 <a<1 kai P polu numo me degp 1. 0 x a 1 P (x) dx Sto eswterikì thc kampôlhc K ε, M èqoume z a 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 ètsi arkeð na upologistoôn ta oloklhr mata epð twn I 1,I 2. EpÐ tou I 1 : z a 1 = e (a 1) log x = x a 1, en epð tou I 1 : z a 1 = e (a 1)(log x+2πi) = x a 1 e 2πi(a 1) 'Etsi {1 e 2πi(a 1) } x a 1 dx =2πi ( ) 0 P (x) k Res z a 1 ; z P (z) k Par deigma 0 dx (1+x) x Pìloc thc P (z) =1+z to 1. ( ) 1 Res (1+z) ; 1 = 1 1 z= 1 z z = 1 = i (1+z) i Opìte 74

75 dhlad (1 e πi dx ) 0 (1 + x) x =2π, 0 dx (1 + x) x = π. (D) Oloklhr mata thc morf c 2π 0 R(cos θ, sin θ)dθ, ìpou R rht sun rthsh. S' aut thn perðptwsh, jewroôme to pragmatikì olokl rwma wc parametrik anapar stash enìc epikampôliou oloklhr matoc epð tou monadiaðou kôklou. Ac jumhjoôme, ìti, jètontac z = e iθ, 0 θ 2π, èqoume Sunep c 2π 0 z =1 f(z)dz = 2π dθ = dz iz, cos θ = eiθ+e iθ sin θ = eiθ e iθ R(cos θ, sin θ)dθ = 0 f(e iθ )ie iθ dθ 2 = 1 2 2i z =1 = 1 2i [ z + 1 z R, 2 ( ) z + 1 z ( ) z 1 z z 1 z 2i To deôtero olokl rwma, mporeð ìpwc gnwrðzoume na upologisteð me th bo jeia tou Jewr matoc Oloklhrwtik n UpoloÐpwn. ] dz iz Par deigma 1 2π 0 dθ 2+cosθ = 2 dz i z =1 z 2 +4z +1 = ( 1 = 4πRes z 2 +4z +1 ; ) 3 2 = 1 = 4π z 2 = +4z +1 z= 3 2 = π. Par deigma 2 2π 0 dθ 5+3sinθ 75

76 z = e iθ sin θ = eiθ e iθ 2i dz =3e iθ dθ = izdθ 2π 0 dθ 5+3 sin θ = C dz iz 5+3(z 1 z ) 1 2i = z 1 z 2i = C 2dz 3z 2 +10iz 3 ìpou C = C(0; 1), h perifèreia tou kôklou me kèntro 0 kai aktðna 1. H 2 3z 2 +10iz 3 èqei pìlouc tic rðzec thc 3z2 +10iz 3 dhlad ta 1 i, 3i. 3 Mìno to 1 i brðsketai entìc thc C. 3 To upìloipo sto z 1 = 1 3 i eðnai: 2 lim (z z 1 ) z z 1 3z 2 +10iz 3 = = 1 4i. Sunep c C 2dz 3z 2 +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 pìlouc thc upì olokl rwsh sun rthshc. Par deigma I e z dz,i: z(t) =1+it, <t< (z +2) 3 'Estw C R to aristerì hmikôklio kèntrou z =1kai aktðnac R>3. Tìte 1+iR 1 ir e z dz (z +2) + 3 C R Sto hmiepðpedo x 1, e z e kai ètsi ìtan R e z dz (z +2) 0, 3 opìte I ( ) e z dz e z (z +2) =2πiRes dz 3 (z +2) ; 2. 3 C R ( ) e z dz e z (z +2) =2πiRes dz 3 (z +2) ; 2. 3 Gia ton upologismì tou oloklhrwtikoô upoloðpou parathroôme ìti ( ) e z = e 2 e z+2 = e 2 (z +2)2 1+(z +2)+ +, 2 76

77 Sq ma 8.5: opìte kai ètsi ( ) e z dz Res (z +2) ; 2 = c 3 1 = 1 2e 2 I e z dz (z +2) = πi 3 e Upologismìc OrÐwn Seir n (A) n= f(n) AnazhtoÔme mia sun rthsh g pou na èqei wc oloklhrwtik upìloipa ta {f(n) : n = 0, ±1, ±2,...}. 'Estw g(z) =f(z)φ(z). H φ ja prèpei na èqei ènan aplì pìlo me oloklhrwtikì upìloipo 1 se k je akèraio. Mia tètoia sun rthsh eðnai h φ(z) =π cot πz afoô h sin πz èqei aplì pìlo se k je akèraio kai Res(π cot πz; n) = Efarmìzoume to Je rhma Oloklhrwtik n UpoloÐpwn sto olokl rwma C N f(z)π cot πzdz 77 π cot πz sin πz =1. z=n

78 ìpou C N mia apl kleist kampôlh pou perièqei touc akèraiouc n =0, ±1, ±2,..., ±N kai touc pìlouc (èstw z k thc f (pou upojètoume ìti eðnai peperasmènou pl jouc.) Sq ma 8.6: Tìte { f(z)π cot πzdz ( ) N =2πi f(n)+ C N n= N n z k k } Res(f(z)π cot πz; z k ). Upojètontac ìti lim zf(z) =0 z ( ) ( ) kai eklègontac wc C N to tetr gwno me korufèc ± N + 1 ± N + 1 i eðnai eôkolo 2 2 na deðxoume ìti lim N c N f(z)π cot πzdz =0. Sunep c (*): Pr gmati: N f(n) = n= N k n z k Res(f(z)π cot πz; z k ) z n Res(f(z)π cot πz; n) = lim(z n)π cot πzf(z) = lim cos πzf(z) =f(n) z n z n sin πz (de l Hospital). 78

79 Par deigma ParathroÔme ìti SÔmfwna me ta prohgoômena n=1 n=1 n=1 1 n 2 = n 2 n= n 0 1 n 2. ( ) 1 π cot πz n = Res ;0. z 2 To an ptugma Laurent thc cot z eðnai cot z = 1 z z 3 z π cot πz = 1 π2 π4 z z 2 z 3 3z 2 opìte ( Sunep c Res π cot πz z 2 ;0 ) 45. = π2 3 kai ètsi n=1 1 n 2 = π2 6. (B) n= ( 1) n f(n) DouleÔoume sto Ðdio tetr gwno ìpwc prohgoumènwc, me th bohjhtik sun rthsh π sin πz f(z). ParathroÔme ìti ( ) π Res sin πz ;0 = 1 cos πn =( 1)n. 'Etsi èqoume ( 1) n= n z k n f(n) = k ìpou z k oi, peperasmènou pl jouc, pìloi thc f. Par deigma n= Res( πf(z) sin πz ; z k) ( 1) n n + a 2,a R Z 79

80 'Estw f(z) = 1 n+a 2. 'Eqei diplì pìlo sto z = a. Tìte ( ) π Res sin πz(z + a) ; a 2 = lim z a = π2 cot πa sin πa { d (z + a) 2 dz = π2 cos πa sin 2 πa. } π sin πz(z + a) 2 'Etsi (G) Parat rhsh AfoÔ to èqoume ìti ( n k n= ) ( n k ( 1) n n + a = π2 cos πa 2 sin 2 πa. eðnai o suntelest c tou z k sto an ptugma thc (1+z) n, ) = 1 2πi c (1 + z) n dz ìpou C tuqoôsa apl kleist kampôlh pou perib llei to mhdèn. Autì to gegonìc epitrèpei ton upologismì twn orðwn seir n me diwnumikoôc suntelestèc. z k+1 80

81 Kef laio 9 SÔmmorfh Apeikìnish S' autì to kef laio ìlec oi kampôlec z(t) jewroôntai ìti eðnai tètoiec, ste z (t) 0gia ìla ta t. Orismìc 'Estw ìti dôo leðec kampôlec C 1 kai C 2 tèmnontai sto z 0 H gwnða apì th C 1 sth C 2 sto z 0 orðzetai wc h gwnða apì thn efaptomènh thc C 1 sto z 0 wc thn efaptomènh thc C 2 sto z 0 kat thn antðjeth apì thc kðnhshc twn deikt n tou rologioô for. Orismìc 'Estw f mia C 1 apeikìnish pou orðzetai se mia perioq tou z 0. H f lègetai sômmorfh sto z 0 an diathreð gwnðec s' autì to shmeðo tìso kat mètro ìso kai kat prosanatolismì. Me lla lìgia an gia k je zeug ri leðwn kampôlwn C 1 kai C 2 pou tèmnontai sto z 0 : (C 1,C 2 )= (Γ 1, Γ 2 ) ìpou Γ 1 = f(c 1 ), Γ 2 = f(c 2 ). An h f eðnai sômmorfh s' ìla ta shmeða enìc sunìlou D, h f lègetai sômmorfh sto D. Orismìc (a) H f eðnai topik 1 1 sto z 0, an gia δ>0 kai z 1 z 2,z 1,z 2 D(z 0 ; δ), èqoume ìti f(z 1 ) f(z 2 ). (g) H f eðnai topik 1 1 sto D an eðnai topik 1 1 se k je shmeðo tou D. (b) H f eðnai 1 1 sto D an gia k je diaforetik z 1,z 2 sto D èqoume f(z 1 ) f(z 2 ). Je rhma 'Estw f analutik sto z 0 kai f (z) 0. Tìte h f eðnai sômmorfh kai topik 1 1 sto z 0. ParadeÐgmata 1. H f(z) =e z eðnai pantoô sômmorfh kai topik 1-1 (den eðnai olik 1-1 afoô f(z+2πi) = f(z)). MporeÐ na apodeiqjeð ìti h e z apeikonðzei tic katakìrufec grammèc x=staj. epð kôklwn me kèntro to O kai tic orizìntiec grammèc y =staj. epð aktðnwn (pou dièrqontai apì to O) (Sq ma 9.1). 81

82 Sq ma 9.1: 2. H f(z) =z 2 eðnai sômmorfh ektìc apì to z =0(Sq ma 9.2). Orismìc 'Estw k N. H f lègetai k proc 1 apeikìnish tou D 1 epð tou D 2, an gia k je a D 2, h exðswsh f(z) =a èqei k rðzec (lambanomènhc up' ìyin thc pollaplìthtac) sto D 1. L mma 'Estw f(z) =z k,k N. Tìte h f megejônei tic gwnðec sto 0 me par gontai k kai apeikonðzei to dðsko D(0; δ), δ>0 epð tou dðskou d(0; δ k ), me k proc 1 trìpo. To Je rhma (9.0.1) sumplhr netai wc ex c: Je rhma 'Estw f analutik sto z 0 kai f (z) =0. Tìte eðte h f eðnai stajer, se k poio ikanopoihtik mikrì anoiqtì sônolo pou perièqei to z 0, h f eðnai k proc 1 apeikìnish kai megejônei tic gwnðec sto z 0 me par gonta k, ìpou k o el qistoc fusikìc gia ton opoðon isqôei ìti f (k) (z 0 ) 0. MporoÔme ètsi na p roume to akìloujo Je rhma 'Estw f 1 1 analutik sun rthsh s' ènan tìpo D. Tìte (a) h f 1 up rqei kai eðnai analutik ston f(d). (b) h f kai h f 1 eðnai sômmorfec sto D kai to f(d), antistoðqwc. To Je rhma autì odhgeð ston epìmeno Orismìc DÔo tìpoi D 1, D 2 lègontai sômmorfa isodônamoi, an up rqei sômmorfh apeikìnish tou D 1 epð tou D 2. 82

83 Sq ma 9.2: 9.1 Je rhma SÔmmorfhc Apeikìnishc tou Riemann Gia k je apl sunektikì tìpo R ( C) kai z 0 R, up rqei monadik sômmorfh apeikìnish φ epð tou anoiqtoô monadiaðou dðskou U, tètoia ste φ(z 0 )=0kai φ (z 0 ) > 0. (Me lla lìgia, k je dôo apl sunektikoð gn sioi upì tìpoi tou migadikoô epipèdou eðnai sômmorfa isodônamoi). Merikèc AxioshmeÐwtec ApeikonÐseic 1. w = z + b (metatìpish) 2. w = e iθ 0 z (strof ) 3. w = az (omojesða) 4. w = 1 z (antistrof ) 5. w = az + b (grammikìc metasqhmatismìc) w = ζ + b, ζ = e iθ 0 ξ, ξ = Az (a = Ae iθ 0 ): omojesðac. sônjesh metatìpishc, strof c kai O grammikìc metasqhmatismìc eðnai 1 1 analutik apeikìnish tou C epð tou C. 83

84 6. w = z a,a R An jewr soume ton kl do tou log z pou eðnai jetikìc ston jetikì hmi xona, tìte h z a apeikonðzei ton jetikì hmi xona epð tou eautoô tou. Akìma, apeikonðzei to S = {z : θ 1 <Argz<θ 2 } sto T = {w : aθ 1 <Argw<aθ 2 }. An, epiplèon, θ 2 θ 1 2π a, h za eðnai sômmorfh apeikìnish tou S epð tou T. Sq ma 9.3: 7. w = e z ApeikonÐzei th lwrðda y 1 <y<y 2 epð tou {z : y 1 <Argw<y 2 }. An y 2 y 1 2π, h apeikìnish eðnai 1 1. Sq ma 9.4: 8. w = az+b,ad bc 0(rhtogrammikìc metasqhmatismìc metasqhmatismìc Möbius) cz+d 84

85 MporeÐ na grafeð wc sônjesh metatìpishc, strof c, omojesðac kai antistrof c. ( ( ) EÐnai 1 1 apeikìnish thc sfaðrac tou Riemann epð tou eautoô thc f d c ), f( ) = a. c = ApoteleÐ om da wc proc th sônjesh. ApeikonÐzei kôkouc eujeðec epð kôklwn eujei n. An z 1,z 2,z 3 C o monadikìc metasqhmatismìc Möbius pou apeikonðzei ta z 1,z 2,z 3 sta, 0, 1, antistoðqwc, eðnai o T (z) = (z z 2)(z 3 z 1 ) (z z 1 )(z 3 z 2 ) Diplìc lìgoc twn z 1,z 2,z 3,z 4, sumb. (z 1,z 2,z 3,z 4 ), eðnai to T (z 4 ). O diplìc lìgoc tess rwn shmeðwn eðnai analloðwtoc apì to metasqhmatismì Möbius. O monadikìc metasqhmatismìc Möbius w = f(z) pou apeikonðzei ta z 1, z 2, z 3 sta w 1, w 2, w 3, antistoðqwc dðnetai apì th sqèsh T (w) =T (z). Mia sômmorfh apeikìnish enìc tìpou epð tou eautoô tou, lègetai automorfismìc tou tìpou autoô. Sqetik isqôoun: ( ) Oi automorfismoð tou monadiaðou dðskou, eðnai thc morf c g(z) =e iθ z a gia a < 1. Oi sômmorfec apeikonðseic h tou nw hmiepipèdou epð tou monadiaðou dðskou eðnai thc morf c ( ) z a h(z) =e iθ z ā Oi automorfismoð tou nw hmiepipèdou eðnai thc morf c f(z) = a, b, c, d R kai ad bc > Apeikìnish tou nw hmiepipèdou se kôklo ( 1 āz az+b cz+d ), me a C me Ima > 0 dedomèno shmeðo pou apeikonðzetai sto w =0. Tìte to ā apeikonðzetai sto w = (lìgw summetrðac). Prèpei sunep c w = β z a,β: stajerì. z ā Gia aujaðreto β, h sqèsh aut apeikonðzei to nw hmiepðpedo se k poio kôklo kèntrou 0. An jèloume apeikìnish ston monadiaðo kôklo, ja prèpei to z =0n' apeikonisjeð sto w 0 me w 0 =1, dhlad βa = ā. Epomènwc β = e iθ,θ R. 'Ara h sun rthsh w = e iθ z a z ā 85

86 apeikonðzei sômmorfa to Ima > 0 ston kôklo w < 1, ètsi ste to a w =0. An Ima < 0, tìte h anwtèrw sun rthsh apeikonðzei to k tw hmiepðpedo ston kôklo w < 1. H antðstrofh thc anwtèrw sun rthshc eðnai h w = zā aeiθ,ima>0 z e iθ pou apeikonðzei ton kôklo z < 1 sto nw hmiepðpedo. 2. Apeikìnish kôklou se kôklo 'Estw ìti to a C me a < 1 apeikonðzetai sto w =0. Tìte to summetrikì shmeðo 1 a apeikonðzetai sto w = kai h zhtoômenh apeikìnish ja èqei th morf : w = β z a z 1 ā = γ z a 1 āz Gia na broôme to γ upojètoume ìti 1 w 0, w 0 =1. Tìte γ 1 a 1 ā = w 0, dhlad γ = e i θ, θ R, opìte paðrnoume w = e i θ z a 1 āz pou apeinonðzei ton kôklo z < 1 ston kôklo w < Apeikìnish hmiepipèdou se hmiepðpedo IsqÔei to akìloujo je rhma: Up rqei monadikìc metasqhmatismìc Möbius pou apeikonðzei trða (diaforetik an dôo) shmeða z 1, z 2, z 3 se trða (an dôo diaforetik ) shmeða w 1,w 2,w 3, antistoðqwc. Aut h apeikìnish dðnetai apì th sqèsh w w 1 w w 3 w2 w 3 w 2 w 1 = z z 1 z z 3 z2 z 3 z 2 z 1 Gia na kataskeu soume th sun rthsh pou apeikonðzei hmiepðpedo se hmiepðpedo paðrnoume trða shmeða x 1 <x 2 <x 3 ston x- xona kai trða shmeða u 1 <u 2 <u 3 ston u- xona, ètsi ste x j u j. Ef' ìson oi x j,u j eðnai pragmatikoð, h anwtèrw sqèsh gðnetai: w = az + β,a,β,γz,δ R. γ + δ An aδ βγ > 0 to nw hmiepðpedo apeikonðzetai sto nw hmiepðpedo. An aδ βγ < 0 to nw hmiepðpedo apeikonðzetai sto k tw hmiepðpedo. 86

87 4. Orismènec qr simec apeikonðseic Sq ma 9.5: w = z m,m 1 2 Sq ma 9.6: w = e πz a 87

88 Sq ma 9.7: w =sin πz a Sq ma 9.8: w =cos πz a Sq ma 9.9: w =cosh πz a 88

89 ( ) Sq ma 9.10: w = a z + 1, (gia a =1:metasqhmatismìc Zukovski) 2 z Sq ma 9.11: w = ( ) 2 1+z 1 z Sq ma 9.12: w = 1 z 89

90 Sq ma 9.13: w =logz Sq ma 9.14: w = 1 2 (ze a + 1 z ea ) Sq ma 9.15: w = ( 1+z m 1 z m ) 2,m

91 ParadeÐgmata 1. JewroÔme to orjog nio R = {(x, y) :x [0, 2], y [0, 1]}. Na prosdioristeð h eikìna tou R sto epðpedo w apì to metasqhmatismì w = 2e πi 4 z. LÔsh e πi 4 = 2 2 (1 + i) w = 2e πi 4 z =(1+i)(x + iy) =x y + i(x + y) =u + iv u = x y, v = x + y x =0 u = y v = y y =0 u = x v = x u = v, x =2 u =2 y v =2+y u + v =4 u = v, y =1 u = x 1 v = x +1 u v =2 O parap nw metasqhmatismìc sunist mia strof kat gwnða π tou R kai mia omojesða 4 mhk n me lìgo Na brejeð mia analutik sun rthsh w = u + iv = f(z) pou apeikonðzei to sônolo 0 <Argz< π epð tou u<1. 3 LÔsh H w 1 = z 3 apeikonðzei to 0 <Argz< π 3 sto 0 <Argw 1 <π. H w 2 = iw 1 = iz 3 (strof kat π 2 afoô i = ei π 2 ) apeikonðzei to 0 <Argw 1 <πsto π 2 <Argw 2 < 3π 2. H w 3 = w 2 +1 metatopðzei thn u =0sthn u =1. Opìte w = w 3 w 2 w 1 = iz 3 +1 f(z) =iz Na brejeð ènac metasqhmatismìc Möbius pou apeikonðzei to nw hmiepðpedo tou epipèdou z sto monadiaðo kôklo tou epipèdou w me tètoio trìpo ste to z = i na apeikonðzetai sto w =0kai to z = sto w = 1. LÔsh ( ) Apì ton tôpo w = e iθ z a,ima>0paðrnoume: z ā ( ) 0=e iθ i a a = i i ā ( ) 1 =e iθ a 1=e iθ ā w =( 1) ( ) z i z + i w = i z i + z. 4. Na brejeð o metasqhmatismìc Möbius pou apeikonðzei to koinì mèroc twn dðskwn z 1 < 1 kai z i < 1 sto 1 o tetarthmìrio tou epipèdou w. LÔsh Oi kôkloi z 1 =1kai z i =1tèmnontai sta shmeða 0 kai 1+i. 91

92 K noume to metasqhmatismì ζ = z z (1 + i) pou apeikonðzei ta shmeða 0, 1+i sta shmeða 0,, antistoðqwc. AfoÔ oi efaptìmenec stouc dôo kôklouc sto 0 eðnai k jetec, oi kôkloi apeikonðzontai se eujeðec k jetec ( ) metaxô touc pou dièrqontai apì thn arq twn axìnwn. Afou ζ(2) = 1 + i, ζ = 1, oi eujeðec èqoun kðseic ±1 to de koinì meroc apeikonðzetai sto qwrðo: 1+i 2 3π 4 <Argζ< 5π 4. H strof w = e 3π 4 i ζ mac odhgeð sto zhtoômeno. Telik. w = 3π e 4 i z z (1 + i) 9.2 Efarmogèc thc SÔmmorfhc Apeikìnishc Probl mata Dirichlet kai Neumann a. Dirichlet Φ := 2 Φ x Φ y 2 =0, ston R, R: apl sunektikìc tìpoc. Φ=f, sth C, C = R:apl kleist kampôlh. b. Neumann Φ = 0, ston R. Φ n = g, sth C. g. To prìblhma Dirichlet me f suneq, kai to prìblhma Neumann me g suneq : g(s)ds =0, èqoun monadik lôsh. C Parat rhsh Ta probl mata v =0,R v = g, C, me g(s)ds =0 C n kai u: suzug c armonik thc v, ston R u = s g(s)ds, sth C, (a: aujaðreto shmeðo thc C) a eðnai isodônama. 92

93 UpenjÔmish: u, v (pragmatikèc sunart seic) suzugeðc armonikèc orc f = u+iv (migadik sun rthsh): analutik u = v =0kai isqôoun oi exis seic Cauchy Riemann. Parat rhsh O tìpoc R mporeð na mhn eðnai fragmènoc, p.q. to R mporeð na eðnai to nw hmiepðpedo, opìte to C eðnai o x- xonac. To prìblhma Dirichlet gia ton monadiaðo dðsko TÔpoc Poisson R = {z C : z < 1}, C= {z C : z =1} { Φ(r, θ) =0, (r, θ) ( 1, 1) (0, 2π) Φ(1, θ)=f (θ), θ [0, 2π) H lôsh dðnetai wc Φ(r, θ) = 1 2π (1 r 2 )F (ϕ)dϕ 2π 0 1 2r cos(θ ϕ)+r 2 To prìblhma Dirichlet gia to hmiepðpedo TÔpoc Poisson R = {z C : Imz > 0}, C= {z C : Imz =0} { Φ(x, y) =0, (x, y) R Φ(x, y) = 1 yf(ξ)dξ Φ(x, 0) = f(x), x R π y 2 +(x ξ) EpÐlush problhm twn Dirichlet kai Neumann me sômmorfh apeikìnish. 1. MetasqhmatÐzoume to prìblhma sunoriak n tim n gia ton apl sunektikì tìpo R se prìblhma sunoriak n tim n gia ton monadiaðo kôklo to hmiepðpedo (jewrhtik autì gðnetai p nta: J. Riemann) 2. LÔnoume to prìblhma sunoriak n tim n gia ton monadiaðo kôklo to hmiepðpedo. 3. Mèsw tou antðstrofou metasqhmatismoô brðskoume th lôsh tou arqikoô probl matoc. Sqetik isqôoun to akìlouja. Je rhma (A) Mèsw tou metasqhmatismoô w = f(z), f analutik kai f (z) 0 ston R, mia armonik sun rthsh apeikonðzetai se mða armonik sun rthsh. Je rhma (B) 'Opwc prin, Φ(x, y) =Φ(x(u, v), y(u, v)) := Ψ(u, v). Φ=a: staj. sto C se mèroc tou C Ψ=a, p nw sthn eikìna C sto w-epðpedo. Ψ sto C se mèroc tou C Φ n =0, n sto C. 93

94 Par deigma 1 2 Φ x Φ y 2 =0,y>0 lim y 0 + Φ(x, y) =f(x) = { 1,x>0 0, x<0 H sun rthsh φ(θ) = Aθ + B, A, B : pragmatikèc stajerèc, eðnai armonik, diìti eðnai to fantastikì mèroc thc A log z + B. 'Eqoume φ =1, gia x>0, dhlad gia θ =0 φ =0, gia x<0, dhlad gia θ = π. 'Ara } 1=A 0+B A = 1 0=A π + B π,b=1, opìte B' Mèjodoc: Φ(θ) =Aθ + B =1 θ π =1 1 π arctan ( y x Φ(x, y) = 1 π = 1 π 0 yf(ξ)dξ y 2 +(x ξ) 2 = y 0dξ y 2 +(x ξ) π ) = 1 π arctan ( ξ x y = 1 1 π arctan ( y x ) 0 0 ) y 1dξ y 2 +(x ξ) 2 (9.1) Par deigma 2 2 Φ x Φ y 2 =0,y>0 lim y 0 + Φ(x, y) =f(x) = T 0, x < 1 T 1, 1 <x<1 T 2, x > 1 H sun rthsh Φ(θ 1,θ 2 )=Aθ 1 +Aθ 2 +C, A, B, C: pragmatikèc stajerèc, eðnai armonik, diìti eðnai to fantastikì mèroc thc A log(z +1)+Blog(z 1) + C. φ = T 2, gia x>1, dhlad gia θ 1 = θ 2 =0 φ = T 1, gia 1 <x<1, dhlad gia θ 1 =0,θ 2 = π φ = T 1 0, gia x< 1, dhlad gia θ 1 = π, θ 2 = π 'Ara T 2 = A 0+B 0+C T 1 = A 0+B π + C T 0 = A π + B π + C A = T 0 T 1 π,b= T 1 T 2,C= T 2 π 94

95 Φ(θ 1,θ 2 ) = Aθ 1 + Bθ 2 + C = T 0 T 1 π B' Mèjodoc: Qr sh tou tôpou Poisson arctan y x +1 + T 1 T 2 y arctan π x 1 + T 2 Par deigma 3 ZhteÐtai armonik sun rthsh sto eswterikì tou monadiaðou kôkou kai { 1, 0 <θ<π f(θ) = 0, π < θ < 2π epð thc monadiaðac perifèreiac z =1. (φ =0sto ÂBC, φ =1sto ĈDE, blèpe sq ma 9.16). Sq ma 9.16: Me ton metasqhmatismì z = i w i+w w = i 1 z 1+z z =1sto nw hmiepðpedo w. φ =0sto ÂBC φ =0sto A B C φ =1sto ĈDE φ =1sto C D E. Autì to prìblhma lôjhke sto Par deigma 1: apeikonðzetai to eswterikì thc perifèreiac ) φ =1 1 π arctan ( v u w = i 1 z u = 2y,v= 1 (x2 +y 2 ) 1+z (1+x) 2 +y 2 (1+x) 2 +y 2 Φ=1 1 ( ) π arctan 2y x=r cos θ y=r sin θ = 1 1 ( ) 3r sin θ 1 (x 2 + y 2 ) π arctan 1 r 2 95

96 Kef laio 10 Efarmogèc 10.1 Hlektrik kukl mata - Hmitonoeid ReÔmata 'Opwc gnwrðzoume mia T-periodik sun rthsh f mporeð - upì k poiec proôpojèseic - na anaptuqjeð se seir Fourier. f(t) =b o + b m cos mωt + m=1 ìpou oi suntelestèc Fourier dðnontai wc m=1 a m sin mωt, ω := 2π T, b 0 = 1 T T 0 f(τ)dτ, b m = 2 T T 0 f(τ)cosmωτdτ, a m = 2 T T 0 f(τ)sinmωτdτ. H melèth thc sumperifor c enìc hlektrikoô kukl matoc sto opoðo kukloforeð èna periodikì reôma i(t), an getai ètsi sth melèth hmitonoeid n reum twn. OmoÐwc, gia mh periodikèc sunart seic to suneqèc} an logo eðnai to olokl rwma Fourier f(t) = 1 ( + ) f(τ)cosω(τ t)dτ dω π 0 kai ètsi h shmasða twn hmitonoeid n reum twn paramènei ousi dhc kai sth melèth hlektrik n kuklwm twn sta opoða to reôma pou ta diarrèei den eðnai periodikì Grafik Anapar stash Hmitonoeid n Sunart sewn JewroÔme èna di nusma OM stajeroô m kouc I pou peristrèfetai perð to O me gwniak taqôthta ω. 'Estw φ h gwnða pou sqhmatðzei to OM me ton xona Ox kat thn arq thc mètrhshc tou qrìnou. Tìte h probol ston xona Oy tou M dðnetai apì thn èkfrash i(t) =I sin(ωt + φ) 96

97 kai to di nusma OM anaparist thn hmitonoeid sun rthsh. Profan c, h prìsjesh dôo hmitonoeid n sunart sewn thc Ðdiac periìdou dðnetai wc (blèpe sq ma 10.1) i(t) =I sin(ωt + φ) =i 1 (t)+i 2 (t) =I 1 sin(ωt + φ)+i 2 sin(ωt + φ 2 ) Sq ma 10.1: H par gwgoc thc i(t) eðnai en to olokl rwma di(t) dt = ωi cos(ωt + φ) =ωi sin(ωt + φ + π 2 ), i(t)dt = I ω cos(ωt + φ) = I ω sin(ωt + φ π 2 ). 'Etsi: Par gwgoc: di nusma m kouc ωi pou sqhmatðzei gwnða π 2 fasik c gwnðac φ. me to di nusma m kouc I kai Olokl rwma: di nusma m kouc I ω pou sqhmatðzei gwnða π 2 fasik c gwnðac φ. me to di nusma m kouc I kai Migadik Anapar stash T ra Ox: pragmatikìc xonac, Oy: fantastikìc xonac kai upenjumðzoume ìti: e iθ = cos θ + i sin θ. 'Etsi, to di nusma OM orðzetai wc I[cos(ωt + φ)+i sin(ωt + φ) =Ie iφ e iωt = Je iωt,j:= Ie iφ 97

98 An h èkfrash i(t) =I sin(ωt + φ) anaparist èna hmitonoeidèc reôma, to J = Ie iφ onom zetai migadikì reôma. 'Oson afor thn prìsjesh, an kai an jèsoume tìte i 1 (t) =I 1 sin(ωt + φ 1 ),i 2 (t) =I 2 sin(ωt + φ 2 ) ĩ 1 (t) =I 1 e iφ 1 e iωt = J 1 e iωt, ĩ 2 (t) =I 2 e iφ 2 e iωt = J 2 e iωt J = J 1 + J 2 kai h hmitonoeid c sun rthsh pou anazhtoôme ja eðnai to fantastikì mèroc thc Je iωt, dhlad to Im(Je iωt ), pou eðnai i(t) =i 1 (t)+i 2 (t). AntistoÐqwc, an i(t) = I sin(ωt+ φ) ( ) di(t) d = Im dt dt (Jeiωt ) = ωi cos(ωt + φ) kai omoðwc gia to olokl rwma. An ìmwc èqoume na k noume pollaplasiasmì Ôywsh se dônamh o aplìc parap nw kanìnac (dhl. na p roume to fantastikì mèroc) den isqôei plèon: P.q. èstw i 1 (t) =I 1 sin(ωt + φ 1 ),i 2 (t) =I 2 sin(ωt + φ 2 ) tìte en gia paðrnoume opìte i 1 (t)i 2 (t) = 1 [ ] 2 I 1I 2 cos(φ 1 φ 2 ) cos(2ωt + φ 1 + φ 2 ), ĩ 1 (t) =J 1 e iωt,j 1 = I 1 e iφ 1, ĩ 2 (t) =J 2 e iωt,j 2 = I 2 e iφ 2 ĩ 1 (t)ĩ 2 (t) =I 1 I 2 e i(φ 1+φ 2 )+2iωt Im(ĩ 1 (t)ĩ 2 (t)) = I 1 I 2 sin(2ωt + φ 1 + φ 2 ). 'Etsi eðnai emfanèc ìti h diadikasða pou perigr yame (me th migadik anapar stash) eðnai sumbat me grammikèc, omogeneðc parast seic pr tou bajmoô, thc morf c aj ĩ j (t)+ dĩ j (t) β j + γ j ĩ j (t)dt dt ìpou oi suntelestèc a j,β j,γ j eðnai anex rthtoi tou t. H antikat stash ĩ k (t) =J k e iωt, me J k = I k e iφ k odhgeð se apotèlesma thc morf c Je iωt = Ie iφ e iωt kai to fantastikì mèroc thc teleutaðac par stashc eðnai h zhtoômenh hmitonoeid c sun rthsh i(t) =I sin(ωt + φ). 98

99 Kukl mata RLC JewroÔme èna kôklwma RLC dhlad èna kôklwma pou perilamb nei mìno puknwtèc (qwrhtikìthtac C), antist seic (wmik c antðstashc R) kai phnða (suntelest autepagwg c L) kai sto opoðo orðzontai dôo akraða shmeða. An to hlektrikì reôma èntashc i(t) =I cos(ωt + a) diarrèei to kôklwma, h anaz thsh hmitonoeidoôc lôshc thc oloklhrwtikodiaforik c exðswshc pou perigr fei to kôklwma odhgeð se diafor dunamikoô sta kra. 'Etsi, h antikat stash reômatoc èntashc sthn exðswsh, dðnei apotèlesma thc morf c u(t) =V cos(ωt + θ) ĩ(t) =Je iωt,j:= Ie ia ũ(t) =Ue iωt,u:= Ve iθ OrÐzoume wc migadik empèdhsh tou en lìgw kukl matoc, thn posìthta Z pou dðnetai apì th sqèsh U = ZJ kai èqei diast seic antðstashc. To mètro thc migadik c empèdhshc eðnai kai to ìrism thc Z = V I argz = θ a = φ EÐnai fanerì ìti o nìmoc tou Ohm (i = U ), mporeð na grafeð gia tic antðstoiqec migadikèc R posìthtec. Par deigma JewroÔme to kôklwma RLC (sq ma 10.2) pou diarrèetai apì reôma èntashc i(t) =cos(ωt + a). H diafor dunamikoô sta kra ja brejeð me to na antikatast soume ĩ(t) =Je iωt 99

100 Sq ma 10.2: sthn èkfrash ap' ìpou kai h migadik empèdhsh eðnai Ri(t)+L di(t) dt + 1 C i(t)dt [ ( U = J R + Lω 1 ) ] i Cω Z = R + ( Lω 1 ) i. Cω H anazhtoômenh hmitonoeid c t sh dðnetai apì tic sqèseic ki ètsi u(t) =V cos(ωt + θ) V = I Z kai θ = a + argz V = I R 2 + ( Lω 1 ) 2 Cω tan(θ a) = Lω 1 Cω. R Parat rhsh: QwrÐc bl bh thc genikìthtac mporeð kaneðc na jewr sei ìti h mètrhsh tou qrìnou arqðzei ìtan ta dedomèna èqoun mhdenik f sh. Sto par deigm mac, mporoôme na p roume ex' arq c a = Oi kanìnec tou Kirchhoff 1oc kanìnac (kanìnac twn kìmbwn) n i k (t) =0 k=1 100

101 Sq ma 10.3: 1oc kanìnac Kirchhoff. An ta i k eðnai hmotonoeid reômata, h antikat stash ĩ k (t) =J k e iωt dðnei n J k =0. k=1 2oc kanìnac (kanìnac twn brìgqwn) An ta u k,k=1, 2,..., n eðnai oi diaforèc dunamikoô sta kra twn kl dwn pou apoteloôn ènan brìgqo (kleistì perðgramma) tìte n u k (t) =0. k=1 Sthn perðptwsh hmitonoeid n diafor n dunamikoô, h antikat stash ũ k (t) =U k e iωt dðnei n U k =0. k=1 An oi diaforèc dunamikoô proèrqontai apì migadik reômata J k me emped seic Z k antistoðqwc, èqoume n Z k J k =0. An oi kl doi perilamb noun hlektrikèc diegèrseic e k (t) =ε k e iωt,tìte n Z k J k ε k =0. (ε k : migadik hlektregertik dônamh) k=1 k=1 ShmeÐwsh: Ta ajroðsmata stou kanìnec tou Kirchhoff eðnai algebrik, dhlad to swstì prìshmo (+ -) k je ìrou prèpei na lhfjeð up' ìyh, afoô orisjeð (aujaðreta) h jetik for ro c sto kôklwma. 101

102 Sq ma 10.4: 2oc kanìnac Kirchhoff Migadikèc Emped seic se Seiriak Par llhlh SÔndesh Seiriak SÔndesh Sq ma 10.5: Seiriak SÔndesh. ( n ) U = k=1 Z k J, J = J 1 = J 2 = J n, opìte Z = n k=1 Z k Par llhlh SÔndesh U = Z 1 J 1 = Z 2 J 2 = = Z n J n = ZJ J = n k=1 J k 'Etsi 1 Z = n k=1 1 Z k. 102

103 Sq ma 10.6: Par llhlh SÔndesh. Parat rhsh: Met ton upologismì thc migadik c empèdhshc enìc kukl matoc mporoôme telik c na gr youme Z(iω) =R(ω)+iX(ω) ap' ìpou èqoume th sqèsh twn plat n Z(iω) kai thc diafor c f shc argz(iω). AntistoÐqwc OrologÐa Par deigma 1 Z(iω) = Y (iω) =G(ω)+iB(ω) B(ω): fainìmenh agwgimìthta R(ω): pragmatik antðstash X(ω): fainìmenh antðstash Y (iω): (migadik ) sônjeth agwgimìthta G(ω): pragmatik agwgimìthta Na upologisjeð h empèdhsh Z tou kukl matoc tou sq matoc 10.7a Sq ma 10.7: 103

104 AnaparistoÔme to kôklwma ìpwc faðnetai sto sq ma 10.7b. Omadopoi ntac tic emped seic èqoume Z = Z 1 + Z ( 2Z 3 1, + 1 = Z 2 + Z ) 3 Z 2 + Z 3 Z 2 Z 3 Z 2 Z 3 ( ) me Z 1 = i,z C 1 ω 2 = R, Z 3 = i Lω i C 3. ω Shmei noume ìti ston kìmbo N èqoume J 1 = J 2 + J 3, en sto brìgqo Z 2 NZ 3 M èqoume J 2 Z 2 = J 3 Z 3. Ex' llou J 1 = U Z GenÐkeush thc ènnoiac thc migadik c empèdhshc JewroÔme èna kôklwma n brìgqwn. Ergazìmaste me touc brìgqouc l kai m (sq ma 10.8). 'Estw L ll,r ll,c ll ta stoiqeða tou brìgqou l. 'Estw L lm = L ml,r lm = R ml,c lm = C ml ta Sq ma 10.8: stoiqeða thc sôzeuxhc twn brìgqwn l kai m. 'Estw e l (t) =E l e iωt h hlektregertik sônamh sto brìgqo l, en e l (t) =0,k=1, 2,..., l 1, l+1,..., n. Tìte to anwtèrw kôklwma perigr fetai apì to sôsthma oloklhrwtikodiaforik n exis sewn [ di 1 (t) L 11 + R 11 i 1 (t)+ 1 dt C 11 ] [ i 1 (t)dt + + L 1n di n (t) dt + R 1n i n (t)+ 1 C 1n ] i n (t)dt = e 1 [ di 1 (t) L n1 + R n1 i 1 (t)+ 1 dt C 11 ]. [ di n (t) i 1 (t)dt + + L nn dt + R nn i n (t)+ 1 C nn ] i n (t)dt = e n Jètoume kai ĩ k (t) =J k e iωt iωl lm + R lm + 1 iωc lm = z lm (iω) 104

105 kai paðrnoume to sôsthma z 11 (iω)j z 1n (iω)j n =0. z l1 (iω)j 1 + +z ln (iω)j n = E l. z n1 (iω)j 1 + +z nn (iω)j n =0 (10.1) SumbolÐzoume me D(iω) thn orðzousa tou sust matoc kai me M lm (iω) thn upoorðzousa pou prokôptei me diagraf thc l-gramm c kai m-st lhc. 'Etsi J m = M l m (iω) D(iω) E l Jètontac èqoume telik Z lm (iω) := D(iω) M lm (iω) E l = Z lm J m H Z lm (iω) onom zetai empèdhsh metafor c apì to brìqo l sto brìqo m. Par deigma JewroÔme to kôklwma tou sq matoc 10.9 Sq ma 10.9: 'Eqoume R 11 = R C 11 = C 1 R 33 = R C 33 = C 3 R 13 = R 31 = R C 31 = C 13 =0 L 13 = L 31 =0 105

106 To sôsthma eðnai 'Etsi kai ( R ) i C 1 ω J 1 RJ 3 = E 1 ( ( ) ) RJ 1 + R + Lω i J 3 =0 Z 13 = E ( 1 = R J 3 i C 1 ω i C 3 ω ) ( 1+ Lω i ) C 3 ω i R R Z 11 = E 1 = R i J 1 C 1 ω R ( 2 ) R + Lω i C 3 i ω Migadikì Di nusma 'Estw a(t) di nusma pou exart tai hmitonoeid c pì to qrìno t: a(t) =(a 1 (t), a 2 (t), a 3 (t)) me a j (t) =A j cos(ωt + φ j ),j=1, 2, 3. MporoÔme na jewr soume ìti ta a j (t) proèrqontai apì migadikèc posìthtec ã j (t) =Âje iωt,j=1, 2, 3,  j = A j e iφ j ki ètsi na eisag goume to migadikì di nusma  =(  1,  2,  3 ) ìpou kai Im( Âe iωt )= Re( Âe iωt )= a(t) ( ( A 1 cos ωt + φ 2 π ) (,A 2 cos ωt + φ 2 π ) (,A 3 cos ωt + φ 3 π )) Epiplèon to suzugèc tou  eðnai Akìma an )  = (A 1 e iφ 1,A 2 e iφ 2,A 3 e iφ 3.  = Â1 + i  2 tìte a(t) =  1 cos ωt  2 sin ωt ( = ( Re ) ( cos ωt Im ) ) sin ωt. (Ta ( dianôsmata )  1, a ). π 2ω Â2 eðnai, antistoðqwc, oi jèseic pou katalamb nei to a(0) kai to 106

107 To pleonèkthma thc qr shc twn migadik n dianusm twn sto logismì twn hmitonoeid n dianusm twn, eðnai ìti den emfanðzetai stouc upologismoôc to t. P.q. gia thn exðswsh tou Maxwell curl H = σ E + ε E t, ( ) H = H(x1,x 2,x 3,t), E = E(x1,x 2,x 3,t) se hmitonoeid E, H antistoiqoôn migadik dianôsmata E, H kai h exðswsh Maxwell gr fetai curl H = iωη E ìpou η = ε σ i eðnai h migadik dihlektrik stajer. ω 10.2 Efarmogèc sto Statikì Hlektrismì O nìmoc tou Coulomb JewroÔme dôo hlektrik fortða q 1,q 2 se apìstash r to èna apì to llo. SÔmfwna me to nìmo tou Coulomb se kajèna apì ta fortða askeðtai apì to llo dônamh me mègejoc F = q 1q 2 kr 2. H dônamh aut eðnai elktik, e n ta dôo fortða eðnai eterìshma (èna jetikì ki èna arnhtikì), kai apwstik, e n ta dôo fortða eðnai omìshma (kai ta dôo jetik kai ta dôo arnhtik ). H stajer k kaleðtai dihlektrik stajer kai exart tai apì to ulikì me to opoðo eðnai gem toc ì q roc. Gia to kenì eðnai k =1, en se llec peript seic eðnai k> 'Entash HlektrikoÔ PedÐou. Hlektrostatikì Dunamikì Mia dedomènh katanom hlektrikoô fortðou (shmeiak fortða suneq c katanom sunduasmìc twn dôo) dhmiourgeð èna hlektrikì pedðo. E n èna jetikì fortðo Ðso me th mon da (pou upojètoume ìti eðnai arket mikrì ste na mhn ephre sei to pedðo) topojethjeð se èna shmeðo A tou pedðou (ìpou de brðsketai llo shmeiakì fortðo), tìte h dônamh pou askeðtai p nw sto fortðo autì kaleðtai èntash tou hlektrikoô pedðou sto shmeðo A kai sumbolðzetai me E. H dônamh aut prokôptei apì to èna dunamikì F pou kaleðtai hlektrostatikì dunamikì. 'Etsi èqoume E = gradφ = Φ E n h katanom tou hlektrikoô fortðou eðnai disdi stath, dhl. h perðptwsh pou mac endiafèrei ed, tìte E = E x + ie y = Φ x i Φ y, ìpou E x = Φ x,e y = i Φ y. (10.2) 107

108 E n E t eðnai h sunist sa thc èntashc tou hlektrikoô pedðou, h efaptìmenh se mia apl kleist kampôlh C tou migadikoô epipèdou z, tìte E i ds = E x dx + E y dy = To je rhma tou Gauss C JewroÔme mia disdi stath katanom hlektrikoô fortðou. E n C eðnai mia apl kleist kampôlh (sto epðpedo z) pou perikleðei sunolikì fortðo q (sthn pragmatikìthta mia kulindrik epif neia apeðrou m kouc pou perikleðei sunolikì fortðo q an mon da m kouc) kai E n h k - jeth sthn C sunist sa thc èntashc tou hlektrikoô pedðou, tìte sômfwna me to je rhma tou Gauss E n ds =4πq. E n to sunolikì fortðo pou perikleðei h C eðnai mhdèn, tìte E n ds = E x dy E y dx =0. C C C apì th sqèsh aut èpetai ìti se ènan tìpo pou den perièqei kajìlou fortðo (dhlad h puknìthta tou fortðou eðnai pantoô mhdèn) èqoume Apì tic 10.2, 10.3 èqoume ìti E x x + E y y 2 Φ x + 2 Φ 2 y =0, 2 =0. (10.3) dhlad h Φ eðnai armonik sun rthsh se k je shmeðo ìpou den up rqei fortðo To migadikì hlektrostatikì dunamikì Apì ta prohgoômena eðnai fanerì ìti up rqei mia armonik sun rthsh Ψ, suzug c armonik thc Φ, tètoia ste h Ω(z) =Φ(x, y)+iψ(x, y) na eðnai analutik se ènan tìpo ìpou den up rqei fortðo. H Ω(z) kaleðtai migadikì hlektrostatikì dunamikì apl migadikì dunamikì kai h 10.2 mporeð na grafeð sth morf E = Φ x i Φ y = Φ x + i Ψ y = dω dz = Ω (z). (10.4) To mètro thc E eðnai E = E = Ω (z) = Ω (z). Oi kampôlec (sthn pragmatikìthta kulindrikèc epif neiec) Φ(x, y) = α kai Ψ(x, y) = β kaloôntai antðstoiqa isodunamikèc grammèc kai grammèc ro c. 108

109 Grammik fortða EÐnai fanerì ìti up rqoun pollèc omoiìthtec metaxô enìc hlektrikoô pedðou kai tou pedðou ro c enìc reustoô. To hlektrikì pedðo antistoiqeð sto pedðo taqut twn tou reustoô, me mình diafor èna antðjeto prìshmo sta antðstoiqa migadik dunamik. Oi ènnoiec jetik c kai arnhtik c phg c sth ro reustoô èqoun an logec ènnoiec sto statikì hlektrismì. 'Etsi, to migadikì (hlektrostatikì) dunamikì pou ofeðletai se èna fortðo q (akribèstera se peirou m kouc grammikì fortðo q an fortðo m kouc) sto shmeðo z 0 sto kenì eðnai Ω(z) = 2q ln(z z 0 ). (10.5) To migadikì autì dunamikì perigr fei jetik arnhtik phg, e n antðstoiqa eðnai q<0 q>0. 'Omoia, orðzoume migadikì dunamikì dipìlou, ktl. Se perðptwsh pou o q roc den eðnai kenìc, antikajistoôme sthn 10.5 to q me q/k AgwgoÐ E n èna s ma eðnai (tèleioc) agwgìc, ìlo to fortðo pou èqei katanèmetai mìno sthn epif nei tou. 'Etsi, se disdi stata probl mata, ìpou h epif neia tou s matoc antiproswpeôetai apì mia apl kleist kampôlh C sto epðpedo z, to fortðo katanèmetai sthn C kai isorropeð. Sunep c h C eðnai mia isodunamik gramm tou pedðou. 'Ena shmantikì prìblhma sto statikì hlektrismì eðnai o prosdiorismìc tou dunamikoô pou ofeðletai se dedomènouc (kulindrikoôc) agwgoôc. To prìblhma autì mporeð na lujeð me th mèjodo thc sômmorfhc apeikìnishc Qwrhtikìthta JewroÔme dôo agwgoôc me antðjeta fortða me apìluth tim q to kajèna. E n h diafor dunamikoô metaxô twn agwg n eðnai V, tìte h posìthta C pou orðzetai me th sqèsh q = CV (10.6) exart tai mìno apì th gewmetrða tou sust matoc twn agwg n kai kaleðtai qwrhtikìthta. To sôsthma twn agwg n apoteleð èna puknwt Ask seic 1. (a ) BreÐte to migadikì dunamikì pou ofeðletai se èna grammikì fortðo q (an mon da m kouc) k jeto sto epðpedo z sto shmeðo z =0. (b ) Ti tropopoðhsh prèpei na gðnei sto prohgoômeno er thma, an to grammikì fortðo eðnai k jeto sto z = α? LÔsh 109

110 Sq ma 10.10: Aktinikì pedðo. (a ) To hlektrikì pedðo pou dhmiourgeðtai apì to grammikì fortðo eðnai aktinikì, dhlad èqei mìno aktinik sunist sa E r (blèpe sq ma 10.10). E n C eðnai kulindrik epif neia me kèntro to z =0kai aktðna r, h k jeth sthn C sunist sa tou hlektrikoô pedðou E n isoôtai me E r kai eðnai stajer p nw sthn C. Apì to je rhma tou Gauss èqoume E n ds = E r ds = E r 2πr =4πq C C kai E r = 2q r. EÐnai ìmwc E r = Φ kai ra Φ= 2qln r, ìpou paraleðyame th r stajer olokl rwsehc. Autì eðnai to pragmatikì mèroc tou Ω(z) = 2q ln z, pou eðnai to zhtoômeno migadikì dunamikì. (b ) E n to grammikì fortðo eðnai to shmeðo x = a, to migadikì dunamikì eðnai Ω(z) = 2q ln(z a). 2. (a ) BreÐte to dunamikì se k je shmeðo tou tìpou tou sq matoc 10.11, an to dunamikì tou xona x eðnai V 0 gia x>0 kai V 0 gia x<0. (b ) ProsdiorÐste tic isodunamikèc grammèc kai tic grammèc ro c. LÔsh (a ) Prèpei na broôme mia armonik sun rthsh pou na paðrnei tic timèc V 0 gia x>0 (dhlad θ =0) kai V 0 gia x<0 (dhlad θ = π). An A kai B eðnai pragmatikèc stajerèc h Aθ + B eðnai armonik. Oi sunj kec A(0) + B = V 0, A(π)+B = V 0 dðnoun A = 2V 0 /π, B = V 0. 'Ara to zhtoômeno dunamikì eðnai ( V ) ( π θ = V π arctan y ) x gia y>0. To dunamikì gia y<0 prokôptei apì th summetrða tou pedðou wc proc ton xona x. 110

111 Sq ma 10.11: ( (b ) Oi isodunamikèc grammèc èqoun exðswsh V arctan y π x) = a y = mx, ìpou m mia stajer. EÐnai eujeðec grammèc pou pernoôn apì thn arq twn axìnwn. Oi grammèc ro c eðnai orjog niec troqièc stic eujeðec y = mx, dhlad oi perifèreiec kôklwn me kèntro thn arq x 2 + y 2 = β. ( 'Allh mèjodoc. Mia sun rthsh suzug c armonik thc V arctan y π x) eðnai h 2V 0 ln π r. Sunep c oi grammèc ro c eðnai r = x 2 + y 2 =staj., dhlad perifèreiec kôklwn me kèntro thn arq twn axìnwn. 3. (a ) BreÐte to dunamikì pou ofeðletai se èna grammikì fortðo q (an mon da m kouc) k jeto sto z = z 0 kai èna llo grammikì fortðo q (an mon da m kouc) k jeto sto z = z 0. (b ) DeÐxte ìti to dunamikì, pou ofeðletai se èna peiro epðpedo ABC (sq ma 10.12b) me stajerì dunamikì mhdèn kai se èna grammikì fortðo q (an mon da m kouc) par llhlo sto epðpedo autì, mporeð na brejeð apì to apotèlesma tou prohgoômenou mèrouc. (a) (b) Sq ma 10.12: 111

112 LÔsh (a ) To migadikì dunamikì pou ofeðletai sta dôo grammik fortða (sq ma 10.12a) eðnai ( ) z z0 Ω(z) = 2q ln(z z 0 )+2qln(z z 0 )=2qln ) z z 0 To zhtoômeno dunamikì eðnai to pragmatikì mèroc tou Ω(z), dhlad { ( )} z z0 Φ=2qR ln. (10.7) z z 0 (b ) ArkeÐ na deðxoume ìti h 10.7 dðnei dunamikì Φ=0ston xona x, dhlad sto epðpedo ABC tou sq matoc (sq ma 10.12b). Autì ìmwc eðnai fanerì, afoô ston xona x èqoume z = x, opìte ( ) ( ) x z0 Ω=2qln kai x z Ω x z0 =2qln = Ω 0 x z 0 dhlad Φ=Re{Ω} =0. 'Etsi, mporoôme na antikatast soume to fortðo q sto z 0 me èna apðpedo ABC pou diathreðtai se stajerì dunamikì mhdèn kai antðstrofa. 4. DÔo peira par llhla epðpeda, se apìstash a to èna apì to llo, eðnai geiwmèna, dhlad èqoun stajerì dunamikì mhdèn. 'Ena grammikì fortðo q (an mon da m kouc) topojeteðtai metaxô twn epipèdwn, par llhlo proc aut kai se apìstash b apì to èna epðpedo. BreÐte to dunamikì se tuqaðo shmeðo metaxô twn epipèdwn. LÔsh 'Estw ìti ABC kai DEF parist noun sto sq ma ta dôo epðpeda (k jeta sto Sq ma 10.13: 112

113 epðpedo z) kai èstw ìti to grammikì fortðo eðnai k jeto sto epðpedo z sto shmeðo tou fantastikoô xona z = bi. H sun rthsh apeikìnishc w = e πz/a apeikonðzei to skiasmèno tìpo tou sq matoc sto p nw misì epðpedo w tou sq matoc. To grammikì fortðo q sto z = bi tou sq matoc apeikonðzetai sto grammikì fortðo q sto w = e πbi/a. To sônoro ABCDEF tou sq matoc (pou brðsketai se dunamikì mhdèn) apeikonðzetai ston xona ua B C D E F (epðshc se dunamikì mhdèn), ìpou ta C kai D sumpðptoun me to w =0. Apì to prohgoômeno prìblhma to dunamikì sto skiasmèno tìpo tou sq matoc eðnai { } w e πbi/a Φ=2qRe. w e πbi/a Sunep c, to dunamikì sto skiasmèno tìpo tou sq matoc eðnai { } e πz/a e πbi/a Φ=2qRe. e πz/a e πbi/a 5. DÔo apeðrou m kouc kulindrikoð agwgoð èqoun koinì xona, aktðnec r 1 kai r 2 (r 2 >r 1 ) kai diathroôntai se stajer dunamik Φ 1 kai Φ 2 antðstoiqa (sq ma 10.14). BreÐte (a ) To dunamikì (b ) thn èntash tou hlektrikoô pedðou ston tìpo metaxô twn kulindrik n agwg n. Sq ma 10.14: LÔsh (a ) JewroÔme th sun rthsh Ω=A ln z + B, ìpou A kai B eðnai pragmatikèc stajerèc. An z = re iθ, èqoume Ω=Φ+iΨ =A ln r + Aiθ + B Φ=A ln r + B, Ψ = Aθ. H Φ ikanopoieð thn exðswsh tou Laplace, dhlad eðnai armonik, gia r 1 <r<r 2 kai dðnei antðstoiqa Φ=Φ 1 kai Φ=Φ 2 gia r = r 1 kai r = r 2. Sunep c Φ 1 = A ln r 1 + B, Φ 2 = A ln r 2 + B apì ìpou A = Φ 2 Φ 1 ln(r 2 /r 1 ),B= Φ 1 ln r 2 Φ 2 ln r 1. ln(r 2 /r 1 ) 113

114 Epomènwc to zhtoômeno dunamikì eðnai Φ=(Φ 2 Φ 1 ln(r 2 /r 1 )lnr + Φ 1 ln r 2 Φ 2 ln r 1. ln(r 2 /r 1 ) (b ) 'Entash hlektrikoô pedðou = E = gradφ = Φ = Φ 1 Φ 2 1 r ln(r 2 /r 1. Ac shmeiwjeð ) r ìti oi dunamikèc grammèc grammèc ro c, pou shmei nontai me diakekommènec sto sq ma 10.15, eðnai orjog niec stic isodunamikèc grammèc. Sq ma 10.15: 6. BreÐte th qwrhtikìthta tou puknwt pou sqhmatðzetai apì touc dôo kulindrikoôc agwgoôc tou prohgoômenou probl matoc. LÔsh An Γ eðnai mia apl kleist kampôlh pou perikleðei ton eswterikì agwgì kai q eðnai to fortðo autoô tou agwgoô, èqoume apì to je rhma tou Gauss E n ds = Sunep c q = Φ 1 Φ 2 2ln(r 2 r 1 ) kai 2π θ=0 C(qwrhtikìthta) = { } Φ1 Φ 2 ln(r 2 /r 1 ) 1 dθ = 2π(Φ 1 Φ 2 ) ln(r r 2 /r 1 ) fortðo diafor dunamikoô = q Φ 1 Φ 2 = =4πq. 1 2ln(r 2 /r 1 ) pou exart tai mìno apì th gewmetrða tou sust matoc twn agwg n. To apotèlesma autì isqôei ìtan o q roc metaxô twn kulðndrwn eðnai kenìc. Ean eðnai gem toc apì ulikì me dihlektrik stajer k, prèpei kai antikatast soume to q me q/k. Sthn perðptwsh aut h qwrhtikìthta gðnetai 1/[2k ln(r 2 /r 1 )]. 7. DÔo kulindrikoð agwgoð peirou m kouc èqoun o kajènac aktðna R kai ta kèntra touc apèqoun apìstash D to èna apì to llo (megalôterh thc diamètrou tou kajenìc, sq ma ). E n diathroôntai se dunamik V 0 kai V 0 upologðste 114

115 (a ) to fortðo kajenìc (an mon da m kouc) kai (b ) th qwrhtikìthta tou puknwt pou sqhmatðzoun. LÔsh Sq ma 10.16: (a ) MporoÔme na jewr soume k je kulindrikì agwgì san mia isodunamik epif neia me sugkekrimèno dunamikì. Jètontac a = V 0 kai a = V 0 (kai epeid k =2q) brðskoume ìti ta kèntra twn kôklwn eðnai sta shmeða (b ) apì ìpou H aktðna k je kôklou eðnai x = a coth(v 0 /2q) kai x = a coth(v 0 /2q) D =2a coth(v 0 /2q). (10.8) R = a cosech(v 0 /2q). (10.9) Diair ntac tic 10.8 kai 10.9 brðskoume 2cosh(V 0 /2q) =D/R kai to zhtoômeno fortðo eðnai q = V 0 2 arccosh(d/2r). C(qwrhtikìthta) = fortðo diafor dunamikoô = q Φ 1 Φ 2 = 1 2ln(r 2 /r 1 ). To apotèlesma autì isqôei ìtan o q roc gôrw apì touc agwgoôc eðnai kenìc. An eðnai gem toc me ulikì dihlektrik c stajer c k, tìte to apotèlesma prèpei na diairejeð me k. Ac shmeiwjeð ìti h qwrhtikìthta exart tai mìno apì th gewmetrða tou sust matoc. H èkfrash thc qwrhtikìthtac pou br kame èqei meg lh shmasða sth metafor enèrgeiac. 115