H selida auth brisketai sth dieu0unsh http://www.csd.uch.gr/~kolount/complex.html opws epishs kai sth dieu0unsh http://www.math.uiuc.edu/~kolount/complex.html

Mporeite na thn prospelasete xrhsimopoiwntas Netscape, Mosaic, Internet Explorer (auta xreiazontai termatika graphics) eite to programma lynx. Auto den apaitei graphics kai mporeite na to trejete apo sxedon opoiodhpote termatiko dinontas p.x. thn entolh
lynx http://www.csd.uch.gr/~kolount/complex.html

Sthn periptwsh pou den exete graphics uparxei periptwsh to termatiko sas na mh xeirizetai kala tous Ellhnikous xarakthres. 0a uparxoun tote sthn korufh ths selidas enallaktikes dieu0unseis (URL) pou 0a einai panta anagnwsimes kai apo tis opoies (analoga me to ti termatiko exete) 0a mporeite na exete kanonika prosbash sth selida.

Oi pio prosfates kataxwrhseis briskontai sto telos ths selidas.

Panepisthmio Krhths -- Ma0hmatiko Tmhma

Migadikh Analush (M 211)

Xeimerino Ejamhno 1997-98

Didaskwn: Mixalhs N. Kolountzakhs

Boh0os metaptuxiakos foithths: Giwrgos Ketsetshs

1. Episkophsh migadikwn ari0mwn
2. Analutikes sunarthseis, sun0hkes Cauchy-Riemann, armonikes sunarthseis
3. Stoixeiwdeis sunarthseis (exp, cos, sin, log, antistrofes trigwnometrikes)
4. To 0ewrhma tou Cauchy, 0ewrhma Morera, oloklhrwtikos tupos tou Cauchy
5. Dunamoseires, seires Taylor kai Laurent
6. Rizes
7. Oloklhrwtika upoloipa
8. Summorfes apeikoniseis

Askhseis Ka0'olh th diarkeia tou ejamhnou 0a dinetai ena fulladio me 5-10 askhseis/ebdomada. Ka0e 2h bdomada kai gia 15 peripou lepta oi foithtes 0a grafoun (me kleistes shmeiwseis) mia askhsh pou exei epilegei apo to didaskonta apo ta prohgoumena 2 fulladia (h mia polu paromoia). H summetoxh twn foithtwn s'auto einai proairetikh alla sunistatai entona h summetoxh sas wste ta diagwnismata (proodos, telikos) na mhn sas er0oun duskola.

Ba0mologiko susthma: Estw I o ba0mos ths ejetashs tou Ianouariou, M o ba0mos ths proodou kai T o ba0mos twn askhsewn.
O telikos ba0mos gia thn periodo tou Ianouariou 0a einai to megisto twn:

1. I
2. 0.6 I + 0.4 M
3. 0.6 I + 0.4 T
4. 0.6 I + 0.2 M + 0.2 T

Askhsh 1: Mporeite na paraleiyete mia apo tis parapanw 4 posothtes xwris na allajei to ba0mologiko susthma gia thn periodo Ianouariou;

Arxh ejamhnou: H selida auth 0a enhmerwnetai taktika gia 0emata pou aforoun th Migadikh Analush (M 211). Edw 0a briskete sunh0ws:

1. Mia polu suntomh perigrafh tou ti eipw0hke ka0e mera sto ma0hma
2. Poies askhseis sunistw na lunete kai, endexomenws, upodeijeis gia th lush tous
3. Palia 0emata ejetasewn
4. Shmantikes anakoinwseis (hmeromhnies diagwnismatwn, statistika stoixeia gia tis epidoseis sta diagwnismata, k.l.p.)
5. Deiktes (links) se alles selides sto Internet me paromoia 0emata
6. Diafora istorika stoixeia sxetika me to ma0hma, k.a.

Giati h selida;
Enas apo tous logous uparjhs auths ths selidas einai sa kinhtro gia thn apokthsh apo to foithth ikanothtas xrhshs tou Internet. Skopos einai na gnwrizei kaneis ti eidous plhrofories mporei na brei sto diktuo ka0ws kai to pws na tis anazhthsei.
Fusika, mia kai uparxei kosmos pou den 0elei na apokthsei tetoies gnwseis kai de 0elei na exei epafh me ton upologisth, h selida auth 0a tupwnetai kai 0a anartatai ejw apo to grafeio mou peripou mia fora th bdomada, wste na mporei kaneis na parei olh thn plhroforia apo ekei.

Merika istorika stoixeia gia ton Kwnstantino Kara0eodwrh , isws to megalutero Ellhna ma0hmatiko twn neoterwn xronwn kai me terastia suneisfora sth Migadikh Analush.

Pe, 18 Sep: Sthn kleisth sullogh ths biblio0hkhs exw topo0ethsei ta ejhs biblia sxetika me to ma0hma (einai ola grammena sta Agglika).

1. J.B. Conway, Functions of one complex variable, Springer (1973).
2. G. Polya and G. Latta, Complex Variables, Wiley (1974).
3. J. Marsden and M. Hoffman, Basic Complex Analysis, Freeman (1987).

Pa, 19 Sep: Mporeite na anazhteite hlektronika biblia sth biblio0hkh ths Sxolhs 0et. Episthmwn. (Dwste library gia onoma login.)

D, 22 Sep: Oi ai0ouses termatikwn sthn pteruga G 0a paramenoun anoixtes 9pm-9mm (ka0hmerines) kai 9pm-3mm (Sabbato) Auta isxuoun apo 29/9 alla oi ai0ouses 0a einai genika anoixtes kai auth thn ebdomada.
Ekei uparxoun apla termatika (xwris graphics) tupou VT (opou ta ellhnika grammata en genei den exoun problhma) kai termatika graphics tupou X-terminals ta opoia (proswrina elpizw) exoun problhma me ta Ellhnika, alla einai safws protimhtea apo ta VT kata ta alla. An de sas peirazei na blepete Ellhnika me latinikous xarakthres tote xrhsimopoieiste auta. Suntoma 0a exoume kai arketa PC pou 0a kanoun thn prosbash sto diktuo akomh pio aneth.
Sta VT xrhsimopoieite to programma lynx enw sta X-terminals kai PC to programma netscape.

Tr, 23 Sep: Kaname episkophsh twn migadikwn ari0mwn (Kef. 1, § 1-7). Thn Pempth 0a moirastei to prwto fulladio askhsewn kai 0a arxisoume na milame gia sunarthseis me mia migadikh metablhth.

Te, 24 Sep: 1h omada askhsewn:
sel. 24: 6, 13, 18
sel. 39: 1, 3, 6, 8
sel. 50: 8, 9, 11

Pe, 25 Sep: Kef. 2, § 9-12. Milhsame gia sunarthseis me mia migadikh metablhth kai eidame merika paradeigmata apeikonisewn xwriwn tou migadikou epipedou mesw sunarthsewn. Eidame epishs kai ton orismo tou oriou mias sunarthshs otan h metablhth sugklinei se ena peperasmeno migadiko ari0mo h sto apeiro.
Elpizw thn Trith na exw na sas moirasw sthn tajh fwtotupies apo ta sxetika tmhmata tou bibliou pou aforoun tis askhseis kuriws. Mporeite epishs na daneizeste to biblio apo thn kleisth sullogh ths biblio0hkhs.

Tr, 30 Sep: Kef. 2, § 13-16. Milhsame gia stereografikh probolh, sunexeia sunarthsewn kai telos dwsame ton orismo ths paragwgou sunarthshs mias migadikhs metablhths. Eidame paradeigmata sunarthsewn pou exoun paragwgo kai paradeigmata mh paragwgisimwn sunarthsewn.

Pe, 2 Okt: Kef. 2, § 16-17. Epanalabame ton orismo ths paragwgou kai ton anadiatupwsame ws
f(z+h) = f(z) + f'(z) h + o(h)
opou h sunarthsh o(h) exei thn idiothta
o(h) / h --> 0, otan to h --> 0.
Apodeijame pws ena poluwnumo tou z den mporei na pairnei mono pragmatikes times se olo to migadiko epipedo. Telos deijame pws mia sunarthsh pou einai paragwgisimh se ena shmeio ikanopoiei tis ejiswseis Cauchy-Riemann.

Merika istorika stoixeia gia tous Cauchy kai Riemann.

Sa, 4 Okt: Ta termatika graphics tupou X-terminals ths pterugas G twra uposthrizoun kai Ellhnikous xarakthres. Etsi mporeite na blepete th selida auth (ka0ws kai tis selides pros tis opoies uparxoun deiktes apo edw) se periballon graphics. Etsi mporei sto mellon na uparxoun kai eikones edw, pragma pou apefeuga na kanw mexri twra epeidh oi perissoteroi apo sas den eixate prosbash se periballon grafikwn.

De, 6 Okt: O Giwrgos Ketsetshs (metaptuxiakos foithths tou Ma0hmatikou Tmhmatos) oristhke boh0os gia to ma0hma. 0a brisketai sto grafeio tou (H 304) ka0e Deutera 6-9mm gia na sas boh0hseis me opoies erwthseis tuxon exete gia to ma0hma.
Epishs, osoi apo sas 0elete, mporeite na grafete tis askhseis pou sas dinw kai na tis dinete sto Giwrgo gia dior0wsh. De 0a pairnete ba0mo apo auto alla 0a sas boh0hsei sto na jerete an exete lusei swsta kapoies askhseis.

Tr, 7 Okt: Kef. 2, § 18,20,21. Apodeijame pws oi sun0hkes Cauchy-Riemann kai h sunexeia twn merikwn paragwgwn sunepagontai thn uparjh migadikhs paragwgou. Apodeijame epishs pws to pragmatiko kai to fantastiko meros mias analutikhs sunarthshs einai armonikes sunarthseis (xrhsimopoihsame xwris apodeijh pws ka0e analutikh sunarthsh exei sunexeis merikes paragwgous 2hs tajhs).
Upen0umizetai pws thn prosexh Pempth, 9 Okt, 0a exoume to prwto diagwnisma panw stis duo prwtes omades askhsewn.

Pe, 9 Okt: Kef. 3, § 22-25. Stoixeiwdeis sunarthseis. Eixame to prwto diagwnisma (askhsh 2 tou 2ou fulladiou) kai moirasthke to 3o fulladio askhsewn.

Ku, 12 Okt: O mesos oros sto prwto diagwnisma htan 6.9/10. To pleon koino la0os htan h euresh twn merikwn paragwgwn twn sunarthsewn u kai v sto (0,0) paragwgizontas ton prwto tupo tou kladikou orismou. Auto einai fusika la0os afou h sunarthsh de dinetai apo auto ton tupo sto (0,0).
Epishs uphrxe mia genikh sugxush ws pros to poies epipleon sun0hkes xreiazontai wste apo tis ejiswseis Cauchy-Riemann na mporoume na sumperanoume thn uparjh migadikhs paragwgou se ena shmeio.

Tr, 14 Okt: Kef. 4, § 30-32. Sunarthseis apo tous pragmatikous stous migadikous, paragwgoi kai oloklhrwmata tous. Parametriseis kampulwn sto C. Mhkos tojou. Epikampulia oloklhrwmata (oloklhrwmata broxou, kata to biblio).

Pe, 16 Okt: Kef. 4, § 33-34. Eidame diafora paradeigmata upologismou epikampuliwn oloklhrwmatwn. Epishs mia basikh anisothta gia to metro enos oloklhrwmatos mesw tou megistou ths oloklhrwteas sunarthshs kai tou mhkous tou dromou. Orisame thn paragousa sunarthshs kai apodeijame pws mia sunexhs sunarthsh orismenh se ena xwrio D exei paragousa sto D an kai mono an to oloklhrwma ths einai 0 panw se ka0e kleisth kampulh sto D (pou einai to idio me to na pei kaneis pws h timh enos oloklhrwmatos den ejartatai apo to dromo pou dialegei kaneis alla mono apo ta akra tou.

4h omada askhsewn:
sel. 111: 5, 7.
sel. 120: 2, 3, 7, 11, 13, 14, 16.

Tr, 21 Okt: Kef. 4, § 35,38: To 0ewrhma tou Cauchy. To apodeijame me thn (mh anagkaia) upo0esh oti h paragwgos ths f einai sunexhs. H apodeijh xwris auth thn upo0esh brisketai stis § 36,37 tis opoies den kaluyame (alla isws na to kanoume sto prosexes mellon).
Thn epomenh Pempth, 23/10, 0a exoume to 2o diagwnisma panw sthn 3h kai 4h omada askhsewn.
Epishs thn Pempth 0a exoume thn enarjh tou foithtikou seminariou. Omilhths o G. Menegakhs me 0ema thn uparjh sunexwn pou0ena paragwgisimwn sunarthsewn me me0odous kathgorias.

Pe, 23 Okt: Kef. 4, § 39: Eidame ton tupo tou Cauchy gia ta oloklhrwmata, o opoios mas epitrepei na ekfrasoume thn timh mias analutikhs sunarthshs f se ena shmeio z enos xwriou D me sunoro mian aplh kleisth kampulh G ws ena epikampulio oloklhrwma panw sth G mias sunarthshs pou ka0orizetai apo tis times ths f panw sth G kai mono. Eidame epishs diafora paradeigmata oloklhrwmatwn pou upologizontai xrhsimopoiwntas ton tupo tou Cauchy.
Moirasthke to 5o fulladio askhsewn kai eixame to 2o 15hmero diagwnisma.

De, 27 Okt: O mesos oros sto 2o diagwnisma htan peripou 5/10. O mesos oros afora mono ta grapta ta opoia parado0hkan (18 auth th fora enanti 32 thn prohgoumenh).
Dedomenou oti h askhsh pou sas zhth0hke na grayete eixe (sxedon h idia) lu0ei mesa sthn tajh thn Trith 21 Okt, ta apotelesmata safws den htan antistoixa me tis dunatothtes sas. 0a h0ela na sas tonisw edw pws den einai idiaitera epituxhmenh h taktikh tou na apofasizete prin to diagwnisma pws "kapoies askhseis apokleietai na pesoun". Epishs 0a eprepe na erxeste sto grafeio mou gia erwthseis arketes meres nwritera kai oxi panta thn prohgoumenh opws sunh0izete, afou auto den sas afhnei arketo xrono na katanohsete tis opoies upodeijeis sas dinontai.

To foithtiko seminario sunexizei tis drasthriothtes tou. Auth thn Pempth, 30 Okt (8mm, 0207), 0a milhsei h Mariza Zumwnopoulou me 0ema Gewmetrikes Pi0anothtes: To problhma twn tessarwn shmeiwn tou J.J. Sylvester.

Pe, 30 Okt: Kef. 4, § 40-41. O oloklhrwtikos tupos tou Cauchy gia thn paragwgo. Ka0e analutikh sunarthsh exei apeires migadikes (kai merikes) paragwgous. To 0ewrhma tou Morera.

6h omada askhsewn: sel. 151: 4,5,9.
Epishs, gia na ejoikeiw0eite me problhmata opws auto tou 2ou diagwnismatos koitajete tis askhseis 1-6 ths selidas 254 (autes oi teleutaies de 0a ejetastoun sto epomeno diagwnisma).

Tr, 4 Noe: Kef. 4, § 42: Arxh megistou. Thn Pempth 6 Noembriou 0a exoume to 3o 150hmero diagwnisma.

Pe, 6 Noe: Kef. 4, § 43: To 0ewrhma tou Liouville kai to 0emeliwdes 0ewrhma ths Algebras (dhl. oti ka0e mh sta0ero poluwnumo mias migadikhs metablhths mhdenizetai toulaxiston se ena shmeio). Akeraies sunarthseis pou einai fragmenes apo mia dunamh tou |z|.

Foithtiko seminario: Kataskeuh sugkekrimenhs sunarthshs h opoia einai pantou sunexhs kai pou0ena paragwgisimh (omilhths: Giannhs Toulopoulos).

7h omada askhsewn: sel. 159, 1-8.

H katanomh ths ba0mologias twn triwn prwtwn askhsewn exei ws ejhs (arista = 30):
31 31 30 29 29 28 26 25 25 25 24 24 23 22 20 20 19 19 17 15 15 15 14 12 10 10 9 7 7 6 4 4 4 3 0
O mesos oros sto teleutaio diagwnisma htan 10.69 ! (apo 26 grapta).

Foithtiko seminario: H katanomh tou klasmatikou merous tou n0 gia arrhto 0. Ta 0ewrhmata twn L.Kronecker kai H.Weyl.
Omilhtries: Iwanna Mprokou kai Basw Flwrou.
Pempth 13 Noembriou, 8mm, Amfi0eatro S. Pneumatikou.

Pe, 13 Noe: Kef. 5, § 44-46. Seires migadikwn ari0mwn, dunamoseires, seires Taylor analutikwn sunarthsewn kai sugklish autwn sth sunarthsh, paradeigmata anaptugmatwn se dunamoseira.

8h omada askhsewn: sel. 174, 1-11.

PROODOS: Kuriakh, 23 Noembriou, 5mm, Amfi0eatra S.P. kai B.J.
H ejetastea ulh einai mexri kai th selida 174.
Den epitrepetai h xrhsh shmeiwsewn para mono mias selidas A4 sthn opoia epitrepetai na grayete o,ti 0elete (ki apo tis duo pleures).

Tr, 18 Noe: Kef. 5, § 47-48. Seires Laurent kai paradeigmata.
Mhn jexasete oti thn Pempth exoume to taktiko mas 150hmero diagwnisma para to oti thn Kuriakh exoume proodo.

Wra askhsewn: Paraskeuh, 21 Noembriou, 5mm, 0207 h 0201. Elate proetoimasmenoi na kanete erwthseis gia pragmata pou den katalabate.

Pe, 20 Noe: Kef. 5, § 49. Apoluth kai omoiomorfh sugklish dunamoseirwn.

To foithtiko seminario exei ws 0ema auth thn Pempth (8mm, Amf. S.P.) to 0ewrhma ths filias. Omilhtries oi Euaggelia Kalligianakh kai Eirhnh Mauritsakh.

9h omada askhsewn:
sel. 182: 1, 3, 4, 7, 9, 10
sel. 198: 1, 4, 5

O mesos ba0mos sto teleutaio diagwnisma htan 5.27 (apo 22 grapta).
Sas zhth0hke na deijete oti an mia akeraia f=u+iv exei u+2v fragmeno panw tote h f einai sta0erh. H apanthsh einai oti h u+2v=Re( (1-2i)f ) kai afou h (1-2i)f einai akeraia, apo to 0ewrhma pou sas do0hke mesa sthn tajh, h (1-2i)f, ara kai h f, einai sta0erh.

Tr, 25 Noe: Kef. 5, § 50-51. Oloklhrwsh kai paragwgish dunamoseirwn. Oi seires Taylor kai Laurent paristanoun analutikes sunarthseis mesa stous diskous h daktulious sugklishs tous.

Pe, 27 Noe: Kef. 5, § 52. Pollaplasiasmos (alla oxi kai diairesh) dunamoseirwn.
Kef. 6, § 53-55. Oloklhrwtika upoloipa kai h xrhsh tous gia upologismo oloklhrwmatwn. Tajinomhsh memonwmenwn anwmaliwn mias analutikhs sunarthshs.

10h omada askhsewn:
sel. 198: 13, 14.
sel. 210: 1, 2, 3, 4, 6, 7, 8

Foithtiko seminario: To 0ewrhma tou Gamou kai Latinika Tetragwna, Amf. S. Pneumatikou, 8mm, 27/11/97.
Omilhtria: Franziska Berger.

Tr, 2 Dek kai Pe, 4 Dek: Kef. 6, § 56-59. Upologismos oloklhrwtikwn upoloipwn se polous kai xrhsh epikampuliwn oloklhrwmatwn gia ton upologismo orismenwn oloklhrwmatwn panw sthn pragmatikh eu0eia.

11h omada askhsewn:
sel. 217: 1, 3, 6, 7, 12.
sel. 228: 1, 2

Foithtiko seminario: Paixnidia me apeires kinhseis kai nikhthries strathgikes: To paixnidi twn S.Banach kai S.Mazur.
0 201, 8mm, 4/12/97.
Omilhtria: Despoina Nika.

Tr, 9 Dek kai Pe, 11 Dek: Kef. 6, § 60, 63. Upologismos pragmatikwn oloklhrwmatwn (sunexeia). Koitajte opwsdhpote tis askhseis 12, 13 ths sel. 236. Upologismos tou ari0mou rizwn mesa se aplh kampulh kai 0ewrhma Rouche.

Egraya kai moirasa 4 selides me askhseis pou einai epilegmenes etsi wste na sas boh0hsoun sthn proetoimasia sas gia to teliko diagwnisma. Parte ena antigrafo apo to grafeio mou prin fugete gia diakopes.

To teleutaio diagwnisma gia to ma0hma 0a ginei thn epomenh Trith 16 Dek. Thn Pempth 18 Dek. 0a ginei kanonika dialejh gia to ma0hma.

Foithtiko seminario: 0ewrhmata tupou Ramsey.
0 201, 8mm, 11/12/97.
Omilhtria: Katerina Xatzhmalh.

Tr, 16 Dek kai Pe, 18 Dek: Kef. 7, § 64-66. H apeikonish w = 1/z. Pws metasxhmatizontai oi kukloi kai oi eu0eies. H apeikonish sto genikeumeno epipedo (mazi me to shmeio sto apeiro). Apeikoniseis me digrammikous metasxhmatismous.
Luste tis askhseis ths sel. 254 kai tis 1-10 ths sel. 261.

To teliko diagwnisma 0a ginei stis 18 Ianouariou, 1998, 5mm.
H ejetastea ulh einai apo thn arxh tou bibliou mexri kai thn § 66 me thn ejairesh twn § 19, 28, 29, 36, 37, 61, 62.
Όpws kai sthn proodo, sto teliko diagwnisma 0a mporeite na exete mazi sas mono mia selida A4 me o,ti shmeiwseis 0elete.

Einai pi0ano na oristei ena diwro askhsewn prin to diagwnisma.

Askhseis, Pempth, 15 Ian., 5mm, 0-207.

H katanomh telikwn ba0mwn gia osous edwsan to diagwnisma Ianouariou exei ws ejhs:
10 10 9 9 8.5 8.5 8.5 8.5 8 8 7.5 7 6.5 6.5 6.5 6.5 6.5 6.5 6.5 6 6 5.5 5.5 5.5 5.5 5.5 5 5 5 5 5 5 5 5 4 3.5 3 3 2