O ZBEKISTON RESPUBLIKASI OLIY VA O RTA MAXSUS TA`LIM VAZIRLIGI. FUNKSIYALAR NAZARIYASI kafedrasi

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1 O ZBEKISTON RESPUBLIKASI OLIY VA O RTA MAXSUS TA`LIM VAZIRLIGI Al-Xorm oml Urgch Dvlt uverstet MATEMATIKA kultet FUNKSIYALAR NAZARIYASI kedrs Kоmplеks o gruvchl ukslr rs d o quv uslub mjmus 546-Mtemtk t lm o lsh uchu Tuuvchlr: dos. B.I.Abdullеv, ss. Z.SH.Ibrgmоv Urgch

2 . Nmuv dstur.. Ishch dstur. 3. Mustql sh mvulr. 4. Kledr-temtk rej. 5. Retg meo. 6. Test topshrqlr. 7. Nort svollr. 8. M ru mt.

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4 Тузувчи: ЎзМУ доктори Шоимқулов Б.А., доц. Лян Г.М. Тақризчилар: ЎзМУ доценти доц.туйчиев Т.Т., ТДАУ доценти Қ. Рўзиметов Дастур Мирзо Улугбек номидаги Ўзбекистон Миллий Университети Илмий-методик кенгашида кўриб чиқилган ва тавсия қилинган. 6 йил сонли мажлис баёни. Мувофиқлаштирувчи Кенгашнинг қайдномаси билан тасдиқлаш учун тавсия этилган.

5 Кириш Ушбу дастур университетларнинг механика мутахассислиги талабаларига мўлжалланган бўлиб, унда "Комплекс ўзгарувчили функциялар назарияси" нинг асослари берилган. Комплекс ўзгарувчили функциялар назарияси математик анализ фанининг узвий давоми хисобланади ва комплекс қийматли функциялар хоссалари ўрганилади. Маърузалар мазмуни Комплекс сонлар ва улар устида амаллар. Комплекс текислик. Риман сфераси. Комплекс текисликда чизиқлар ва сохалар. Комплекс ўзгарувчили функциялар. Функция лимити ва узлуксизлиги, дифференциалланувчилиги. Коши-Риман шартлари. Голоморф функция тушунчаси. Хосила модули ва аргументининг геометрик маъноси. Конформ акслантиришлар. Каср чизиқли функция ва унинг хоссалари доиравийлик, симметрияни сақлаш хоссалари. Жуковский функцияси, даражали ва Курсаткичли функциялар. Комплекс ўзгарувчили функциядан олинган интеграл ва унинг хоссалари. Коши теоремаси. Бошлангич функция ва унинг мавжудлиги хақидаги теорема. Кошининг интеграл формуласи. Морера теоремаси. Даражали қаторлар. Тейлор қатори, Коши тенгсизлиги ва Лиувилль теоремаси. Ягоналик теоремаси ва голоморф функциянинг ноллари. Лоран қаторлари. Яккаланган махсус нукталар. Яккаланган махсус нуқталар ва Лоран қаторлари орасида богланиш, чегирмалар назарияси. Чегирмалар назариясининг интеграл хисоблашга татбиқлари. Жордан леммаси. Аналитик давом эттириш. Монодроми теоремаси. Лагорифмик чегирма. Аргумент принципи. Сохани сақлаш принципи. Модулнинг максимум принципи. Шварц леммаси. Риман теоремаси. Амалий машгулотлар мавзуси Комплекс сонлар ва улар устида амаллар. Комплекс текисликда соха ва эгри чизиқлар. Коши-Риман шарти. Хосила модули ва аргументнинг геометрик маъноси. Каср чизиқли акслантиришлар. Жуковский функцияси, даражали, Курсаткичли ва тригонометрик функциялар. Интеграл. Даражали қаторлар. Лоран қатори.

6 Яккаланган махсус нуқталар. Чегирмалар ва улар ёрдамида интеграл хисоблаш. Илова: Талаба камида 7-8% ни ўзлаштириши шарт.. Лабаратория иши. Стереографик проекция. Каср чизиқли акслантиришлар. Гармоник функциялар.. Лабаратория иши. Функцияларни Тейлор ва Лоран қаторларига ёйиш. Чегирмалар ва унинг татбиқлари. Илова: Талаба камида 7-8% ни ўзлаштириши шарт. Мустақил иш мавзулари Интегралларни чегарма ёрдамида хисоблаш. Кристофил-Шварц формуласини тадбиқи. Аналитик функцияларни механикада қулланишига оид мисоллар. Коши ва Шварцнинг чегаравий масалани ечиш учун интеграл формулалари. Илова: Талаба камида 7-8% ни ўзлаштириши шарт. Дарсликлар ва ўқув қўлланмалар рўйхати Асосий. Шабат Б.В. Введение в комплексный анализ. Т., М. Наука, 985. Худайберганов Г., Ворисов А.К., Мансуров Х.Т., Комплекс анализ. Т. Университет Садуллаев А., Худойберганов Г., Мансуров Х., Ворисов А., Туйчиев Т. Математик анализ Курсидан мисол ва масалалар тўплами комплекс анализ 3 қисм. "Ўзбекистон" й. 4. Волковўский Л.И., Лунц Г.А., Араманович И.Г. Сборник задач по теории функций комплексного переменного. М., "Наука"975. Қўшимча 5. Сиражиддинов С.Х., Салохитдинов М.С., Максудов Ш. Комплекс ўзгарувчининг функциялари назарияси. Т. "Ўқитувчи" Привалов И.И. Введение в теорию функций комплексного переменного. М. "Наука"977.

7 7. Сидоров Ю.В., Федорюк И.В., Шабунин М.И. Лекции по ТФКП. М., Наука Бицадзе А.В. Основў теории аналитических функций комплексного переменного. М., Наука. 97. Евграфов М.А., Сидоров Ю.В., Федорюк М.В., Шабунин М.И., Бежанов К.А. Сборник задач по теории аналитических функций. М., Наука. 97.

8 TASDIQLAYMAN: Kedr mudr DASTUR BAJARILISHINING KALENDARLI REJASI m ru,lbrtor,mlot mshgulotlr, kurs shlr Fkultet mtemtk Kurs 3 Akdemk guru 3, 3 mtemtk Fg om Kompleks o gruvchl ukslr rs M ru o gd dots. Abdullev B. Mslt v ml mshg ulotor olb bord Tjrb mshg ulotlr olb boruvch Ishlb chqrsh mlot olb bord t r Msh g ulot Turlr Mvu om v ortlr Turlr 3 4 V SЕMЕSTR Kоmplеks sоlr. M ru Kоmplеks sо tushuchs. Kоmplеks sоg ko rshlr. Kоmplеks sо drjg ko trsh v ud ld chqrsh. M ru Kоmplеks sоlr gеоmеtrk tsvrlsh. Kоmplеks tеkslk. Rm sеrs. 3 M ru Kоmplеks tеkslkd chqlr v sоhlr. Kоmplеks sоlr kеtm-kеtlg v ug lmt. Kоmplеks o gruvchlrg ukss. 4 M ru Kоmplеks rgumеtl uks tushuchs.ulrg lmt v ulukslg. 5 M ru Kоmplеks rgumеtl uksg dеrеtslluvchlg. Kоsh-Rm shrtlr 6 M ru Hosl mоdul v rgumеtg gеоmеtrk mоs. Kооrm ksltrshlr. Elеmеtr ukslr v ulr ordmd bjrldg kооrm ksltrshlr. 7 M ru Chql uks v ksr chql uks. 8 M ru Ksr chql uks v ug хоsslr 9 M ru Drjl uks.jukоvsk ukss. M ru Ko rstkchl v lоgrmk ukslr. M ru Trgооmеtrk v gpеrbоlk ukslr. Kоmplеks rgumеtl uksd оlg tеgrl. M ru Itеgrl tushuchs v ug хоsslr. 3 M ru Kоsh tеоrеms. 4 M ru Bоshlg ch uks tushuchs. Kоshg Ajr tl g So t Bjrlglg qd m lumot O v ku 5 So tl r so 6 Oqtuvc h mos 7

9 tеgrl оrmuls. Sоl qtоrlr v drjl qtоrlr. 5 M ru Sоl v uksоl qtоrlr.drjl qtоrlr.abеl tеоrеms. 6 M ru Kоsh-Admr tеоrеms. Gоlоmоr ukslr v ulrg хоsslr. 7 M ru Gоlоmоr ukslrg stlg trtbdg hoslg eg bo lsh. 8 M ru Fuks Tеlоr qtоrg osh. 9 M ru Luvll tеоrеms. Mоrеr tеоrеms. M ru Ygоlk tеоrеms. Vееrshtrss tеоrеms. M ru Gоlоmоr uksg оllr. Lоr qtоrlr. M ru Lоr qtоr. 3 M ru Fuksg Lоr qtоrg olmsg gоlg 4 M ru Mхsus uqtlr v ulrg turlr. Sохоtsk tеоrеms. Chеgrmlr rs. 5 M ru Chеgrmlr v ulr hsоblsh. Chеgrmlr hqd tеоrеmlr. 6 M ru Chеgrmlr rsg b tdbqlr. 7 M ru Jоrd lеmms v ug tdbqlr. JAMI 54 ADABIYOTLAR Г. Худайберганов, А. Ворисов, Х. Мансуров. Комплекс анализ. Тошкент, «Университет», 998. Б. В. Шабат. Введение в комплексный анализ. част. М.Н И. И. Привалов. Введение теории функции комплексного переменного. М. Госиззат физ мат литературы, 977 Yetkch proessor dotset mo

10 O`ZBEKISTON RESPUBLIKASI OLIY VA O`RTA MAXSUS TA LIM VAZIRLIGI URGANCH DAVLAT UNIVERSITETI «TASDIQLAYMAN» Urgch dvlt uverstet o`quv shlr bo`ch prorektor: dots. S. Xo jov l «KОMPLЕKS O ZGARUVCHILI FUNKSIYALAR NAZARIYASI» ISHCHI DASTUR d Blm sоhs: T lm sоhs: Bklvr o lsh: 4 - F 46 Mtemtk v sttstk 546-Mtеmtk

11 URGANCH - l Ushbu shch dstur bklvrtg 546 «Mtеmtk» o lsh uchu O`.R.OO`MTV tomod vgust 6 l 86 rqm ostd tsdqlg v 6 l vgustd BD rqm ostd ro`tg olg muv dstur sosd tulg. Tuuvchlr: Tqrch:.-m..., dots. B. I.Abdullev ss. M.D. Vsov.-m..., dots. R. M. Mdrmov Ishch dstur «Fukslr rs» kеdrsd vgust ld muhоkm qld boоm Kеdr mudr :.-m..., dots.r.m. Mdrmov Ishch dstur mtеmtk kultеtg lm kеgshd vgust ld tsdqld Fkultet dek:.-m..., dots B.I. Abdullev

12 . FANNING AHAMIYATI, UNING MAQSADI VA MASALALARI Kоmplеks o gruvchl ukslr rs mtemtk l g uv dvom hsobld v kоmplеks qmtl ukslrg хоsslr o rgld. Kоmplеks o gruvchl ukslr rs mtеmtkg mtеmtk k tеglmlr, dеrеsl tеglmlr, tеgrl tеglmlr v r kg ko pg msllr еchshd lоhd hmt ksb etd.. TALABALAR BILIM, KO NIKMA VA MALAKALARIGA QO YILADIGAN TALABLAR. Tlb Kоmplеks o gruvchl ukslr rs d qudg tushuchlrg eg bo lsh shrt: kоmplеks sоlr, uks kоmplеks dеrеtslluvchlg, kооrm ksltrshlr, Kоshg tеgrl оrmuls, gоlоmоr uks, gоlоmоr uksg mхsuslklr, Tеlоr v Lоr qtоrlr, qоldqlr v ulrg tdbqlr. Shu bl brg tlb Rm sеrs, elеmеtr kооrm ksltrshlr, Kоshg tеgrl оrmuls, mхsus uqtlr klsskss, Lоr qtоr хоsslr, ltk dvоm qldrsh, sоdd sоhlrg kооrm ksltrshlr tоpsh, Kоsh-Rm shrtlr qo ll blsh, kоtur bo ch tеgrllr hsоblsh mlklrg eg bo lsh kеrk. FANNING TARKIBI Sеmеstrlr bo`ch sotlr tqsmot: Sеmеstr. M ru. Aml Semr Mustql Jm mshg`ulot t lm V Jm «KОMPLЕKS O ZGARUVCHIlI FUNKSIYALAR NAZARIYASI»DAN MA RUZALAR KURSI MA RUZA MAVZULARI Ajr tlg sоt V SЕMЕSTR Kоmplеks sоlr. Kоmplеks sо tushuchs. Kоmplеks sоg ko rshlr. Kоmplеks sо drjg ko trsh v ud ld chqrsh. Kоmplеks sоlr gеоmеtrk tsvrlsh. Kоmplеks tеkslk. Rm sеrs 3 Kоmplеks tеkslkd chqlr v sоhlr. Kоmplеks sоlr kеtmkеtlg v ug lmt.

13 Kоmplеks o gruvchlrg ukss. 4 Kоmplеks rgumеtl uks tushuchs.ulrg lmt v ulukslg. 5 Kоmplеks rgumеtl uksg dеrеtslluvchlg. Kоsh-Rm shrtlr 6 Hosl mоdul v rgumеtg gеоmеtrk mоs. Kооrm ksltrshlr. Elеmеtr ukslr v ulr ordmd bjrldg kооrm ksltrshlr. 7 Chql uks v ksr chql uks. 8 Ksr chql uks v ug хоsslr 9 Drjl uks.jukоvsk ukss. Ko rstkchl v lоgrmk ukslr. Trgооmеtrk v gpеrbоlk ukslr. Kоmplеks rgumеtl uksd оlg tеgrl. Itеgrl tushuchs v ug хоsslr. 3 Kоsh tеоrеms. 4 Bоshlg ch uks tushuchs. Kоshg tеgrl оrmuls. Sоl qtоrlr v drjl qtоrlr. 5 Sоl v uksоl qtоrlr.drjl qtоrlr.abеl tеоrеms. 6 Kоsh-Admr tеоrеms. Gоlоmоr ukslr v ulrg хоsslr. 7 Gоlоmоr ukslrg stlg trtbdg hoslg eg bo lsh. 8 Fuks Tеlоr qtоrg osh 9 Luvll tеоrеms. Mоrеr tеоrеms. Ygоlk tеоrеms Vееrshtrss tеоrеms. Gоlоmоr uksg оllr. Gоlоmоr uksg оllr. Lоr qtоrlr. Lоr qtоr. 3 Fuksg Lоr qtоrg olmsg gоlg 4 Mхsus uqtlr v ulrg turlr. Sохоtsk tеоrеms. Chеgrmlr rs. 5 Chеgrmlr v ulr hsоblsh. Chеgrmlr hqd tеоrеmlr. 6 Chеgrmlr rsg b tdbqlr. 7 Jоrd lеmms v ug tdbqlr. JAMI AMALIY MASHG ULОTLAR RO YХATI AMALIY MASHG ULОT MAVZUSI V SЕMЕSTR Kоmplеks sо. Kоmplеks sоg ko rshlr v gеоmеtrk Ajrt. sоt

14 tsvr. Kоmplеks tеkslkd chq v sоhlr.,5,,5,,3,36 4,4., v, 5,5,65,68,7,7,78,8 [4], I-bob Kоmplеks rgumеtl ukslr v ulrg ulukslg. Fuks dеrеtslluvchlg. Kоsh Rm shrtlr.,3,5,8,,3, 68,73,75,78,94,95,8,3 [4], II-bob Hosl mоdul v rgumеtg gеоmеtrk m оs. Kооrm ksltrshlr. 3 73,75,77,8,88,95,,,5. [4],II-bob Chql v ksr chql ukslr. 4,3,5,7,,,4,35,37,6,64,66,7,74. [4], III-bob Drjl uks. Jukоvsk ukss. 5 4,45,47,49,54,56,7,74,78,8,85. [4], III-bob Ko rstkchl v trgооmеtrk ukslr. 6,4,6,3,38,4,4,97,99,33,35. [4], III-bob Itеgrl tushuchs. Kоsh tеоrеms. 7,3,5,7,,5,8,73,75,78,8,84,88,98,. [4], IV-bob Bоshlg ch uks tushuchs. Kоshg tеgrl оrmuls. 8 84,86,89,98,,5,7,9,,5. [4], IV-bob Drjl qtоrlr. Gоlоmоr uksg оllr. 9 89,9,93,96,98, 78,8,8,84,88,93,95,97. [4],V-bob Lоr qtоr. 394,396,398,4,4,46,4,44,46. [4], V-bob Fuksg kklg mхsus uqtlr. 493,495,54,56,55,57,5,5. [4],V-bob Chеgrmlr v ulr hsоblsh.,3,5,7,9,3,5,7,37,39. [4],VI-bob Itеgrllr chеgrmlr ordmd hsоblsh. 3 86,88,9,9,94,96,98,. [4], VI-bob Aq tegrllr chegrmlr ordmd hsoblsh 4 5,5,56,58,6,63,8,8. [4], VI-bob Argumеt prtsp. Rushе tеоrеms 5 55,57,59,6,63,65,79 [4] Sоhg sqlsh prtsp 6 59, 64,67,6,9,46,47,38,3 [4], III-bob Mоdulg mksmum prtsp. 7 37,374, ,379,383,384,388 [4], V-bob Grmоk ukslr. 8 3,6, 9, 3,3,36,38,4. [4], II-bob JAMI 36

15 5. MUSTAQIL ISHLAR RO YXATI MUSTAQIL ISH MAVZULARI V SЕMЕSTR Kompleks solr. Kompleks solrg ko rshlr v geometrk tsvr. Kompleks tekslk. Rm sers. 3 Kompleks tekslkd chqlr v sohlr. 4 Kompleks solr ketm-ketlg. 5 Kompleks rgumetl ukslr. 6 Fuks lmt v ulukslg 7 Fuksg kompleks dereslluvchlg. 8 Kosh-Rm shrtlr 9 Golomor uks v ug osslr Xosl modul v rgumetg geometrk m olr. Koorm ksltrshlr. Chql ukslr. 3 Ksr chql ukslr. 4 Drjl ukslr.. 5 Jukovsk ukss 6 Ko rstkchl ukslr. 7 Trgoometrk ukslr. 8 Elemetr ukslr ordmd mlg oshrldg koorm ksltrshlr. 9 Itegrl tushuchs.. Kosh teorems Umumlshg Kosh teorems Boshlg ch uks v ug mvjudlg hqdg teorem 3 Koshg tegrl ormuls.. 4 Kosh tpdg tegrl 5 Fuksol qtorlr. 6 Drjl qtorlr. 7 Telor qtor 8 Lor qtorlr. 9 Kosh tegslg v Luvll teorems. 3 Ygolk teorems 3 Golomor uksg ollr. 3 Ykklg msus uqtlr. 33 Ykklg msus uqtlr v Lor qtorlr orsdg bog lsh 34 Sotsk teorems. 35 Chegrmlr v ulr hsoblsh. 36 Itegrllr chegrmlr ordmd hsoblsh. Ajr t. sоt

16 37 Xosms tegrllr chegrmlr ordmd hsoblsh. 38 Argumet prsp. Rushe teorems. 39 Meromor ukslr sodd ksrlr qtorg v cheks ko ptmlrg osh. 4 Grmok ukslr. 4 Drle msls. 4 Kompleks sog ld. 43 Ko p qmtl ukslr. 44 Teskr trgoometrk ukslr. 45 Algebrk teglmlr echsh. 46 Ko p qmtl ukslr odmd mlg oshrldg koorm ksltrshlr. 47 Smmetr prspd odlb koorm ksltrshl mlg oshrsh. JAMI: TALABALAR BILIMINI BAHOLASH. Al-Хоrm оml Urgch Dvlt uverstet Ilm kegshg setbrdg g lshd tsdqlg «Tlblrg blm оrt qlsh v bhоlshg rеtg tm to g rsd»g оmg sоs «Fukslr rs» kеdrsd o`qtlotg kompleks o gruvchl ukslr rs d tlblr blm оrt qlsh v bhоlsh MЕZОNI. Aml mshg ulоt: tlbg drsd msоl shlg uchu - 7bll Bud qudg muv mеоlr оbtg оld: Хulоs v qrоr qbul qlsh bll b Ijоd krl оlsh bll c Mustql mushохd urtsh bll d Amld qo ll оlsh bll e Mохt tushush bll Blsh, tb bеrsh bll j Tsvvurg eg bo lsh bll

17 tlbg drsdg оllg uchu bll 3 tlbg ug vs uchu bll 4 tlbg dtr urtsh uchu bll

18 O bеkstо Rеspublks Оl v o rt mхsus t lm vrlgg еvrl l 34 sоl burug v Al-Хоrm оml UrDU g Tlb mustql sh tshkl qlsh to g rsd g burugg sоs Fukslr rs kеdrs bo ch shlb chqlg «Tlb mustql sh bhоlsh» MЕZОNI Mustql t lm bllk tmd bhоld bll lо 7 85bll хsh 55 7bll o rt Mustql shg o lshtrsh kurstkch 86 bll bl bhоld gr, Mvug оd r tushuchlr mukmml o lshtrls. Mvug d tch tushuchlr охl оls. 3 Drsd utlg mvu bl mmu uvlg ug mtq dvоm ok mmu to ldrsh mоht bls. 4Nr blmlr mustql trd mlotg tdbq ql оls. 5Mvu tхll qlb ud rur hulоslr chqr оls. Mustql shg o lshtrsh kurstkch 7-85bll bl bhоld gr. Mvug оd r blmlr to lq o lshtrls. Mvug оd tch tushuchlr охls. 3Drsd o tlg mvu bl mkur mvu mmu bоglsh jrod hmtg mоlk bulmg хtоlklrg ul qo ls. 4Nr blmlr o qtuvchg ko rstms ordmd mlotg tdbq ql оlgd. 5Msоl v msllr еchg hоld ug оd r blmlr sbоtl оlmgd. 6Ish mqsdg muvоq rsmlshtrlmgd Mustql shg o lshtrsh ko rstkch 55-7bll bll bl bhоld, gr. Mvug оd r blm o lshtrls. Mvug оd tch tushuchlr uqtuvch ordmd охls.

19 3Nr blmlr mlotg qsm tdbq ql оls. 4Drsd o tlg mvu bl mustql bоgl оlms. 5Msоl v msllr еchsh jrod b хtоlklrg ul quls. 6Ish mksdg muvоq rsmlshtrls. Mustql shg o lshtrsh ko rstkch 55bll d pst bll bl bhоld, gr. Mvug оd r blmlr qsm o lshtrs Tch tushuchlr mохt jхtd охls 3Nr blmlr mlotg qsm tdbq ql оls. 4Mustql sh mqsdg muvоq rsmlshtrlms. 5Mvug оd kurstmlr, tеgshl tvsоmlrd оdlg hоld hm mu tоpshrq bjr оlms. Jоr bhоlshd umum mshg`ulоt, sеmr, lbоrtоr, bhоlrg o rtchs оld. ОRALIQ BAHОLASH MЕZОNI bhо bll hsоbd std ml mustql shlrd оlg Nоrt vrtlr t r svоl, t msоld bоrt bo lb, r svоllr 4bll d, msоllr bll d bhоld. Nr svоllr - 4bll gr tlb svоllrg v tushuchlrg jvоb bеr оlms bll b gr tlb kt tushuchlr t rlshd v tеоrеmlr sbоtlshd хtоlklrg o l qo s bll- bll v gr tlb tushuchlr to g r t rlb bеrb, tеоrеmlr tulk sbоtl оlms bll- 3 bll g gr tlb gr tlb tushuchlr to g r t rlb bеrb, tеоrеmlr tulk sbоtl оls 3bll - 4bll b Msоllr - bll gr tlb msоl еchshg umum хrkt klms bll b gr tlb msоl еchshg хrkt klb, tj оto g r hsоbls bll- bll v gr tlb jоd odshg hоld kchk хtоlklrg o l qo s bll-5bll g gr tlb jоd odshg hоld msоl tjs to g r hsоbls 5 bll - bll

20 YAKUNIY BAHОLASH MЕZОNI Nоrt vrtlr t r svоl, 3 t msоld v 5 t sоdd svоllrd bоrt bulb, r svоl 3bll bl, msоllr bll d, sоdd svоllrd хr br bll bl bhоld. Nr svоl - 3bll gr tlb svоllrg v tushuchlrg jvоb bеr оlms bll b gr tlb kt tushuchlr t rlshd v tеоrеmlr sbоtlshd хtоlklrg ul kus bll- bll v gr tlb tushuchlr to g r t rlb bеrb, tеоrеmlr tulk sbоtl оlms bll- bll g gr tlb gr tlb tushuchlr to g r t rlb bеrb, tеоrеmlr tulk sbоtl оls bll - 3bll b Msоllr - bll gr tlb msоl еchshg umum хrkt klms bll b gr tlb msоl еchshg хrkt klb, tj оto g r hsоbls bll- bll v gr tlb jоd odshg hоld kchk хtоlklrg ul kus bll - 5 bll g gr tlb jоd odshg hоld msоl tjs to g r hsоbls 5 bll - bll v Sоdd svоllr-bll tlb t r v tushuchlr tulk, to g r t rl оlms bll- bll b tlb t r v tushuchlr tulk, to g r t rl оls bll-bll Fukslr rs kеdrsd o`qtlotg kompleks o gruvchl ukslr rs d rеtg tm:. Оrlq оrt 3% kоetsеt-,3. Jоr оrt 4% kоetsеt-,4 3. Yku оrt 3% kоetsеt-,3 Mksml bll% M ru Aml mshg`ulоt sеmr mustql shlrg jrtlg sоtlr bll hsоbd.

21 5-sеmеstr uchu jm bl bl хsоbld Tlb оrlq v jоr оrtlrd оlg bllr g ds 77,77 bld pst bo`ls tlb ku оrtg krtlmd v kdеmk qrdоr ҳsоbld. Agr tlbg jоr v оrlq оrtd to`plg bllr g ds 77,77 bl v ud uqоr bo`ls ku оrtg krtld v tlb rеtg jоr, оrlq v ku оrtlrd to`plg bllr g ds bl ҳsоbld. Agr tlb rеtg оrtlrd to`plg bllr g ds, blld pst bo`ls qоqrs,. d 43,4 gch bo`ls qоqrl 43,4 d 73,7 gch хsh 73,7 d gch o`lshtrsh lо ҳsоbld. Qud ushbu bllr tqsmоt jdvl kеltrm. sеmе Mksm Jоr Оrlq Jоr Jоr JОY Tlb rеtg str l bll % Mksm l bl Mksm l bl оrlk mksm оrlk srlsh Srls R JОY 4% 3% l bl bl h bl 7% 7х55% 55% 8,8 6,6 4,4 77,77, R<, qоqrs, R < 43,4 Qоqrl 43,4 R < 73,7 Yхsh 73,7 R lо

22 Asоs 7. ADABIYOTLAR.. Г. Худайберганов, А. Ворисов, Х. Мансуров. Комплекс анализ. Тошкент, «Университет», Б. В. Шабат. Введение в комплексный анализ. част. М.Н Л. И. Волковский, Г. Л. Лунц, И. Г, Абраманович. Сборник задач по теории функции комплексного переменного. 4. А. С. Саъдуллаев ва бошкалар. Математик анализ курсидан мисол ва масалалар тўплами. 3 том. Тошкент. «Ўзбекистон.» й. Qo shmch:. Ш. Максудов, М. Салохиддинов, С. Сирожиддинов. Комплекс ўзгарувчиларнинг функциялари назарияси. Т. «Ўқитувчи» 996. Ю. В. Сидиров, М. В. Федорюк, М. И. Шабунин. Лекции по теории функции комплексного переменного. М. Наука, И. И. Привалов. Введение теории функции комплексного переменного. М. Госиззат физ мат литературы, М. А. Евграфов и др. Сборник задач по теории аналитических функции. М. Наука Г. Худайберганов, А. Ворисов, Х. Мансуров. Комплекс анализ.карши. «Насаф», Бицадзе А.В. Основы теории аналитических функций комплексного переменного. М., Наука. 97.

23 O bеkstо Rеspublks Оl v O rt mхsus t lm vrlgg 3 l 9 vrdg 9-sоl burug bl tsdqlg "Оl t lm to g rsd"g Nоm Al-Хоrm оml UrDU «Tlblrg blm оrt qlsh v bhоlshg rеtg tm to g rsd» g оmg sоs «Fukslr rs» kеdrsd o qtlotg Kompleks o gruvchl ukslr rs d tlblr blm оrt qlsh v bhоlsh MЕZОNI. Aml mshg ulоt: tlbg drsd msоl shlg uchu - 7% Bud qudg muv mеоlr оbtg оld: Хulоs v qrоr qbul qlsh % b Ijоd krl оlsh % c Mustql mushоhd urtsh % d Amld qo ll оlsh % e Mоht tushush % Blsh, tb bеrsh % j Tsvvurg eg bo lsh % tlbg drsdg оllg uchu % 3 tlbg ug vs uchu % 4 tlbg dtr urtsh uchu %. Lbоrtоr sh: tlbg lbоrtоr mshgulоtlrg rs v uslubt o lshtrshg - 4% tlbg lbоrtоr mshgulоt bjrshg 5% 3 mshgulоtd оlg v o quv mехt tоmg % 3. Sеmr: tlbg sеmrdg m rus uchu - 4% tlbg sеmrdg оllg uchu 3% 3 tlbg sеmr mvus ortshd dbotlrd оdlglg uchu 3%.

24 Ubеkstо Rеspublks Оl v o rt mхsus t lm vrlgg еvrl 5 l 34 sоl burug v Al-Хоrm оml UrDU g Tlb mustql sh tshkl qlsh to g rsd g burug g sоs Fukslr rs kеdrs buch shlb chqlg «Tlb mustql sh bhоlsh» MЕZОNI Mustql t lm bllk tmd bhоld % lо 7 85% хsh 55 7% o rt Mustql shg o lshtrsh ko rstkch 86 % bl bhоld gr, Mvug оd r tushuchlr mukmml o lshtrls. Mvug od tch tushuchlr охl оls. 3 Drsd o tlg mvu bl mmu uvlg ug mtq dvоm ok mmu to ldrsh mоht bls. 4 Nr blmlr mustql trd mlotg tdbq ql оls. 5 Mvu thll qlb ud rur hulоslr chqr оls. Mustql shg o lshtrsh ko rstkch 7-85% bl bhоld gr. Mvug оd r blmlr to lq o lshtrls. 3 Mvug оd tch tushuchlr охls. 4 Drsd o tlg mvu bl mkur mvu mmu bоg lsh jrod hmtg mоlk bo lmg хtоlklrg o l qo ls. 5 Nr blmlr o qtuvchg ko rstms ordmd mlotg tdbq ql оlgd. 6 Msоl v msllr еchg hоld ug оd r blmlr sbоtl оlmgd. 7 Ish mqsdg muvоq rsmlshtrlmgd Mustql shg o lshtrsh ko rstkch 55-7% bll bl bhоld, gr. Mvug оd r blm o lshtrls. 3 Mvug оd tch tushuchlr o qtuvch ordmd охls. 4 Nr blmlr mlotg qsm tdbq ql оls. 5 Drsd o tlg mvu bl mustql bоg l оlms. 6 Msоl v msllr еchsh jrod b хtоlklrg o l quls. 7 Ish mqsdg muvоq rsmlshtrls. Mustql shg o lshtrsh ko rstkch 55% d pst bll bl bhоld, gr. Mvug оd r blmlr qsm o lshtrs Tch tushuchlr mоht jхtd охls 3 Nr blmlr mlotg qsm tdbq ql оls. 4 Mustql sh mqsdg muvоq rsmlshtrlms. 5 Mvug оd ko rstmlr, tеgshl tvsоmlrd оdlg hоld hm mu tоpshrq bjr оlms. Jоr bhоlshd umum bhо % hsоbd std ml mshg`ulоt, sеmr, lbоrtоr, mustql shlrd оlg bhоlrg o rtchs оld.

25 ОRALIK BAHОLASH MЕZОNI Nоrt vrtlr t r svоl, t msоld bоrt bo lb, r svоllr 4% d, msоllr % d bhоld. Nr svоllr - 4% gr tlb svоllrg v tushuchlrg jvоb bеr оlms % b gr tlb qt tushuchlr t rlshd v tеоrеmlr sbоtlshd хtоlklrg o l qo s %- % v gr tlb tushuchlr to g r t rlb bеrb, tеоrеmlr to lq sbоtl оlms % - 3 % g gr tlb tushuchlr to g r t rlb bеrb, tеоrеmlr to lq sbоtl оls 3% - 4% b Msоllr - % gr tlb msоl еchshg umum хrkt qlms % b gr tlb msоl еchshg хrkt qlb, tj оto g r hsоbls %- % v gr tlb jоd odshg hоld kchk хtоlklrg o l qo s % - 5 % g gr tlb jоd odshg hоld msоl tjs to g r hsоbls 5 % - % YAKUNIY BAHОLASH MЕZОNI Nоrt vrtlr t r svоl, 3 t msоld v 5 t sоdd svоllrd bоrt bo lb, r svоl 3% bl, msоllr % d, sоdd svоllrd hr br % bl bhоld. Nr svоl - 3% gr tlb svоllrg v tushuchlrg jvоb bеr оlms % b gr tlb qt tushuchlr t rlshd v tеоrеmlr sbоtlshd хtоlklrg o l qo s % - % v gr tlb tushuchlr to g r t rlb bеrb, tеоrеmlr to lq sbоtl оlms % - % g gr tlb gr tlb tushuchlr to g r t rlb bеrb, tеоrеmlr to lq

26 sbоtl оls % - 3% b Msоllr - % gr tlb msоl еchshg umum хrkt qlms % b gr tlb msоl еchshg хrkt qlb, tj оto g r hsоbls %- % v gr tlb jоd odshg hоld kchk хtоlklrg o l qo s % - 5 % g gr tlb jоd odshg hоld msоl tjs to gr hsоbls 5 % - % v Sоdd svоllr-% tlb t r v tushuchlr to lq, to g r t rl оlms %-% b tlb t r v tushuchlr to lq, to g r t rl оls %-% Fukslr rs kеdrsd o qtlotg lrd rеtg tm: 4. Оrlq оrt 35% kоesеt-,35 5. Jоr оrt 5% kоesеt-,5 6. Yku оrt 5% kоesеt-,5 Mksml bll% M ru Aml mshg`ulоt sеmr lbоrtоr mustql shlrg jrtlg sоtlr bll hsоbd. Srlsh bll mksml bllg 55 % mqdоrd

27 Kоmplеks o gruvchl ukslr rs tеst svоllr. kоmplеks sоg hqq qsm tоpg. ½. 3 kоmplеks sо trgооmеtrk ko rshg kеltrg. π π π π π π π π cos s ; cos s ; cos s cos s Kоmplеks tеkslk C d Re,, tеgslk qоtltruvch uqtlr gеоmеtrk o r tоpg. to g r chq v ug o g tmоdg uqtlr to plm. to g r chqd uqоrdg uqtlr to plm. to g r chq v ud pstdg uqtlr to plm. to g r chq v ud chpdg uqtlr to plm. to g r chqd chpdg uqtlr to plm. π e 4. hsоblg. p 5. < p < uks uchu tоpg. p p p p p U, grmоk uksg qo shm grmоk uks tоpg. c c 7. Shud uqtlr to plm tоpgk, shu uqtlrd W ksltrshg cho lsh kоesеt g tеg bo ls. 8. D { < }, sоhg w chql ksltrshdg ks tоpg. w < 4; w < 4 w < w < 4 9. w ksltrshd Re chqg ks tоpg. Im w, Re w, Im w, Im w. D { < } sоhg w ksr-chql ksltrshdg ks tоpg. Im w > Re w > Imw Im w < Im w > Re w. w 4, w, w shrtlr qоtltruvch ksr -chql w ksltrsh tоpg.

28 w 4 w 4 w w 4. Re > to g r chqg w ksltrsh ordmdg ks tоpg. Re w Im w Re w Im w 4 4 Re 4 Im w w Re w Im w 4 3. Kоmplеks tеkslkd uqt uchu Re chqq sbt smmеtrk bo lg uqt tоpg. 4. { <} dоr { >} sоhg ksltruvch kооrm ksltrsh tоpg. w w w w 5. w Juqovsk ukss ordmd lg ks tоpg. 9 9 u v, w u v 6 6 u v, w u v u v, w u v 6 6 u v, w u v w Juqovsk ukss ordmd > sоhg ks tоpg. 6 6 u v, w u v 6 6 u v, w u v u v, w u v 6 6 u v, w u v Re chqg w e ksltrshdg ks tоpg. w e w w e w e e 8. π < Im < sоhg w e ksltrshdg ks tоpg. w, w, w ; w ; π π 9. D { < Re < } sоh w tg ksltrshdg ks tоpg. 4 4 w < w < w < w <. D { Im < π, Re > } sоh bеrlg w sh ksltrshdg ks tоpg. w ;], w [ ; ] w ;], w [ ; ] w ;], w [; ] w [ ;], w [; ]

29 . e kоmplеks sоg mоdul v rgumеt tоpg. e e rg e e e rge e e rg e e e rg e. e ukssg π uqtdg qmt tоpg. 3. s π kоmplеks sоg hqq qsm tоpg. 4. t t, uks bl qlg egr chq tоpg. {, 3 } -to g r chq kеsms. {, } -to g r chq kеsms. {, 3 } -to g r chq kеsms. { 3, } -to g r chq kеsms. d 5. tеgrl hsоblg, bu еrd Г chq mrk uqtg Г jоlshg хtor l. π π π π d 6. tеgrl hsоblg. 7. cos uks uqtg trоd drjl qtоrg og.! 8. 3!!! uks uqtg trоd drjl qtоrg og Lоr qtоrg qlshsh sоhs qlg. 4 < < 4 < < 4 4 < < < < 4 e 3. uks uqt trоd Lоr qtоrg og!!!

30 π 3. cos uksg muhm mхsus uqts tоpg. 3. uksg chеkl qutb mхsus uqtsdg qоldg tоpg. C C C 4 C uks qutb mхsus uqtlr tоpg. s π e d 34. Kоsh tеоrеmsg sоslb, tеgrl hsоblg. 3 e π e π e π 35. e π 4 d 3 e tеgrl qоldqlr ordmd hsоblg. π π π 3 π π cos d 36. < p < tеgrl hsоblg. p s p kоmplеks sоg hqq qsm tоpg sо trgооmеtrk ko rshg kеltrg. 4π 4π π π π π 4π 4π cos s cos s cos s cos s Kоmplеks tеkslk C d < Im < shrt qоtltruvch uqtlr gеоmеtrk o r tоpg. v to g r chq hmd ulr оrsdg uqtlr to plm to g r chqd uqоrdg uqtlr to plm v to g r chqlr оrsdg uqtlr to plm to g r chqd uqоrdg uqtlr to plm 4. e hsоblg. cos s cos s p 4. e e cos s e cos s, < p < uks uchu p p e p 4 p e p e p tоpg p e 4 p

31 4. U, grmоk uksg E { < } sоhd ko shm bo lg grmоk uks tоpg. C C C C Shud uqtlr to plm tоpgk, shu uqtlrd w ksltrshg cho lsh kоesеt g tеg bo ls D {Re < } sоhg w chql uks ordmdg ks tоpg. Re w Im w < 3 Re w Im w < Re w Im w < Re w Im w < 45. Im to plm w ksltrsh ordmdg ks tоpg. Re w Re w Im w Im w 46. D { > } sоhg ksr chg w ksltrsh ordmdg ks tоpg. Re w < Im w < Im w > Re w > 47. w, w, w 4 shrtlr qоtltruvch ksr chql w ksltrsh tоpg. w w w w 48. Im > to plmg w ksltrshdg ks tоpg. Rew w Im Re w Im w Re w Im w Kоmplеks tеkslkd uqt uchu Im chqq sbt smmеtrk bo lg uqt tоpg. 5. Qudg ksltrshlrd qs br { > } sоh {Im w > } uqоr rm tеkslkk kооrm ksltrd? w w w w 5. to plmg w ksltrshdg ks tоpg. 6 6 u v, w u v 6 6 u v, w u v Imw Re w

32 9 9 u v, w u v 9 u v, w u v < to plmg w ksltrshdg ks tоpg. 6 6 u v >, w u v 6 u v >, w u v u v >, w u v 6 u v >, w u v π 53. Im to plmg w e ksltrshdg ks tоpg. rg w π 3π π rg w rg w π rg w π π 54. < Im < to plm w e ksltrshdg ks tоpg. Re w > Re w > Re w > Re w > 3 π 55. D { Im < } sоh w th ksltrsh ordmdg ks tоpg. 4 w < w < w < w < 56. D { < Re < π, Im > } sоh w s ksltrsh ordmd ks tоpg. w [ ; ], w [; ], w [ ;] w [; ], w [ ;] w [; ] w [ ;], w [; ] e kоmplеks sоg mоdul v rgumеt tоpg e e rg e π 3 e e rg e π e e rg e π 3 e e rg e π e uksg π uqtdg qmt tоpg. 59. s π kоmplеks sоg hqq qsm tоpg. 4 3 sh sh sh sh 3 d 6. tеgrl hsоblg, bu еrd Г chq 3 ld bоrt. Г 6 π 8π 8 π π 4

33 Kоmplеks o gruvchl ukslr rs d оrt svоllr. Kоmplеks sо tushuchs. Kоmplеks sоg ko rshlr.. Kоmplеks sо drjg ko trsh v ud ld chqrsh. 3. Kоmplеks sоlr gеоmеtrk tsvrlsh. Kоmplеks tеkslk. 4. Rm sеrs. Kоmplеks tеkslkd chqlr v sоhlr. 5. Kоmplеks sоlr kеtm-kеtlg v ug lmt 6. Kоmplеks rgumеtl uks tushuchs.ulrg lmt v ulukslg. 7. Kоmplеks rgumеtl uksg dеrеslluvchlg. Kоsh-Rm shrtlr 8. Hosl mоdul v rgumеtg gеоmеtrk mоs. 9. Kооrm ksltrshlr.. Chql uks v ksr chql uks. Ksr chql uks v ug хоsslr. Drjl uks.juqovsk ukss 3. Ko rstkchl v lоgrmk ukslr. 4. Trgооmеtrk v gpеrbоlk ukslr. 5. Itеgrl tushuchs v ug хоsslr 6. Kоsh tеоrеms. 7. Bоshlg ch uks tushuchs. 8. Kоshg tеgrl оrmuls 9. Sоl v uksоl qtоrlr Drjl qtоrlr.abеl tеоrеms. Kоsh-Admr tеоrеms.tеlоr qtоr Gоlоmоr ukslrg stlg trtbdg hoslg eg bo lsh. 3. Fuks Tеlоr qtоrg osh. 4. Luvll tеоrеms. 5. Mоrеr tеоrеms. 6. Ygоlk tеоrеms. 7. Vееrshtrss tеоrеms. Gоlоmоr uksg оllr 8. Lоr ktоr. 9 Fuksg Lоr qtоrg olmsg gоlg. 3. Mхsus uqtlr v ulrg turlr. Sохоsk tеоrеms. 3. Chеgrmlr v ulr hsоblsh. 3. Chеgrmlr hqd tеоrеmlr. 33. Chеgrmlr rsg b tdbqlr. 34. Jоrd lеmms v ug tdbqlr.

34 -M ru. Kompleks o gruvch ukslr rs d Kоmplеks sо tushuchs. Kоmplеks sоg ko rshlr. Kоmplеks sо drjg ko trsh v ud ld chqrsh mvusdg m ru mshg ulot drs shlms F: Kompleks o gruvch ukslr rs. Mvu: Kоmplеks sо tushuchs. Kоmplеks sоg ko rshlr. Kоmplеks sо drjg ko trsh v ud ld chqrsh. Mqsd: Tlblrg kompleks o gruvch ukslr rs sos tushuchlrd br bo lg kompleks so tushuchs, kompleks sog ko rshlr, kompleks so drjg ko trsh v ulrd ld chqrsh hqd to l tsvvur hosl qlsh qoblt shklltrsh, kompleks so bl bog lq msllr tushub, mustql, jod krlshg ko kmlr trkb toptrsh. Mshg ulot metod: Utrlg m ru, «Aql hujum» Mshg ulot tur: M ru Mshg ulot jho: Proektor, m ru preetss, Sldlr, trqtm mt mterllr, vtm, Mrkerlr Ajrtlg vqt: 8 mut Mshg ulot borsh: Ish bosqchlr v mmu - bosqch. Torlov 5 mut -bоsqch. Krsh O qtuvch olt Tlblr olt Kutlotg tjlr Mvu e lo qld Aql hujum metodg moht qlshtrd Oldg o tlg mvulr bo ch svollr ob qo ld. Yg mvu mt trqtld. Yg mvu tdg tushuchlr bo ch svollr berld Thll qlsh lom bo lg Aql Tlblr mustql rvshd svollr thll qld v jvoblrg torlshd. Mummоg mоht tushub Tlblrd kеltrldg o quv хbоrоtlrg sbt kuchl mоtvts shklltrsh. Tlblrd «Aql hujum»

35 Aql hujum Yg mvu tdg tushuchlr bo ch 5 mut hujum teologs mld qo llsh Ug o quvchlr tоmоd o quv хbоrоtlr o lshtrb оlsh v o quv mtlr ustd muvqtl shlsh jrodg hmt tushutrb bеrd. оldlr, rur hоllrd o qtuvchg svоllr bl murоjt qlshlrg ershld. tехоlоgs ml оltd muvqtl qo llsh mlklr trkb tоptrsh, ulrd hmkоrlkd jоd mехt qlsh, mustqllk, ksb mхоrt shklltrsh. 3-bоsqch. M ru Yg mvu bo 5 m Utrlg m ru teologs bo ch Yg mvu - qsm bo etld. Kompleks so tushuchs ochb berld, msollr keltrld. Tlblr dqqt bl eshtd. Thll qlb bord. Yg tushuchg eg bo lshd. 4-bоsqch. Aql hujum 5 mut Yg mvu tdg tushuchlr bo ch svollr berld Bud tch tushuchlr oldg bosqchdg m rud hm berld. Fkrlrg tlblr tоmоd krtlsh tshkl qld, ulr оlt kutd, rur хоllrd to g r o lsh bеrb bоrd. Mvu bo ch, Kоmplеks sо tushuchs. Kоmplеks sоg ko rshlr. Kоmplеks sо drjg ko trsh v ud ld chqrsh tushuchlr br bor tkrorlb musthkmlb olshd. Tlblr mummо o rgsh jrod оllg оshrsh, ulrg mustql krlsh qоbltlr rvоjltrsh, duoqrsh kеgotrsh, hmkоrlkd shlshg o rgtsh. Ksb оltg mukmml torlsh. Mumо tlq jro kutd v tlblr tоmоd o quv хbоrоtg q

36 drjd mukmml o lshtrlsh хususd q хulоslr qlb bоrd. 5-bоsqch. M ru Yg mvu - qsm bo etld Yg mvu - qsm bo etld Kompleks so ko rsh, ochb berld, msollr keltrld. Tlblr dqqt bl eshtd. Thll qlb bord. Yg tushuchg eg bo lshd. mutgch 6-bоsqch. Aql hujum 5 mut Yg mvu tdg tushuchlr bo ch svollr berld Bud tch tushuchlr oldg bosqchdg m rud hm berld. Mvu bo ch Tlblrd mustqllk, оllk, jоdkоrlk, ms ultllk, vtprvrlk, tshbbuskоrlk hss trkb tоptrsh v rvоjltrsh. Fkrlrg tlblr tоmоd krtlsh tshkl qld, ulr оlt kutd, rur хоllrd to g r o lsh bеrb bоrd. Mumо tlq jro kutd v tlblr tоmоd o quv хbоrоtg q drjd mukmml o lshtrlsh хususd q хulоslr qlb bоrd. Tlblr mummо o rgsh jrod оllg оshrsh, ulrg mustql krlsh qоbltlr rvоjltrsh, duoqrsh kеgotrsh, hmkоrlkd shlshg o rgtsh. Ksb оltg mukmml torlsh 7-bоsqch. M ru Yg mvu 3- qsm bo etld 5 Yg mvu 3- qsm bo etld. Kompleks so drjg ko trsh v ulrd ld chqrsh Tlblr dqqt bl eshtd. Thll qlb bord. Yg tushuchg eg bo lshd.

37 mutgch 8-bоsqch. Yku qsm mutgch Tlblr tomod berlg svollrg jvob berd. Tlblr tоmоd mlg оshrlg оlt thll qld, mshg ulоtg ku sd. O оltlrg bhо bеrdlr, eg mkbul kr vrtlr ko rstdlr, krlr hmоs usd qo shmchlr, qlklr krtdlr v ulr sbоtldlr. Tlblrd mustqllk, оllk, jоdkоrlk, ms ultllk, vtprvrlk, tshbbuskоrlk hss trkb tоptrsh v rvоjltrsh.

38 Mvu: Kоmplеks sо tushuchs. Kоmplеks sоg ko rshlr. Kоmplеks sо drjg ko trsh v ud ld chqrsh. -qsm. Kоmplеks sо tushuchs. Tеkslkd Dеkrt kооrdtlr sstеms bеrlg bo ls. Abssslr o qd jоlshg uqtlr to plm R, оrdtlr o qd jоlshg uqtlr to plm R оrql bеlgllk. Iхtor R, R хqq sоlrd, jutlk hоsl qlm. Bud, gr bo ls,, dеb qrm. Bud jutlklrd tshkl tоpg C{, : R, R} to plmd rmеtk mllr krtlsh mumk. Agr, C,, C jutlklr uchu, bo ls, bu jutlklr o rо tеg dеld v,, kb bеlgld., Cва, C jutlklrg g ds qudgch qld:,,,, Cва, C jutlklrg rms hm qudgch qld:,,, Ko ptrsh v bo lsh mllr hm mоs rvshd qudgch qld.,,, Shud qlb, C to plm elеmеtlr ustd to rt ml qo shsh, rsh, ko ptrsh v bo lsh mllr krtld. Bu mllr qudg хоsslrg eg:. Kоmmuttvlk. Assоstvlk 3. Dstrbutvlk Yuqоrd kеltrlg C {, : R, R} To plm elеmеtlr ustd rmеtk mllrg bjrlsh v ulrg 3 хоsslrg eg eklg, tb rvshd C to plm elеmеtlr sо dеb qrsh mkо ug kеltrd. Оdtd, C to plm elеmеt, jutlk kоmplеks sо dеld v u btt хr bl bеlgld., Dеmk, C to plm kоmplеks sоlr to plm оdlr ek. M lumk, R uchu, bu es хqq sо kоmplеks sоg хusus hоl ek bldrd. -qsm. Kompleks sog ko rshlr.. Algеbrk ko rsh.

39 , jutlk оlb, u bl bеlglm, v bu bеlg mvhum brlk dеb tm. bo ld. Hqqt hm,,, bеlgs ordmd, kоmplеks sо lgеbrk shkld ko rshd osh mumk. Chuk,,,,,, bo ls, kоmplеks sоg хqq qsm dеld v Re kb bеlgld. kоmplеks sоg mvhum qsm dеld v Im kb bеlgld. kоmplеks sо bеrlg bo ls, kоmplеks sо u qo shms dеld v оrql bеlgld:. Qudg tеglklr o rldr: Esltm: t,,..., kоmplеks sоlrg g ds hmd ko ptms uqоrdgdеk krtld v ulr uchu mоs хоsslr hmd tеglklr o rl bo ld. Jumld,......,......, bo ld.. Trgоmеtrk ko rsh. Iхtor kоmplеks sо оllk. Tеkslkd, kооrdtlr v bo lg M, uqt qrm.

40 M lumk, OM shu M uqtg rdus-vеktоr dеld. Bu rdus-vеktоrg uulg r, ug O o q bl tshkl etg burchg ϕ bo ls. Chmd tsvrlg OMB to g r burchg uch burchkd tоpm: r cosϕ, r sϕ ϕ < π Ud ko rshdg kоmplеks sо qudgch r cosϕ r sϕ r cosϕ sϕ оdld. Оdtd kоmplеks sоg bu оds ug trgоmеtrk ko rsh dеld. Bud r musbt sо kоmplеks sоg mоdul dеlb, kb bеlgld: r, ϕ burchk es kоmplеks sоg rgumеt dеlb, rg kb bеlgld:. ϕ rg OMB d, Pgоr tеоrеmsg ko r r r < 3 hmd tg ϕ, ϕ rctg 4 bo lsh tоpm. Dеmk, kоmplеks sоg mоdul 3 оrmul, rgumеt es 4 оrmul ordmd tоpld. Msоl.. 3 kоmplеks sоg mоdul hmd rgumеt tоpg. Bud, 3 bo ld. 3 v 4 g ko r 3 tg ϕ. Ushbu 3, r π ϕ bo ld. 3 3 kоmplеks sо trgomеtrk ko rshd оdlg. 3 Bud, bo lb 3 3 r 3 π ϕ rg rctg rctg 3 3 u hоld оrmulg ko r bеrlg kоmplеks sо qudg

41 3 π cos s π 3 3 trgomеtrk ko rshg eg bo ld. 3. Ko rstkchl ko rsh. Fr qllk, C ϕ ϕ < π bo ls. Ud bu kompleks so sog modul r < cosϕ sϕ r r rgumet es trgometrk ko rshg eg bo ld. Kompleks l kursd muhm bo lg qudg ϕ e cosϕ sϕ 5 Elеr оrmulsd bu оrmul kеg m rud sbоtlm оdlsk, kоmplеks sоg ushbu ϕ re оdsg kеlm. Bu kоmplеks sоg ko rstkchl оds dеld. ϕ ϕ re, re bo ls, u hоld r r r r ϕ ϕ e ϕ ϕ e Yuqordg muоsbtlrd qudg muоsbtlr kеlb chqd:. rg rg rg. rg rg rg. 3-qsm. Kоmplеks sо drjg ko trsh v ud ld chqrsh. Atlk,,..., kompleks solr berl bo ls. kkt kompleks solr ko ptms sgr bu t kompleks solr ko ptms. ϕ ϕ... ϕ... r r... r e bo ld. Bud husus k ϕ k r e, k,,...,. k...

42 bo ls, tеglk ushbu ϕ r e ko rshg eg bo lb, bu kоmplеks sоg -drjs dеld. Rvshk, ϕ r e r cos ϕ s ϕ dеmk, r cos ϕ s ϕ 3 оdtd 3 оrmuls Muvr оrmuls dеld. Atlk, C kоmplеks sо v tlg N sоlr bеrlg bo ls. Ushbu ξ 4 tеglk qоtltruvch ξ kоmplеks sо kоmplеks sоd оlg -drjl ld dеld v u kb bеlgld: ξ bеrlg kоmplеks sо qudg r cos ϕ s ϕ 5 trgоmеtrk ko rshd bo ls. ξ kоmplеks sо ushbu ξ ρ cos ψ s ψ 6 ko rshd lm. Ud 4, 5, v 6 muosbtlrg ko r bo ld. Ed. [ ρ сos ψ s ψ ] r cos ϕ s ϕ ormul e tborg olb, qudg ρ tеglkk kеlm. Ud [ ρ сos ψ s ψ ] ρ cos ψ s ψ cos ψ s ψ r cos ϕ s ϕ ρ cos ψ r cos ϕ 7 ρ s ψ r s ϕ bo lsh kеlb chqd. Bu tеglklr kvdrtg ko trb, so g ulr хdlb qo shb tоpm: ρ ψ s ψ cos r cos ϕ s ϕ ρ r ρ Tоplg ρ g qmt 7 tеglklrdg ρ g o rg qo sk, Ushbu tеglmlr hоsl bo ld. Agr m lum bo lg cos ψ cos ϕ s ψ s ϕ r

43 tеglklr e tbоrg оlsk, ud bo lsh tоpm. ψ α k π cos α α k π s α cos s ϕ kπ ψ ϕ kπ k, ±, ±,... Dеmk lotg ξ ρ cos ψ s ψ rgumеt es bo lr ek. Dеmk, k, bo ld. kоmplеks sоg mоdul ξ ρ ψ r ϕ kπ ϕ kπ ϕ kπ r cos s 8 Tch bоrlr. Kоmplеks sо, kоmplеks sоg ko rshlr, kоmplеks sоg mоdul v rgumеt, Elеr оrmuls, kоmplеks sоg хqq qsm, kоmplеks sоg mvhum qsm, qo shm kоmplеks sо. Kоmplеks sо drjs, kоmplеks sо ld, Muvr оrmuls, O -o tеkshrsh uchu svоllr:. Kоmplеks sо t r tg.. Kоmplеks sоg lgеbrk ko rsh tg. 3. Kоmplеks sоg trgmеtrk ko rsh tg. 4. Kоmplеks sоg ko rstkchl ko rsh tg. 5. Kоmplеks sо drjg ko trsh dеgd m tushus? 6. Kоmplеks sоd ld chqrsh dеgd m tushus? Адабиётлар: [3] 5-4 бетлар, [4] 3-5 бетлар, [5] 3-8 бетлар, [6] 5-4 бетлар, [7] 7-7 бетлар.

44 -M ru. Kompleks o gruvch ukslr rs d Kompleks so geometrk tsvrlsh. Kompleks tekslk. Rm sers mvusdg m ru mshg ulot drs shlms F: Kompleks o gruvch ukslr rs. Mvu: Kompleks so geometrk tsvrlsh. Kompleks tekslk. Rm sers. Mqsd: Tlblrg kompleks o gruvch ukslr rs sos tushuchlrd br bo lg Kompleks so geometrk tsvrlsh. Kompleks tekslk. Rm sers to l tsvvur hosl qlsh qoblt shklltrsh, Kompleks so geometrk tsvrlsh bl bog lq msllr tushub, mustql, jod krlshg ko kmlr trkb toptrsh. Mshg ulot metod: Utrlg m ru, «Aql hujum» Mshg ulot tur: M ru Mshg ulot jho: Proektor, m ru preetss, Sldlr, trqtm mt mterllr, vtm, Mrkerlr Ajrtlg vqt: 8 mut Mshg ulot borsh: Ish O qtuvch olt bosqchlr v mmu - bosqch. Torlov 5 mut Mvu e lo qld Aql hujum metodg moht qlshtrd Oldg o tlg mvulr bo ch svollr ob qo ld. Yg mvu mt trqtld. Yg mvu tdg tushuchlr bo ch svollr berld Tlblr olt Tlblr mustql rvshd svollr thll qld v jvoblrg torlshd. Kutlotg tjlr Tlblrd kеltrldg o quv хbоrоtlrg sbt kuchl mоtvts shklltrsh. -bоsqch. Krsh Aql hujum Yg Thll qlsh lom bo lg Aql hujum teologs mld qo llsh Ug o quvchlr tоmоd o quv хbоrоtlr Mummоg mоht tushub оldlr, rur hоllrd o qtuvchg Tlblrd «Aql hujum» tехоlоgs ml оltd muvqtl qo llsh

45 mvu tdg tushuchlr bo ch 5 mut o lshtrb оlsh v o quv mtlr ustd muvqtl shlsh jrodg hmt tushutrb bеrd. svоllr bl murоjt qlshlrg ershld. mlklr trkb tоptrsh, ulrd hmkоrlkd jоd mехt qlsh, mustqllk, ksb mхоrt shklltrsh. 3-bоsqch. M ru Yg mvu bo 5 m Utrlg m ru teologs bo ch g mvug - qsm bo etld. Kompleks so geometrk tsvrlsh ochb berld, msollr keltrld. Tlblr dqqt bl eshtd. Thll qlb bord. Yg tushuchg eg bo lshd. 4-bоsqch. Aql hujum 5 mut Yg mvu tdg tushuchlr bo ch svollr berld Bud tch tushuchlr oldg bosqchdg m rud hm berld. Fkrlrg tlblr tоmоd krtlsh tshkl qld, ulr оlt kutd, rur хоllrd to g r o lsh bеrb bоrd. Mumо tlq jro kutd v tlblr tоmоd o quv хbоrоtg q drjd mukmml o lshtrlsh хususd q хulоslr qlb bоrd. Mvu bo ch kompleks so geometrk tsvrlsh. Kompleks tekslk. Rm sers tushuchlr br bor tkrorlb musthkmlb olshd. Tlblr mummо o rgsh jrod оllg оshrsh, ulrg mustql krlsh qоbltlr rvоjltrsh, duoqrsh kеgotrsh, hmkоrlkd shlshg o rgtsh. Ksb оltg mukmml torlsh. 5-bоsqch. M ru Yg mvu - qsm bo etld Yg mvu - qsm bo etld Kompleks tekslk, mvus ochb berld, msollr keltrld. Tlblr dqqt bl eshtd. Thll qlb bord. Yg tushuchg eg bo lshd.

46 mutgch 6-bоsqch. Aql hujum 5 mut Yg mvu t dg tushuchlr bo ch svollr berld. Bud tch tushuchlr oldg bosqchdg m rud hm berld. Fkrlrg tlblr tоmоd krtlsh tshkl qld, ulr оlt kutd, rur хоllrd to g r o lsh bеrb bоrd. Mummо tlq jro kutd v tlblr tоmоd o quv хbоrоtg q drjd mukmml o lshtrlsh хususd q хulоslr qlb bоrd. Mvu bo ch, kompleks tekslk, Rm sers Tlblrd mustqllk, оllk, jоdkоrlk, ms ultllk, vtprvrlk, tshbbuskоrlk hss trkb tоptrsh v rvоjltrsh. Tlblr mummо o rgsh jrod оllg оshrsh, ulrg mustql krlsh qоbltlr rvоjltrsh, duoqrsh kеgotrsh, hmkоrlkd shlshg o rgtsh. Ksb оltg mukmml torlsh 7-bоsqch. M ru Yg mvu 3- qsm bo etld Yg mvu 3- qsm bo etld. Rm sers, ochb berld, msollr keltrld. Tlblr dqqt bl eshtd. Thll qlb bord. Yg tushuchg eg bo lshd. 5 mutgch 8-bоsqch. Yku qsm mutgch Tlblr tomod berlg svollrg jvob berd. Tlblr tоmоd mlg оshrlg оlt thll qld, mshg ulоtg ku sd. O оltlrg bhо bеrdlr, eg mkbul kr vrtlr kurstdlr, Fkrlr хmоs usd Tlblrd mustqllk, оllk, jоdkоrlk, ms ultllk, vtprvrlk, tshbbuskоrlk hss trkb tоptrsh v

47 kushmchlr, qlklr krtdlr v ulr sbоtldlr. rvоjltrsh. Mvu: Kompleks so geometrk tsvrlsh. Kompleks tekslk. Rm sers. -qsm. Kompleks so geometrk tsvrlsh. Iхtor C kоmplеks sо оllk. Bu, jutlk bl qls:, R,. R Tеkslkd bssss g, оrdts es g tеg bo lg uqt kоmplеks sоg gеоmеtrk tsvr dеld. Хusus,, ko rshdg kоmplеks sоg gеоmеtrk tsvr bssslr o qd jоlshg uqt bo ld., ko rshdg kоmplеks sоg gеоmеtrk tsvr es оrdtlr o qd jоlshg uqt bo ld. Abssslr o`q хqq o`q, оrdlr o`k es mvхum o`k dеb urtld. Dеmk, C to plmd оlg hr br kоmplеks sоg tеkslkd, bu sо gеоmеtrk tsvrlоvch btt uqt mоs kеlr ek. Ed tеkslkd хtor uqt оllk. Ug bssss, оrdts bo ls. Bu sоlrd tulg, jutlk btt kоmplеks sо qld. Оlg uqtg shu kоmplеks sо mоs qo sh bl tеkslkdg hr br uqtg btt kоmplеks sо mоs kеlsh qlm. Shud qlb, C bl tеkslkdg brch uqtlr to plm оrsd o rо br qmtl mоslk o rtld. Bu es C to plmg gеоmеtrk tsvr tеkslk dеb

48 qrsh mkо bеrd. Bud tеkslk kоmplеks sоlr tekslg dеb tld v u hm S kb bеlgld. 3 Kоmplеks sо bоshqch hm tsvrlsh mumk. Bug uchu R оd Dеkrt kооrdtlr sstеms оlb, ud mrk,, uqtd, rdus g tеg bo lg ushbu 3 S ξ, η, ζ R ; ξ η ζ 4 sеr qrm. Rvshk, bu sеr O ξ o q,, hmd N,, uqtlrd kеsd. N,, uqt shmоl qutb dеb tm. v o qlr mоs rvshd ξ v η o qlrg ustm-ust qo m. O kоmplеks tеkslkdg uqt bl shmоl N kutb ur ordmd tutshtrm. Ntjd Ν ur S sеr qddr Z uqtd kеsd. B Z mоslkk eg bo lm. Shud qlb, kоmplеks tеkslkdg brch uqtlr to plm bl sеrg S \ { Ν } uqtlr to plm o rо br qmtl mоslkd bo lr ek. Kоmplеks tеkslkdg uqt kооrdt bоshd uоqlsh bоrg sr ug sеrdg tsvr N uqtg qlsh bоrd. Agr kоmplеks tеkslkd uqt оls v u sеrdg N g mоs kеluvch uqt dеb qrls, ud C C { } to plm bl S sеr uqtlrd bоrt to plm o rо br kml mоslkd bo ld. S ~ Bu mоslk kоmplеks tеkslkd stеrеоgrk prоеkss dеld. Оdtd C to plm kеgtrlg kоmplеks tеkslk, S srt es Rm sеrs dеb tld. Sеrdg uqt kооrdtlr bl kоmplеks tеkslkdg mоs uqtlr kооrdtlr оrsdg bоg lsh tоplk. Rvshk, N,, hmd C uqtlr оrql o tuvch to g r chq tеglms qudgch C

49 ξ t η t ζ t bo ld, bud t d N uqt, t d uqt hоsl bo ld. Kоmplеks tеkslkdg uqt kооrdtlr m lum bo lgd Z uqt kооrdtlr ξ, η, ζ lr qudgch qld. M lumk ξ, η, ζ uqt hm S sеrd otd. Shu e tbоrg оlb, ξ t, η t, ζ t lr sеr tеglms dg ξ η ζ, ζ lrg o rg qo b tоpm. ξ η, t t Dеmk, ξ t t t 4 4 t t η ζ 3 bo ld. Agr ξ, η, ζ lr m lum bo ls v lr qudgch qld: to g r chq tеglmsd t bo lsh tоpb, u g brch kkt tеglmsdg t o rg qo m: ξ ζ η ζ bulrd ξ η, ζ ζ bo lsh kеlb chqd. B C d t mеtrk krtm. Оdd Еvkld mеtrks:, C uqtlr оrsdg msо dеld. Sеrk mеtrk:, C uchu ρ ζ, 4

50 bu оrmul C g osh mumk. ρ, Kоmplеks tеkslkd chqlr v sоhlr. Kоmplеks sоlr kеtm-kеtlg v ug lmt. Ktоrlr. Egr chq tеkslkd uqtg uluks hrkt tjsd qоldrg dеb qrsh mumk. Xrktdg uqtg kооrdtlr х v dеls, rvshk ulr brоr t o gruvchg uluks ukslr bo ld: t t α t β A ptd, jutlk kоmplеks sо оdlg sbbl, u ko rshd osh mumk. Ntjd, t t t bo ld. Dеmk, t α t β uks [ α, β ] sеgmеt kоmplеks tеkslk uqtlrg ksltrd v bu uqtlr to plm es kоmplеks tеkslkd egr chq оdlr ek. Bud α egr chqg bоshlg ch uqts, β es egr chqg охrg uqts bo ld. Agr α β bo ls, bud egr chq opq dеld. Agr t egr chqd t o gruvchg kkt turl t v t t t qmtlrg mоs kеldg t v t uqtlr hm turlch bo ls, u hоld egr chq Jоrd chg dеld. Agr t v t ukslr [, b] sеgmеtd uluks dеrеslluvch bo lb, t t t shrt qоtltrs, t t t egr chq sllq egr chq dеld. -3 qsmlr. Kоmplеks tеkslk. Rm sers.. Kompleks tekslkd ochq v opq to plmlr. Sohlr. Brоr C uqt v ε > sо bеrlg. -t r: Ushbu U, ε { C : < ε} to plmg C uqtg ε trо dеld. Shug o хshsh C uqtg ε trо tushuchs krtld: U, ε { C : ρ, < ε} Ushbu { C : < < ε} { C : < ρ, < ε} to plm C uqtg o lg trо dеld. C

51 Fr qllk C d brоr D to plm bеrlg bo ls. -t r: Agr D uqt o g brоr trо bl shu D to plmg tеgshl bo ls, uqt D to plmg chk uqts dеld. 3-t r: Brch uqtlr chk uqtlrd bоrt to plm оchq to plm dеld. Agr C C uqtg хtor o lg trоd D C D D to plmg kmd btt uqts bo ls, uqt D to plmg lmt uqts dеld. 4-t r: Agr D to plmg brch lmt uqtlr shu D to plmg tеgshl bo ls, D to plm opq to plm dеld. Msоllr:. Ushbu D { C : < r } to plm qrlk. Bud b bеrlg uqt, r es musbt sо. M lumk ; b Dеmk, b b < r b < r Bu es, mrk, b uqtd bo lg r rdusl lg brch chk uqtlrd bоrtdr. Shud qlb, bu tеgslkg gеоmеtrk m оs mrk uqtd bo lg r rdusl dоrd bоrt ek.. Ushbu D { C : r < < r } to plm qrlk. Bud C bеrlg uqt, r v r lr musbt sоlr. Bu to plm оchq to plm bo ld. D to plm mrk uqtd, rduslr r v r r < r bo lg llr bl chеgrlg хlq оdld. Хqqt hm, ; b bo ls, < < r r < b < r r < b r < r bo ld.. Ushbu D { C : r} opq to plm bo ld. D C to plm bl bu to plmg brch lmt uqtlrg g dsd bоrt to plmg D to plmg opg dеld v D kb bеlgld. 5-t r: D C D C to plm bеrlg bo ls. Agr D UD D, D I D, D I D shrtlr qоtltruvch, bo sh bo lmg D v D to plmlr mvjud bo lms, D to plm bоg lml to plm dеld.

52 6-t r: Agr D C D C to plmg хtor kkt v uqtlr D to plmd to lq otuvch uluks chq bl tutshtrsh mumk bo ls, D to plm chql bоg lml dеld. 7-t r: Agr D C D C to plm hm оchq hm bоg lml bo ls, u sоh dеb tld. Оchq to plmlr uchu bоg lmllk tushuchs bl chql bоg lmllk tushuchs ustm-ust tushd. 8-t r: D C D C sоhg o g tеgshl bo lmg lmt uqts ug chеgrv uqts dеld. D sоhg brch chеgrv uqtlr to plmg ug chеgrs dеld v D ko rshd bеlgld. Agr D sоhg chеgrs bоg lml to plm bo ls, D sоh br bоg lml dеld, ks hоld u ko p bоg lml dеld.. Kоmplеks sоl kеtm kеtlklr v qtоrlr. Bg,,...,,... kоmplеks sоlr kеtm-kеtlg v C sо bеrlg bo ls. 9-t r: Agr shud M > sо mvjud bo lsk, N uchu M bo ls, { } kеtm-kеtlk chеgrlg dеld. -t r: Agr ε > sо оlgd hm shud ε N tоplsk, > uchu < ε tеgslk bjrls, C sо { } kеtm-kеtlkg lmt dеld v lm ko rshd bеlgld. Chеkl lmtg eg bo lg kеtm-kеtlk qlshuvch kеtm-kеtlk dеld. Yqlshuvch kеtm kеtlklr хоsslr.. { }. kеtm-kеtlk qlshuvch bo ls, u hоld u chеgrlg bo ld. Agr } v } kеtm-kеtlk qlshuvch bo ls, u hоld ± }, { { { { }, kеtm-kеtlklr hm qlshuvch bo ld v bo ld. lm ± lm ± lm lm lm lm lm lm lm

53 Bu хоsslr хqq sоlr kеtm-kеtlg uchu qd sbоtls, хudd shud sbоtld. -t r: Agr ε > so olgd hm shud ε N toplsk, > uchu v p N solr uchu < ε p tegslk bjrls, { } udmetl ketm-ketlk deld. Teorem: Kosh krters { } ketm-ketlk qlshuvch bo lsh uchu ug udmetl bo lsh rur v etrl. Ushbu Isbot: mustql k оdg kompleks sоl qtоr dеld, bu еrd,,...,,... lr bеrlg chеkl kompleks sоlr. qtоrg brch t hdg g ds S dеb bеlgllk, S k k gr { S } kеtm-kеtlk qlshuvch bo ls, qtоr qlshuvch dеld, ks hоld bu qtоr uоqlshuvch dеld. Agr S lm S bo ls, S sо qtоrg k g ds dеld. qtоr bl brg k qtоr qrm. Agr k qtоr k k qlshuvch bo ls u hоld qtоr bsоlut qlshuvch dеld. Agr qtоr qlshuvch bo lb, k qtоr uоqlshuvch bo ls, k qtоr shrtl qlshuvch dеld. Tch bоrlr: Kоmplеks sо gоmеtrk tsvr, kоmplеks tеkslk, Rm. sеrs, stеrеоgrk prоеks, Еvkld mеtrks, sеrk mеtrk, Egr chq, Jоrd chg, sllq chq, lmt uqt, chk uqt, opq to plm, оchq to plm, bоg lml to plm, sоh, sоhg chеgrs, uqt trо, uqt o lg trо, br bоg lml sоh, ko p bоg lml sоh, sоlr kеtm-kеtlg, kеtm-kеtlk lmt, qlshuvch kеtm-kеtlk, uоqlshuvch kеtm-kеtlk, udmеtl kеtm-kеtlk, sоl qtоr. O -o tеkshrsh uchu svоllr:. Kоmplеks sоg gеоmеtrk tsvr tushutrg.. Rm sеrs tushutrg. 3. Egr chq t r tg. 4. Jоrd chq dеb mg tld. 5. Sllk chq dеb mg tld. 6. Ichk uqt t r tg.

54 7. Lmt uqt t r tg. 8. Sоh t r tg. 9. Bоg lml sоh t r tg. Adbotlr: [3] bеtlr, [4]. 3-6 bеtlr, [5] bеtlr, [6] bеtlr. [7] bеtlr.

55 3-M ru. Kompleks o gruvch ukslr rs d Kоmplеks tеkslkd chqlr v sоhlr. Kоmplеks sоlr kеtm-kеtlg v ug lmt mvusdg m ru mshg ulot drs shlms F: Kompleks o gruvch ukslr rs. Mvu: Kоmplеks tеkslkd chqlr v sоhlr. Kоmplеks sоlr kеtmkеtlg v ug lmt. Mqsd: Tlblrg kompleks o gruvch ukslr rs sos tushuchlrd br bo lg kоmplеks tеkslkd chqlr v sоhlr, kоmplеks sоlr kеtm-kеtlg v ug lmt to g rsd to l tsvvur hosl qlsh qoblt shklltrsh, geometrk tsvrlsh bl bog lq msllr tushub, mustql, jod krlshg ko kmlr trkb toptrsh. Mshg ulot metod: Utrlg m ru, «Aql hujum» Mshg ulot tur: M ru Mshg ulot jho: Proektor, m ru preetss, Sldlr, trqtm mt mterllr, vtm, Mrkerlr Ajrtlg vqt: 8 mut Mshg ulot borsh: Ish O qtuvch olt bosqchlr v mmu -bosqch. Torlov 5 mut Mvu e lo qld Aql hujum metodg moht qlshtrd Oldg o tlg mvulr bo ch svollr ob qo ld. Yg mvu mt trqtld. Yg mvu tdg tushuchlr bo ch svollr berld Tlblr olt Tlblr mustql rvshd svollr thll qld v jvoblrg torlshd. Kutlotg tjlr Tlblrd kеltrldg o quv хbоrоtlrg sbt kuchl mоtvts shklltrsh. -bоsqch. Krsh Thll qlsh lom bo lg Aql hujum teologs mld Mummоg mоht tushub Tlblrd «Aql hujum» tехоlоgs

56 Aql hujum Yg mvu tdg tushuchlr bo ch 5 mut qo llsh Ug o quvchlr tоmоd o quv хbоrоtlr o lshtrb оlsh v o quv mtlr ustd muvqtl shlsh jrodg hmt tushutrb bеrd. оldlr, rur hоllrd o qtuvchg svоllr bl murоjt qlshlrg ershld. ml оltd muvqtl qo llsh mlklr trkb tоptrsh, ulrd hmkоrlkd jоd mехt qlsh, mustqllk, ksb mхоrt shklltrsh. 3-bоsqch. M ru Yg mvu bo 5 m Utrlg m ru teologs bo ch g mvug - qsm bo etld. kоmplеks tеkslkd chqlr v sоhlr, ochb berld, msollr keltrld. Tlblr dqqt bl eshtd. Thll qlb bord. Yg tushuchg eg bo lshd. 4-bоsqch. Aql hujum 5 mut Yg mvu tdg tushuchlr bo ch svollr berld Bud tch tushuchlr oldg bosqchdg m rud hm berld. Fkrlrg tlblr tоmоd krtlsh tshkl qld, ulr оlt kutd, rur хоllrd to g r o lsh bеrb bоrd. Mumо tlq jro kutd v tlblr tоmоd o quv хbоrоtg q drjd mukmml o lshtrlsh хususd q хulоslr qlb bоrd. Mvu bo ch Kоmplеks tеkslkd chqlr v sоhlr. Kоmplеks sоlr kеtmkеtlg v ug lmt tushuchlr br bor tkrorlb musthkmlb olshd. Tlblr mummо o rgsh jrod оllg оshrsh, ulrg mustql krlsh qоbltlr rvоjltrsh, duoqrsh kеgotrsh, hmkоrlkd shlshg o rgtsh. Ksb оltg mukmml torlsh. 5-bоsqch. M ru Yg mvu - qsm bo etld Tlblr dqqt bl eshtd. Thll qlb Yg tushuchg eg bo lshd.

57 Yg mvu - qsm bo etld mutgch Kоmplеks sоlr kеtm-kеtlg v ug lmt ochb berld, msollr keltrld. bord. 6-bоsqch. Aql hujum 5 mut Yg mvu t dg tushuchlr bo ch svollr berld. Bud tch tushuchlr oldg bosqchdg m rud hm berld. Fkrlrg tlblr tоmоd krtlsh tshkl qld, ulr оlt kutd, rur хоllrd to g r o lsh bеrb bоrd. Mummо tlq jro kutd v tlblr tоmоd o quv хbоrоtg q drjd mukmml o lshtrlsh хususd q хulоslr qlb bоrd. Mvu bo ch, Kоmplеks tеkslkd chqlr v sоhlr. Kоmplеks sоlr kеtmkеtlg v ug lmt Tlblrd mustqllk, оllk, jоdkоrlk, ms ultllk, vtprvrlk, tshbbuskоrlk hss trkb tоptrsh v rvоjltrsh. Tlblr mum-mо o rgsh jrod оllg оshrsh, ulrg mustql krlsh qоbltlrrvоjltrsh, duoqrsh kеgotrsh, hmkоrlkd shlshg o rgtsh. Ksb оltg mukmml torlsh 7-bоsqch. M ru Yg mvu 3- qsm bo etld 5 mutgch Yg mvu 3- qsm bo etld, msollr keltrld. Tlblr dqqt bl eshtd. Thll qlb bord. Yg tushuchg eg bo lshd. 8-bоsqch. Yku qsm mutgch Tlblr tomod berlg svollrg jvob berd. Tlblr tоmоd mlg оshrlg оlt thll qld, mshg ulоtg ku sd. O оltlrg bhо bеrdlr, eg mkbul kr vrtlr kurstdlr, Fkrlr хmоs usd kushmchlr, Tlblrd mustqllk, оllk, jоdkоrlk, ms ultllk, vtprvrlk, tshbbuskоrlk hss trkb tоptrsh v rvоjltrsh.

58 qlklr krtdlr v ulr sbоtldlr.

59 Mvu: Kоmplеks tеkslkd chqlr v sоhlr. Kоmplеks sоlr kеtm-kеtlg v ug lmt. -qsm. Kоmplеks tеkslkd chqlr v sohlr. Egr chq tеkslkd uqtg uluks хrkt tjsd kоldrg dеb qrsh mumk. Хrktdg uqtg kооrdtlr х v dеls, rvshk ulr brоr t o gruvchg uluks ukslr bo ld: t t α t β ptd, jutlk kоmplеks sо оdlg sbbl, u ko rshd osh mumk. Ntjd, t t t bo ld. Dеmk, t α t β uks [α,β] sеgmеt kоmplеks tеkslk uqtlrg ksltrd v bu uqtlr to plm es kоmplеks tеkslkd egr chq оdlr ek. Bud α egr chqg bоshlgch uqts, β es egr chqg охrg uqts bo ld. Agr α β bo ls, bud egr chq opq dеld. Agr t t egr chqd t o gruvchg kkt turl t v t qmtlrg mоs kеldg t v t uqtlr hm turlch bo ls, u хоld egr chq Jоrd chg dеld. Agr t v t ukslr [,b] sеgmеtd uluks dеrеslluvch bo lb, t t t shrt qоtltrs, t t t t egr chq sllq egr chq dеld.. Kоmplеks tеkslkd оchq v opq to plmlr. Sоhlr. t Brоr C uqt v ε > sо bеrlg. -t r: Ushbu U, ε { C : - < ε } to plmg C uqtg ε - trо dеld. Shug o хshsh С uqtg ε - trо tushuchs krtld: U,ε{ С :ρ, <ε} Ushbu { C : < - < ε } { С : < ρ, < ε } to plm C С uqtg o lg trо dеld. Fr qllk C d brоr D to plm bеrlg bo ls. -t r: Agr D uqt o g brоr trо bl shu D to plmg tеgshl bo ls, uqt D to plmg chk uqts dеld. 3-t r: Brch uqtlr chk uqtlrd bоrt to plm оchq to plm dеld.

60 Agr C С uqtg хtor o lg trоd D C D D to plmg kmd btt uqts bo ls, uqt D to plmg lmt uqts dеld. 4-t r: Agr D to plmg brch lmt uqtlr shu D to plmg tеgshl bo ls, D to plm opq to plm dеld. Msоllr:. Ushbu D { C : - < r } to plm qrlk. Bud b bеrlg uqt, r es musbt sо. M lumk ; b dеmk, - --b b < r - -b < r bu es, mrk, b uqtd bo lg r rdusl lg brch chk uqtlrd bоrtdr. Shud qlb, bu tеgslkg gеоmеtrk m оs mrk uqtd bo lg r rdusl dоrd bоrt ek.. Ushbu D { C : r < - < r } to plm qrlk. Bud C bеrlg uqt, r v r lr musbt sоlr. Bu to plm оchq to plm bo ld. D to plm mrk uqtd, rduslr r v r r < r bo lg llr bl chеgrlg хlq оdld. Хqqt hm, ; b bo ls, r < - <r r < bo ld.. Ushbu b < r r < - -b < r D { C : - r } opq to plm bo ld. D C to plm bl bu to plmg brch lmt uqtlrg gdsd bоrt to plmg D to plmg opg dеld v D kb bеlgld. 5-t r: D C D C to plm bеrlg bo ls. Agr D UD D, D I D, D I D shrtlr qоtltruvch, bo sh bo lmg D v D to plmlr mvjud bo lms, D to plm bоg lml to plm dеld. 6-t r: Agr D C D C to plmg хtor kkt v uqtlr D to plmd to lq otuvch chq bl tutshtrsh mumk bo ls, D to plm chql bоg lml dеld. 7-t r: Agr D C D C to plm hm оchq hm bоg lml bo ls, u sоh dеb tld. Оchq to plmlr uchu bоg lmllk tushuchs bl chql bоg lmllk tushuchs ustm-ust tushd. 8-t r: D C D C sоhg o g tеgshl bo lmg lmt uqts ug chеgrv uqts dеld. D sоhg brch chеgrv uqtlr to plmg ug chеgrs dеld v D ko rshd bеlgld. Agr D sоhg chеgrs bоg lml to plm bo ls, D sоh br bоg lml