BITIRUV MALAKAVIY ISHI
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- Διόνυσος Φθα Ζυγομαλάς
- 6 χρόνια πριν
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1 O'ZBEKISTON ESPUBLIKASI OLIY VA O'TA AXSUS TA'LI VAZILIGI ALISHE NAVOIY NOIDAGI SAAQAND DAVLAT UNIVESITETI EXANIKA ATEATIKA FAKULTETI atematk fka a fksoal aal kafedas ade Olmos 5 - matematka ta'lm o'alsh bo cha bakala daajas olsh ch BITIUV ALAKAVIY ISHI a: Gamok fksag asos ossala a tegal ko sh Ilm ahba: dots: E.N.Sattoo Bt malaka sh matematk fka a fksoal aal kafedasda bajald.kafedag 5 l majlsda mhokama qld a hmoaga tasa etld baooma Fakltet deka: dots. X.o mado Kafeda md: dots. A..Xalo jae Bt malaka sh YaDAKg 5 l dag majlsda hmoa qld a << >> ball bla baholad baooma. YaDAK as: A ola: Samaqad 5
2 Ksh.asalag qo lsh. Kompleks o gachl fksala aaas fag mm ksda ma lm bo lga b qmatl aaltk fksag mhm ossalada b shda boatk aga b fksag ol bo lmasa g modl logam gamok fksada boat bo lad. Komples o gachl fksala aaas asosa g geometk qsm-kofom akslatsh aaas-fk tasda pado bo lga a ojlaga. Akscha kompleksl o gachl fksala aaasg ojlash matematkag tl o alshlada mhm amal ahamatga ega bo lga masalala echshda ag slla aatlshga olb keld. Ikk o gachl gamok fksala b qmatl aaltk fksag haqq ok mahm qsmda boat bo lb Laplas teglamasg echm bo lad. Laplas teglamas ch qo lga chegaa masala kofom akslatsh oqal aq sohalada echmg aqqol ko shga taslash mhmd. Gamok fksa tekslkdag ekto mado potesal a chegaa masalala bla bog lqd..ag dolablg. atematk fka teglamala echshda Laplas teglamas o g amal ahamat jhatda mhm o egallad. Teglama echm topshda echmg majdlk a agoalg qo lga chegaa shatla qaoatlatsh tekshsh mhmd. Tekslkda qaalaotga Laplas teglamas ch Dl masalasg echm topshda echm gamok fksa bo lsh keaklgda kompleks o gachl b qmatl aaltk fksada bog lqd. Qaalaotga soha makkab ko shga ega bo lgada b kofom akslatsh oqal echm topsh mmk bo lga sohaga o tkalad. Kofom akslatlga sohada Laplas teglamas ch qo lga Dle masalas echm G fksas oqal qla ko shda fodalaad. Ushb malaka bt shda matematk fka teglamas echshda kompleks o gachl fksala aaas slda
3 fodalash qla mkoatlaga ega bo lshlg shb mag dolablg fodalad..ishg maqsad a afala. Bt malaka shg maqsad kompleksl o gachl fksala aaas ksg mhm tshchala asosda gamok fksag asos ossala o gashda boat. 4.Ilm tadqqot slla. Kompleksl o gachl fksala aaas ksda b qmatl aaltk fksag haqq a mahm qsm Koshg tegal fomlas o ta qmat haqdag teoema kofom akslatsh tshchas gamok fksala ch G fomlas matematk fkag Laplas teglamas ch qo lga chegaa masalalada Dle masalas echsh. 5.Ishg lm ahamat. Bt malaka shda olga atjala efeat aakteda bo lshga qaamasda kelgsda ellptk tpdag teglamala ch qo lga tl l chegaa masala echshda odam bead. 6.Ishg amal ahamat. Bt malaka shda olga atjala matematk fka teglama ch qo lga chegaa masalala kompleksl o gachl fksala aaas elemetla oqal echsh ahamatga ega. 7.Ishg tlsh. Bt malaka sh ksh ta bob losa qsm a fodalalga adabotla o atda boat. Ushb sh matl sahfa tashkl topga a ha b bob paagaflaga ajatlga la o g omelash hamda belglashga ega. Ishg -bobda gamok fksag asos ossala Kosh-ma shatla qo shma gamok fksala o galga. -bobda gamok fksag tegal tas odamda a o ta qmat haqdag teoema odamda kofom akslatsh oqal doa a qo am tekslkda Dle masalas echsh. 8. Olga atjag qsqacha mam... da. Teoema[]. f fksa qtada dffeesallach bo lsh ch
4 a fksala qtada dffeesallach; qtada. Kosh ma shatg bajalsh a a etald. f hosla ch f. fomla o ld...da gamok fksag asos ossala o galga.-ossa. Ito gamok fksa o g agmetla aaltk fksasda boat bo lad a D sohag ha b qtag g ds ko shda taslaad. m m Cm.9.-ossa o ta qmat haqda. Aga fksa maka qtada ads ga teg bo lga opq doada lks a b doada gamok bo lsa holda e d...-ossa. O gamasda faql bo lga gamok fksa o g eg katta a eg kchk qmatga aqlash sohasg chk qtasda eshsh mmk emas..4-ossa. Aga D sohada gamok fksa hech bo lmagada qo ok qda chegaalaga bo lsa holda o gamas..5-ossa. Aga fksa D sohada lks a etalcha kchk - la ch to qtada e d bo lsa holda fksa D sohada gamokd
5 .6-ossa. D sohada gamok a D da lks bo lga gamok fksala ketma-ketlg belga bo ls. Aga k k qato D g chegaasda teks aqlashsa holda b qato D -g chda ham teks aqlashad a g g ds ham D sohada gamok fksa bo lad..7-ossa. Aga fksa b bog laml D sohada gamok a o g ss hoslas bla D da lks bo lsa holda D ds b eda -omal bo cha hoslas ds -og dffeesal.. da Laplas teglamasg qtb sldk a sfek koodataladag fodas a.8..9 s s s.. la gad j k k j k j dgad oqal fodalashda hosl qlga. Bch bobg 4 paagafda Laplas teglamasg elemeta ok fdametal echm olsh ko satlga:
6 .-ta f. [][6] Laplas teglamasg belga sohag ajalga mass qtala ok o -o kesmadga sllq stlada masslkka ega bo lga echmga fdametal echm delad. Laplas eglamasg sohada masslkka ega bo lmaga a og kch tatbl ss hoslala bla lks echmga esa egla echm delad. Bda olad: l aga bo'lsa aga bo' lsa Gg asos tegal fomlas keltlga: Aga 4 aga aga aga d D bo'lsa bo'lsa D D bo'lsa. d.. gamok fksa bo lsa. fomla qdag ko sh d.4 Qaalaotga soha tekslkda bo sllq opq chq bla chegaalaga D sohada boat bo lsa holda qodag mlohaalada o da Laplas teglamasg tekslkdag fdametal echm l fksa shlatsak. a.4 ga o shash fomlala olam: C l l ds S d.
7 l l ds..4 C II-bob. Gamok fksa ch Dl masalas echshga bag shlaga bo lb.. da Laplas teglamas ch Dle masalasg qo lsh echmg majdlg a agoalg ko satlga... Laplas teglamas kofom akslatshga sbata aatlg ko satlga.. doada Laplas teglamas ch Dle masalas o galga.-teoema. fksa doada gamok opq doada lks bo ls. U holda Passo fomlas o l bo lad. e e d.8 cos.4. Yqo am tekslkda Laplas teglamas ch Dle masalasga bag shlaga..-teoema. fksa Im qo am tekslkda gamok a chegaalaga chekl sodag qtalada tashqa Im to g chqqacha lks bo ls. U holda Passo fomlas t t dt.7 o ld b eda.
8 -Bob. Gamok fksa a g ossala.. Kosh-ma shat. Qo shma gamok fksala w f fksa qtag bo atofda aqlaga bo ls. Aga w f fksag sbat da aq chekl lmtga ega bo lsa b lmtga qtadag hoslas delad a f belglaad w f fksaga qtada dffeesallach delad. Shda qlb f f f lm.. w f fksa sohada dffeesallach delad aga sh sohag ha b qtasda dffeesallach bo lsa.. Teoema[]. f fksa qtada dffeesallach bo lsh ch a fksala qtada dffeesallach; qtada. Kosh ma shatg bajalsh a a etald. f hosla ch f. fomla o ld..-msol. a f Yechsh. f e e e cos s e cos e e fksa dffeesallachalkka tekshg. bt kompleks tekslkda dffeesallach chk s fksa e cos e s fksala. Kosh ma shat qaoatlatad a. fomlaga ko a e cos e s.
9 e e e e e e e e f s cos s cos. Demak e e f. b f fksa dffeesallachalkka tekshg. Yechsh. f. a. shatda 4 4 bda f fksa = qtada dffeesallach eka. f fksa D sohada dffeesallach a fksala kkch shatgacha lks ss hoslalaga ega bo ls beksto holda. teglkg bchs bo cha kkchs beksto bo cha dffeesallab hosl qlam. B teglkla qo shb a hoslala lks ekalgda teglg obatga olb..4 Hdd shga o shash. teglkg bchs beksto bo cha kkchs bo cha dffeesallab Hosl qlam. B teglkla bchda kkchs ab
10 ega bo lam..-taf. D sohada lks kkch tatbl ss hoslaga ega bo lga a.4 teglama qaoatlatch haqq fksaga D sohada gamok fksa.4 teglamaga esa Laplas teglamas delad. Odatda - Laplas opeato fodalad. B qmatl aaltk fksag ossalada ko;adk D sohada dffeesallach fksa sh sohada to tatbl hoslaga ega a haqqatada too tatbl lks ss hoslaga ega bo lad. Shg ch D sohada dffeesallach f fksag haqq a mahm qsmla sh sohada gamok fksalada boatd..-taf. O ao Kosh-ma shat bla bog laga a gamok fksalaga qo shma gamok fksa delad. Shda qlb sohada dffeesallach fksag haqq a mahm qsmla sh sohada qo shma gamok fksalada boatd. Teskas aga D bo lsa.-teoemaga ko a f fksa D sohada dffeesallachd. Bda qdag teoema o ld..-teoema. f fksa D sohada dffeesallach bo lsh ch a fksala sh sohada qo shma gamok bo lsh a a etald. B bog laml sohada fksalada b blga holda kkchs topsh mmk..-teoema. B bog laml D sohada gamok fksa ch ga qo shma bo lga gamok fksa to o gamas qo shlch aqlgda topsh mmk. Isbot. fksa b bog laml D sohada gamok ekalgda
11 d d foda to o gamas C qo shlch aqlgda aqlaadga b qmatl fksag to la dffeesalda boat a d d d ekalg etboga olb d d d d d C D D.5 b eda tegal a qtala ttashtch chqda bog lq emas aga qta fkslaga bo lsa holda faqat qtada bog lq..5 da bda kelb chqadk D sohada fksaga qo shma bo lga gamok fksa.. a. teoemalada kelb chqadk aga b bog laml D sohada gamok fksa belga bo lsa holda o gamas qo shlch aqlgda D sohada dffeesallach f fksa topsh mmk a belga haqq ok mahm fksa tklash mmk. Aga D soha ko p bog laml bo lsa.5 tegal bla aqlach hdd shda f fksa b qmatl bo lmaslg mmk. Belga fksa akscha fksa topshga.5 fomlaga ko a Kosh-ma shatda fodalash ham qlad..-msol. Aga f fksa belga bo lsa f dffeesallach fksa topg.
12 Yechsh. fksa bt kompleks tekslkda gamok ekalg a ko sh mmk. bda g g d.6.6 da 6 g.7 topam. Ikkch tomoda. ga ko a a.8 la teglashtb gamas haqq a c C g da g g ' da C hosl qlam. Ilaaotga fksa C C f bt kompleks tekslkda dffeesallach... Gamok fksag asos ossala
13 .-ossa. Ito gamok fksa o g agmetla aaltk fksasda boat bo lad a D sohag ha b qtag g ds ko shda taslaad. m m Cm.9 Isbot..-teoemaga ko a qtag atofda b qmatl aaltk f fksag haqq qsm sfatda qaash mmk. B qtag atofda B eda C f C.. qatog mm had haqq qsm! Absolt qmat bo cha...! C.... Oshmad Abel teoemasga []9-. ko a. qato to doada absolt aqlashad a aqlashad mm had. bo lga qato C qato da bo lgada absolt aqlashad. B qato fksa ch qato fodalad. Bg hadla ghlaga talab qlga.9 qatola hosl qlam. Teoema sbot bo ld..-ossa o ta qmat haqda. Aga fksa maka qtada ads ga teg bo lga opq doada lks a b doada gamok bo lsa holda e d..
14 B ossag sbot b qmatl aaltk fksa ch o ta qmatl haqdag [] teoemada haqq qsm ajatshda osoga kelb chqad..-ossa. O gamasda faql bo lga gamok fksa o g eg katta a eg kchk qmatga aqlash sohasg chk qtasda eshsh mmk emas. Isbot. Xossa maksmm qta bo lga hol ch sbotlash etal chk gamok fksag mmm qtas fksa maksmm qtas bo lad. Teskada faa qlb gamok fksa chk qtada o g maksmmga eshs. qtag atofda shda b qmatl f fksa qamk e f fksa aaltk a o gamas g modl e fksa bg faamga ko a sohag chk qtasda maksmmga eshad. B esa modlg maksmm pspga []5- dd. Xossa sbot bo ld..-msol. fksag 4 doadag etemal qtala a etemal qmatla topg. Yechsh. Belshga ko a fksa sola tekslgda ssa belga doada lks a stalga tatbl lks ss hoslalaga ega. Dastlab belga fksa 4 doada gamoklkka teksham. Bg ch g ss hoslala hsoblam: U holda b fksa ch. bo lb Laplas teglamasg egla echm a tekslkdag bacha qtalada ssa 4 doada ham gamok fksa bo lad. Demak b fksa ch gamok fksa ch maksmm qmat psp qo llash mmk. B fksa gamok bo lga 4 doa
15 chegaas 4 alaada boat bo lb chegaa qtalada teglk o l bo lb b qtala to plamda qaalaotga fksa w ko sh olad. B kadat fksa bo lb qtada w 4 5 maksmm qmatga a qtada esa w 8 mmm qmatga eshad. Ekl o gach g qmatga g qmatga esa qmat mos kelad. Shda qlb belga chegaa qtalada fksa 5 9 ma ; eg katta maksmm qmatga a ; chegaa qtada esa m eg kchk mmm qmatga eshad. ; 8 5 a 5 ;.4-ossa. Aga D sohada gamok fksa hech bo lmagada qo ok qda chegaalaga bo lsa holda o gamas. Isbot. fksa D sohada qoda chegaalaga bo ls: d. Bt D sohada shda b qmatl aaltk f fksa qamk e f. Shatga ko a w f fksag bacha qmatla am tekslkda otad. f fksa o gamas demak ham o gamasd. Xossa sbot qld. Qdag o ta qmat haqdag teoemaga teska teoemalada boatd..5-ossa. Aga fksa D sohada lks a etalcha kchk - la ch to qtada e d bo lsa holda fksa D sohada gamokd.
16 Qdag ossa kompleks o gach fksoal qatola ch o galga Veetass teoemasga o shashd..6-ossa. D sohada gamok a D da lks bo lga gamok fksala ketma-ketlg belga bo ls. Aga k qato D g chegaasda teks aqlashsa holda b qato D -g k chda ham teks aqlashad a g g ds ham D sohada gamok fksa bo lad..7-ossa. Aga fksa b bog laml D sohada gamok a o g ss hoslas bla D da lks bo lsa holda D ds b eda -omal bo cha hoslas ds -og dffeesal... Laplas teglamasg qtb sldk a sfek
17 koodataladag fodas Bga ma lmk gad j k k j k j dgad deb belglam a Laplas opeato delad.. Laplas teglamas delad. Laplas teglamas sldk a sfek koodatalada mos ashda qdag ko shda bo lad.4 s s s.5 sldk koodatala sstemasda.4 g echm Bessel fksala.5 teglama sfek koodatla sstemasdag echm sfek fksada boat bo lad. a lmk Dekat koodatala sstemasda gad j k ko shda bo lad. Sldk koodatalada k j gad sfek koodatalada esa
18 ko shda bo lad. gad Dekat koodatala sstemas degesas j k s a a a da ko shda bo lad. а=а х + а у j + а k da ecto mado Sldk a sfek koodatalada а ecto mado degesas mos ashda qdag da a da a a a sa s a s ko shda bo lad. Laplas teglamasg qtb koodatala sstemasda ko sh fodalash ch matematk aal ksda ma lm bo lga Dekat koodatala sstemasda eg chql qtb koodatala sstemasga o tsh fomlala cos s.6 ko shga o tsh a. Bda - koodata boshda belga A qtagacha masofa bo lb odatda qtag ads ekto delad - esa OX o qg msbat o alsh bla qtag ads ekto oasdag soat stelkas haakatga teska o alshda aqlaga bchak bo lb odatda belga qtag bosh agmet delad. cos s deb belglam. Atlgalaga asosa bo lgada.6 o ao b qmatl akslatsh bo lb ga mos teska almashtshla qdagcha aqlaad [4]: actg..7
19 koodatala sstemasda Laplas teglamasg ko sh topsh ch dastlab bch tatbl a ss hoslala hsoblam: cos s s cos B ss hoslala odamda Laplas teglamas ch keakl bo lga kkch tatbl a ss hoslala hsoblam: cos cos cos s s cos s s s s s s s s cos cos s cos s cos cos s Toplga b fodala a Laplas teglamsga qo b g qtb koodataladag ko sh olam: s s s s cos s cos cos s s. Ushb teglama soddalashtsak qdag teglamaga teg kchl bo lad: Aga dffeesallah ch. teglkg o l ekalg hsobga olsak qodag teglama....8
20 ko shda osh mmk bo lad. Odatda.8 teglama Laplas teglamasg qtb koodatala sstemasdag tas hsoblaad. Kosh-ma shatg qtb koodatala oqal fodas ko satam. Bg ch cos s ag f fksag modl a agmet oqal fodalam. U holda deb olb a f f fksa ga sbata dffeesallach bo lad. b fksala Kosh-ma sstema qaoatlatad. B sbotlash ch hoslala hsoblam bda.5 sstemada fodalab: cos s s cos = cos s cos s. qodag sstema bajalsh sbot bo ld..4-msol []. Belga c os f e e fksaga ko a b qmatl aaltk f fksa tklag. Yechsh. Belga fksada l f cos topam. Ed l f l f Agf cos b qmatl aaltk fksa ch g haqq qsm cos belga a l f fksa tklash keak. Bda f e. fomlada fksa tklam: Demak s c l f cos s c c e a
21 f s c e cos d s s s c cost f e. Ed ch o lchol faodag bo sohada qaalga Laplas teglamasg sldk a sfek koodataladag tasla keltb chqaam. Ushb teglamala qaalaotga soha sldsmo a shasmo ko shda bo lgada Laplas teglamasg echmla topshda qla hsoblaad. a lmk Dekat koodatala sstemasda eg chql sldk koodatala sstemasga o tsh fomlala cos s ko shga ega bo lb qodag hsoblashlaga asosa Laplas teglamasg sldk koodataladag ko sh kab ekalgga shoch hosl qlam...9 Xddd sh kab hsoblashla Dekat koodatala sstemasda eg chql sfek koodatala sstemasga o tsh fomlala kab bo lb ga teska almashtsh actg s cos s s cos accos fomla bla aqlaad. B holda ham dd qtb koodoataladag kab s cos s s cos w belglash ktb keakl s hoslala hsoblash bla Laplas teglamasg sfek koodataladag ko sh olam:
22 s.. s s.5-msol []. ch s fksaga qo shma gamok bo lga fksa Kosh-ma sstemasda fodalab topg. Yechsh..5 Kosh-ma sstemasda fodalab fksada cos ch olam.bda esa ga ega bo lam. s ch s ch sh cos fksa bo cha dffeesallab belga fksa a.5 fomla e tboga olb qdagga ega bo lam. Bda s sh sh s ' cost. Shda qlb fksa s sh c ko shga ega bo lad..6-msol: topg. bo lsa qo shma gamok fksa Yechsh:.5 Kosh-ma sstemasga ko a ga teg. Bda = = ga ega bo lam. Ilaaotga fksa gamok fksa bo lga ch Laplas teglamas qaoatlatad a la etboga olb qdagla = + = olam. Bda topam.shda qlb qo shma gamok /6 c c fksa = c c 6 ko shsda bo lad..7-msol. Aga f aaltk fksa g e f s ch haqq qsm belga bo lsa holda b bog laml D sohada eg chql tegallash odamda f aaltk fksa topg. fksala Yechsh. a lmk f bo lb a
23 Kosh-ma sstemas qaoatlatad. Demak.7 fomlaga ko a qdagga ega bo lam: d d c s shd cos chd cos sh cos sh cos sh cos sh Shda qlb f aaltk fksa qdag ko shda tlad..8-msol. aaltk fksa topg. maqsadda c cos sh f s ch cos sh cos sh c e s e f cos sh c. fksa a f shat oqal Yechsh. fksa gamok fksa ekalg tekshsh shs a shs hoslala topam a bacha la ch Laplas teglamas qaoatlatlshga shoch hosl qlam a teglk bajalad. Demak fk sa gamok fksad. Bda a.7 fomlada hamda qta talab qdagga ega bo lam. sh t cos dt ch s s ds ch cos Shda qlb f aaltk fksa qdag f ch cos sh s c ko shda tlad. Demak laaotga f fksa ko'shda bo lad. f ch cos sh s c.4. Laplas teglamasg fdametal echm a G fomlala
24 Faodag sldk hamda shasmo sohalada belga Laplas teglamasg faqat ads ektolada a.8 ok.9 teglamag faqat ok da bog lq bo lga qolga o gachlaga bog lq bo lmaga echmga sldk a sfek smmetk echm deb tlad. Ushb echmla gamok fksala a mma ellptk tpl dffeesal teglamala aaasda mhm ahamatga ega. Shg ch ham b Laplas teglamasg sfek a sldk smmetk echmlag ko sh topsh masalas bla shg llaam..-ta f. [][6] Laplas teglamasg belga sohag ajalga mass qtala ok o -o kesmadga sllq stlada masslkka ega bo lga echmga fdametal echm delad. Laplas eglamasg sohada masslkka ega bo lmaga a og kch tatbl ss hoslala bla lks echmga esa egla echm delad. Faa qlalk Laplas teglamasg sldk smmetk faoda ok doa smmetk tekslkda echm topsh lom bo ls. Atlgalaga asosa b holda.8 teglamag faqat da bog lq echm topsh lom. Ushb hollada bo lgalg ch.8 teglama ko shga kelad. U tegallab ok C C teglamaga kelam. U tegallash atjasda Laplas teglamasg sldk smmetk echmg mm ko sh C l C
25 hosl qlam. Aga shb mm echmda C C deb talab Laplas teglamasg sldk ok doa smmetk echmlada bttas hosl qlam: l. Ushb echmga odatda tekslkda Laplas teglamasg fdametal echm delad. Xdd sh kab Laplas teglamasg sfek smmetk echm topam. B holda Lapals teglamasg sfek koodataladag.9 ko shda fodalaam. Qaalaotga holda bo lb Laplas teglamas qdag ko shga kelad: B teglama tegallab. C C mm echm hosl qlam. Aga bda C C deb faa qlsak Laplas teglamasg faodag fdametal echm deb atalch echm hosl qlam. Shda qlb Lapals teglamasg elemeta ok fdametal echm mm holda qdag ko shda olsh mmk dega losaga kelam: bda l aga bo'lsa aga bo' lsa. B fksa ch chekslkda
26 O baho o ld. Aga bo lsa holda fksa chekslkda chegaalaga bo lad..4-msol. E fksa teglama ch fdametal echm bo lsh sbotlag. Isbot. Belga E fksa bo lgada bo cha ham bo cha ham belga teglama qaoatlatad. Haqqata ham E E E E E E B fodala belga teglamag chap tomoga qo b qdagga 5 E E E 5 ega bo lam. Shda qlb E fksa bo lgada belga Laplas teglamas qaoatlatad. Demak.-ta fga ko a E fdametal echmd..5-msol []. E fksa Kosh ma opeatog fdametal echm ekalg sbotlag. Isbot. E fksa da Kosh ma teglamas qaoatlatad. Haqqata
27 E E fodaga ko a E teglkga ega bo lam. Shda qlb E fksa Kosh ma teglamas qaoatlatad. Demak b fksa fdametal echmd. Faa qlalk D soha a g chegaasda o g bch tatbl ss hoslala bla lks D sohada esa kkch tatbl lks s hoslalaga ega bo lga a fksala belga bo ls. Bga matematk aal ksda ma lm bo lga a bo hajm bo cha tegal g st bo cha olga tegalga keltch Ostagadsk fomlas qdagcha ed: d Q P D d Q P cos cos cos. Bda d a d mos ashda blk hajm a o elemetla. Ushb fomlada Q P deb olsak Gg -fomlas deb atalch T D d d d.. Aga b fomlada a lag o almashtb hosl bo lga teglk. da asak Gg -fomlas hosl qlam d d D.. Gg shb -fomlas gamok fksala ch o ta qmat haqdag teoema sbotlashda mhm ahamatga ega. Faqa qlalk qta sllq st chda ots. U U esa maka
28 qtada ads ga teg bo lga a btla D sohada otch sha bo ls. Gg -fomlas. U D \ sohada a fksala ch tatbq etshm mmk bda D. Shada omal bo cha hosla ads ekto bo cha hoslaga teg bo lgalg sfeada fksag o ta qmat hsobga olsak a da lmtga o tb Gg asos fomlas deb atalch qdag fomlaga kelsham: D d d.. Bda bo'lsa. aga bo'lsa aga bo'lsa aga 4 D D Aga gamok fksa bo lsa. fomla qdag ko sh olad: d.4 Agada qaalaotga soha tekslkda bo sllq opq chq bla chegaalaga D sohada boat bo lsa holda qodag mlohaalada o da Laplas teglamasg tekslkdag fdametal echm l fksa shlatsak. a.4 ga o shash fomlala olam: S C d ds l l. C ds l l..4 II-bob. Gamok fksa ch Dl masalas
29 .. Laplas teglamas ch Dle masalasg qo lsh. Yechmg majdlg a agoalg. Ko pga statsoa aqtda bog lq bo lmaga fka masalala ma lm chegaa shatla qaoatlatch gamok fksala topshga keltlad. Chegaalaga D sohag chegaasda lks fksa belga bo ls. Gamok fksala to plam b eg sodda kkch tatbl s hoslal dffeesal teglamalada b bo lga.4 Laplas teglamasg bacha echmla to plamd. Odd dffeesal teglamala kab aq btta echm ajatb olsh ch qo shmcha shat qo lgadek Laplas teglamas echm ham to la aqlash ch qoshmcha shat talab qlad. Laplas teglamas ch chega shat ko shda a belga mosabat laaotga echm sohag chegaasda qaoatlatsh keak. aa shda shatlada eg soddas laaotga gamok fksa chegag ha b qtasdag qmat belshd. Shda qlb bch chega masala ok klassk Dle masalas Lej Dle ems matematg: chegas D D sohada gamok g chegas da D D D gacha lks a g - qmat qabl qlch fksa topg: B eda a ke D;. D. haqq fksala - Laplas opeato. Klassk Dle masalas. g echm majd a agoad. Yechmg majdlg matematk fk teglamala ksda sbotlaga []. Yechshg psp.-ossa da kelb chqad. Umma olgada lks fksala bo lb D sohada gamok chegas D gacha D D holda D sohada
30 gamok chegaas D gacha lks a D olga teg..- ossaga ko a D a D Qdag msolda ko sh mmkk Dle masalas qaashda lach fksa chegaalagalk shat beko qlshda agoalk teoemas o l bo lmad..-msol. chegaagacha lks a sh shat qaoatlatad. fksa am tekslkda gamok da olga teg. fksa ham.. Laplas teglamas kofom akslatshga sbata aatlg
31 .-teoema. g egla fksa G soha D sohaga kofom ~ akslats a D sohada gamok bo ls. U holda g fksa G sohada gamokd. Isbot. G G b bog laml soha qaam. g kofom akslatshda G sohag aks D D b bog laml soha bo lad. f fksa D sohada egla bo ls holda e f bda f fksag majdlg.-teoemaga asosa. U holda f f g fksa ~ ~ G sohada egla a shg ch e f G da gamok fksa. G -G sohag to b bog laml qsm sohas bda G sohada gamok fksa..-teoema qdagcha ham sbotlash mmk. ξη=egζ ξη=imgξ ζ=ξ +η deb belglam. U holda =gζ akslatsh ko shda osh mmk. ~ ~ =ξη =ξη. gζ- egla fksa demak ξηξη fksala Kosh-ma shat qaoatlatad. Shg ch. o gachla almashtshda ~ ~ g... fomlada kelb chqadk aga o gach bo cha ~ gamok bo lsa holda g fksa ξη o gachla bo cha gamok a Laplas teglamas cofom akslatshga sbata aatd. B o abatda Dle masalas kofom akslatsh odamda echsh metod ch asos bo lad..-msol. D -Jm< + >l soha bo ls b eda l>> B sohada kosetk bo lamaga halqa to g chq- ads cheksga
32 teg bo lga alaa. Im= to g chqqa sbata smmetklkka ko a b qtala ±a boat a>/ +l = alaaga sbata smmetaga ko a l a l a a l. Dle masalas echam. D;.4 T cost.5 Jm l Bg ch D soha k: < ζ < kosetk halqaga h a a qaam b eda a l l a / l a. B akslatshda Jm= to g chq ζ = alaaga o tad +l = alaa- ζ = alaaga o tad. Z=gζ fksa ζ=h fksaga teska fksa bo ls..-teoemaga ~ ko a g k halqada gamok fksa :.5 shatda ~.6 k ~ ~ T..7 Shda qlb.4-.5 masala.6-.7 Dle masalasga keltld. B masala echam. ζ =ξ+η =ρe θ bo ls. ξ = ρcosθ η=ρsθ almashtshda ke.6 Laplas teglamas ~ ~ ~ ko shda olad..7 shatda chegaa fksa θ da bog lq emas holda tabk.6-.7 masalag echm ham θ da bog lq bo lmad deb faa qlsh ~ mmk a fksa faqat btta ρ o gachg fksas bo lad. Dle masalas echmg agoalgda sbotlaadk.6-.7 masala ~ echm θda bog lq bo lmad. fksa θ da bog lq bo lmagada.6 odd dffeesal teglamada boat d ~ d~ d d
33 B teglama mm echm l l ~ C C C C..7 shatda l T C C topam a l l ~ T fksa.6-.7 masala echm toppsh ch =+ koodataga o tsh keak bo lad. ~ a a a a a h a h Ekalgda.4-.5 masalag echm 4 l l a a a T Fksada boatd b eda. a l a l l a.- Doada Laplas teglamas ch Dle masalas
34 .-teoema. fksa doada gamok opq doada lks bo ls. U holda Passo fomlas o l bo lad. e e d.8 cos Isbot. bo lgada.8 fomla. bla mos tshad gamok fksa ch o ta qmat haqdag teoema. ham fksag qmat kofom akslatsh odamda o ta qmat haqdag teoema oqal topsh ko satam. e h kofom akslatsh qaam. qta fkslab a doa doaga.9 da h.9 h. Z=gζ fksa doa doaga shda cofom akslatadk g=. fksa doada gamok opq doada lks ~ haqqatda g fksa doada gamok.-teoema a opq doada lks. Gamok fksa ch o ta qmat haqdag teoemaga asosa ~ ~ e d Aalg o gachga qatam.. tegalda.. da topam e e h e. e d d d. e e cos
35 e e ga almashtb.-. da.8 fomla hosl qlam. Passo fomlas boshqacha ko shga ham almashtsh mmk. Ko sh mmkk e e e e cos Shg ch.8 fomla e d e e e.4 Ko shda osh mmk chk e - haqq fksa..4 tegalda e =ζ deb e d d d.5 Hosl qlam. Shda qlb.8 Passo fomlas.5 ko shda osh mmk. Passo fomlas odamda < doada Laplas teglamas ch Dle masalamas echsh mmk. Xss holda aga chegaa fksa sφ a cosφ bog lq atsoal fksa bo lsa holda.5 fomladag tegal qoldq odamda hsoblaad..-msol. 4cos 5 s ;.6 asalag echm topg b eda =e φ Yechsh..5 fomlada fodalaam. ζ =e θ bo ls holda d J a cos 5 s cos s B eda alaa soat stelkasga teska o alshda o alga. Itegal ostdag f fksa sohada btta chekl mass qta bch tatbl
36 ξ =- - qtb a ξ = tatb bo ladga mass qtaga ega. Qoldqla aaasg asos teoemasga ko a [] 8 J es F es F. es F lm F 4 esf J 4.5 fomla oqal.6 masalag echm hosl qlam. e s. 4 cos.4-. Yqo am tekslkda Laplas teglamas ch Dle masalas.
37 .-teoema. fksa Im qo am tekslkda gamok a chegaalaga chekl sodag qtalada tashqa Im to g chqqacha lks bo ls. U holda Passo fomlas t t dt.7 o ld b eda. Isbot. h = doaga kofom akslatsh qaam..8 da qta fekslab Jm> qo tekslk ζ < h.8 g.9 topam. g fksa ξ < doa Im qo am tekslkka ~ shda kofom akslatadk g=..-teoemaga ko a g ζ < doada chegaalaga a chekl sodag qtalada tashqa ζ = alaagacha lks. O ta qmat haqdag teoemaga ko a Aag o gachga qatam.. tegalda e ~ ~ d. almashtsh olam. t t ~ e h t. ~ e ~ h t t. t. da dt. d t ga almashtb.-. da.7 hosl qlam. t e t t ekalgda.7 Passo fomlas
38 ko shda osh mmk. e t dt t..7 ok. fomlala odamda Laplas teglamas ch Dle masalas Im qo am tekslkda echsh mmk. ;.4 asala qaam b eda haqq atsoal fksa haqq o qda qtb o q a da e. B masalag echm. ga ko a t e dt t b eda Im. B tegal qoldq odamda hsoblash mmk. e es. k Jm k.5 Qoldq ξ fksag Jmξ < am tekslkdag bacha qtbla bo cha olad..4-msol. masala qaam..5 fomlaga ko a ; e es e Ito b bog laml sohada Laplas teglamas ch Dle masalas echsh sh soha doa ok qo am tekslkka kofom akslatsh a Passo fomlas oqal echlad..5-msol. ;.6
39 bo' lg ada bo' lg ada.7 masala echg. Yechsh. e fksa <<π o lak η> qo tekslkka kofom akslatad. Bda.7 shat.7 fomlaga ko a ~ bo'lg ada bo' lg ada ~ dt t actg. e cos e s almashtsh olb.6-.7 masalag echm hosl qlam. actg e cos. s
40 X U L O S A Kompleks o gachl fksala aaas ks o g qo llash jhatda afaqat matematkag o alshla balkm fka meaka a boshqa fa o alshladag ko plab masalala echshda asos match ostas o tad. Kompleks o gachl fksala aaasg asos sh ostas hsoblaga b qmatl aaltk fksa g haqq a mahm qsm qo shma gamok fksala tshchas o g qo llash jhatda mhm o egallad. Gamok fksa matematk fkag asos teglamalada b bo lga Laplas teglamasg echmd. Laplas teglamas ch qo lga chegaa masalala Dle Nema echshda G fksas mhm ol o ad. Ushb malaka bt sh gamok a sbgamok fksalag asos ossala o gashga bag shlaga bo lb sbgamok fksa b qmatl aaltk fksag modl ok modlg logafm sfatda qaalad. Sh maqsadda shg I-bobda gamok fksa b qmatl aaltk fksag haqq ok mahm qsm sfatda qaalb asos ossalada o ta qmat haqdag teoema maksmm psplaga do msolla echb o galga. B b tomoda kompleks o gachl fksala aaasg asos tshchala chqoq o gab mstaql tada boshqa masalala echshga qollash mkoat besa kkch tomoda b ottlga tajbala asosda mm o ta ta lm akademk ltse a kasb-a kollejlada b blmla keg qamoda qo lla olsh mkoatga ega bo lad. Ishg II-bobda gamok fksa a Laplas tglamas ch qo lga chegaa Dle masalasg echm sfatda kofom akslatsh oqal echm tegal ko shda fodalash masalala o galga. Xlosa qlb atgada kompleks a haqq o gachl fksala oasdag bo lash b fksala oqal chl o gash mkoat bead.
41 Fodalalga adabotla :. Laet e.a. Shabat B.V. etod teo fkskompleksogo peemeogo: Ucheboe posobe dla esteto..: Naka Gl.ed.f.- mat. lt s.. Sdoo Y.V. Fedok.V. Shab М.I. Leks po teo fks kompleksogo peemeogo: Ucheb. dla o. М.: Naka Gl. ed. f. mat. lt s.. Saloddo.S. atematk fka teglamala. T. «O beksta» s. 4. Sobole S.L. Uaea matematchesko fk =44 s. 5. Bsade A.V. Kalcheko D.F. Sbok adach po aeam matematchesko fk.. 6. s. 6. Too A.P. Samask A.A. Uaea matematchesko fk s. 7. Koshlako B.C. Glps E.B. Smo.. Osoыe dffeesalыe aea matematchesko fk b. 8. Xdabegao G. Voso A.K. aso X.T. Kompleks aal. T. Uestet b. 9. Sadllae A. Xdobegao G. aso X. Voso A. Tche T. atematk aalda msol a masalala tplam Kompleks aal - qsm «O beksto» s.. Sojddo S. X. Salohddo. S. aqsdo Sh. Kompleks o gachg fksala aaas. T. O qtch s.. Egafo.A. Sdoo Y. V. Fedok. V. Shab.I. Bejao K. A. Sbok adach po teo aaltchesk fks.. Naka 97.. Sattoo E.N. Kompleks o gachl fksala aaas. Uslb qo llama Samaqad SamDU b..iteet essla:
42 UNDAIJA Ksh.. I-Bob.GAONIK FUNKSIYA VA UNING XOSSALAI.. Kosh-ma shat. Qo shma gamok ksala Gamok fksag asos ossala.4.. Laplas teglamasg qtb sldk a sfek koodataladag fodas.8.4. Laplas teglamasg fdametal echm a G fomlala...5 II-bob. Gamok fksa ch Dl masalas.. Laplas teglamas ch Dle masalasg qo lsh. Yechmg majdlg a agoalg.. Laplas teglamas kofom akslatshga sbata aatlg..- Doada Laplas teglamas ch Dle masalas Yqo am tekslkda Laplas teglamas ch Dle masalas..8 Xlosa 4 Fodalalga adabotla..4
43 A.Nao omdag SamDU meaka-matematka fakl tet matematka bo lmg IV-ks talabas ade Olmosg Gamok fksag asos ossala a tegal ko sh masdag bt malaka shga B A H O Ushb bt malaka sh gamok fksalag asos ossala o gash a tegal ko shda taslashga bag shlaga bo lb ksh a kkta bobda boat. Kompleks o gachl fksala aaasg asos elemetlada b b qmatl aaltk fksag haqq a mahm qsmla bo lga qo shma gamok fksalag ossala o gash mhm ahamat kasb etad. Kompleks o gachl fksala aaas o g qo llash jhatda haqq o gachl fksala aaasga qaagada ko poq mkoatga egad. B qmatl aaltk fksag haqq a mahm qsmla Kosh-ma shat qaoatlatsh qo shma gamok fksalada boat ekalgda matematk fka teglamala ch qo lga chegaa masalala tekslkdag sohalada echshda b qato qlalkla aatad. Ellptk tpdag teglamalag eg sodda ko shlada b bo lga Laplas teglamas o g qo llash jhatda mhm o egallasa kkch tomoda b teglamag echm gamok fksda boatd. Laplas telamas ch qo lga Dle masalasg echm topshda b qmatl aaltk fksala oqal makkaboq ko shdag sohalada qaalaotga masala echm topshda kofom akslatsh odamda soddaoq ko shdag sohaga o tkalb echsh qlalk tg dad.. Ishg II-bobda Laplas teglamas ch qo lga Dle masalas echm majdlg a agoalg gamok fksa kofom akslatshga sbata aatlg doa a qo am tekslkda Dle masalas echmg tegal ko sh olga. alaka bt sh bajashda talaba O.ade l daomda belga topshqla o aqtda bajashga haakat qld. Ushb sh malaka bt shla ch qo lga bacha talablaga jaob bead a g bajachs ade Olmosg blm ash deb baholash mmk.
44 Fka-matematka fala omod dotset E.N.Sattoo A.Nao omdag SamDU meaka-matematka fakl tet matematka bo lmg IV-ks talabas ade Olmosg Gamok fksag asos ossala a tegal ko sh masdag bt malaka shga T A Q I Z Bt malaka sh кompleks o gachl fksala aaasg asos tshchalaga b b qmatl aaltk fksag haqq a mahm qsmda boat bo lga qo shma gamok fksa fodaloch gamok fksag asos ossala a tegal ko shda taslashga bag shlaga. alaka bt sh ksh a kkta bobda boat bo lb -bobda gamok fksa tshchas b qmatl aaltk fksa bla bog lash g asos ossala o ta qmat haqdag teoema maksmm pspla o galga. Laplas teglamasg fdametal echm sh asosda gamok fksag tegal tas G fomlas o galga. II-bobda sbgamok fksag asos ossalada g chegaa qmat oqal tegal ko sh a Dle masalasg doa qo am tekslkdag echm kofom akslatsh oqal qlga. B b tomoda kompleks o gachl fksala aaasg asos tshchala chqoq o gab mstaql tada boshqa masalala echshga qo llash mkoat besa kkch tomoda b ottlga tajbala asosda mm o ta ta lm akademk ltse a kasb-a kollejlada b blmla keg qamoda qo lla olsh mkoatga ega bo lad. alaka bt sh bajashda talaba ade Olmos maga do b qato matealla o gagalg ma lm bo ld. Ushb sh malaka bt shla ch qo lga bacha talablaga jaob bead a g bajachs ade Olmosg blm mffaqatl hmoada so g ash deb baholash mmk.
45 SamDU dffeesal teglamala kafedas assset F.Tso
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