kompleks sonlar nazariyasi 2

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1576494070.doc ,...) 3 , 2 , 1 ( , = n b a n n , 1 1 a = s n n s s a a a a a + + + = + = ... ,..., 2 1 2 1 2 ¥ ® n s s s s n n = ¥ ® lim ...; ... 2 1 + + + + = n s a a a = + + = n n s a a ... 1 ) ... ( ) ... ( ) ( ... ) ( ) ( 2 1 2 1 2 2 1 1 n n n n b b b i a a a ib a ib a ib a + + + + + + + = + + + + + + , ... 2 1 n n a a a + + + = s n n b …

3 deb olsak, bo`ladi. shuning uchun, agar da, va bo`lib, bunda va lar aniq chekli son bo`lsa, qatorning ikkalasi ham yaqinlashuvchidir. teorema. agar (1.4), (1.5) qatorlarining ikkalasi ham yaqinlashuvchi bo`lsa, (1.1) ham yaqinlashuvchi bo`ladi. ta`rif. agar (1.6) qator yaqinlashuvchi bo`lsa, (1.1) qator ham yaqinlashuvchi bo`lib, u absolyut yaqinlashuvchi qator deyiladi. 22-misol. qatorning yaqinlashuvchiligi tekshirilsin. yechish. bunda va qatorlar yaqinlashuvchi, chunki ; bo`lib, chekli son bo`gani uchun berilgan qator yaqinlashuvchi. endi qatorning absolyut yaqinlashishini tekshiramiz: demak, berilgan qator absolyut yaqinlashuvchi. teorema. (qator yaqinlashining zaruriy sharti) (1.1) qatorning yaqinlashishi uchun bo`lishi zarurdir. isboti. (1.1) yaqinlashuvchi bo`lsin. lekin, ba`zan umumiy had nolga intilsa ham u qator uzoqlashuvchi bo`ladi. 23-misol qatorda bo`lib, qator yaqinlashishining zaruriy sharti bajarilgani bilan qator uzoqlashuvchi. demak, berilgan qator ham uzoqlashuvchi. 2. funksional qatorlar agar qatorning har bir hadi biror sohada aniqlangan o`zgaruvchilarning bir qiymatli funksiyasidan iborat bo`lsa, ya`ni ga funksional qator deyiladi. agar ga tegishli biror o`zgarmas kompleks sonni …

uvchi bo`lsa, u qatorning yig`indisi ham sohada uzluksiz bo`ladi. 3. darajali qatorlar funksional qatorlarning xususiy ko`rinishi bo`lgan darajali qatorlar amalyotda ko`proq ishlatiladi. ushbu ko`rinishdagi. (3.1) yoki (3.2) qatorlar darajali qatorlar deyiladi. bunda , kompleks sonlardir. teorema. (abel teoremasi). agar (3.1) qator biror nuqtada yaqinlashuvchi bo`lsa, bu qator doiraning hamma ichki nuqtalarida absolyut yaqinlashadi (13-chizma). natija. agaar (3.1) qator biror nuqtada uzoqlashuvchi bo`lsa tegsizlikni qanoatlantiruvchi ixtiyoriy nuqtada uzoqlashuvchi bo`ladi. 23-misol qator ixtiyoriy nuqtada absolyut yaqinlashuvchi, chunki har qanday bo`lsa ham biror katta raqamdan boshlab bo`ladi, yoki , qator yaqinlashuvchi. umuman (3.1) yoki (3.2) ko`rinishidagi darajali qatorlaring yaqinlashish doirasining radiusi dalamder – adamar formulasi yordamida topiladi. 1. koshi formulasi: (3.3) 2. dalamber formulasi: (3.4) 3. koshi-adamar forulasi: (3.5) bo`lsa, yaqinlashish doirasi bo`ladi. 24-misol. qatorining yaqinlashish radiusini toping. yechish. demak, qatorning yaqinlashish radiusi (sohasi): bo`lgani uchun yaqinlashish sohasi butun kompleks tekislikdan iborat. 25-misol. qatorini yaqinlashish sohasini toping. yechish. , , , bo`lgani uchun …

. (4.7) 3. (4.8) 4. (4.9) 5. (4.10) 6. (4.11) 26-misol. berilgan funksiyani darajali qatorga yoying. a) , b) yechish. a) (4.8) formuladan foydalansak: b) 5. manfiy darajali qatorlar. loran qatori ta`rif. ning manfiy darajalari bo`yicha yoyilgan ushbu qator (5.1) ga manfiy darajali qator deyiladi. bu qatorning yaqinlashish sohasi doira tashqarisidan iborat. bunda (5.2) 27-misol. bo`lsa , yaqinlashish radiusi topilsin. yechish. demak, yaqinlashish sohasi bo`ladi. ta`rif. ushbu ko`rinishdagi (5.3) qator loran qatori deyiladi, bunda (5.4) ga loran qatorining bosh qismi deyiladi va (5.5) ga loran qatorining to`g`ri qismi deyilib, da yaqinlashadi. shuning uchun loran qatori (5.6) halqada yaqinlashuvchi bo`ladi. loran qatorining koeffitsientlarini (5.7) formula bo`yicha ham topish mumkin. esa halqaga tegishli ixtiyoriy markazli aylanadan iborat. 28-misol. da loran qatorining yaqinlashish sohasi aniqlansin. yechish. a) demak, b) , . demak, 29-misol. halqada loran qatoriga yoying. yechish. buning uchun berilgan funksiyani eng sodda kasrlarga ajratib olamiz bo`lganda bo`lganda berilgan halqaga binoan ish …

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1576494070.doc ,...) 3 , 2 , 1 ( , = n b a n n , 1 1 a = s n n s s a a a a a + + + = + = ... ,..., 2 1 2 1 2 ¥ ® n s s s s n n = ¥ ® lim ...; ... 2 1 + + + + = n s a a a = + + = n n s a a ... 1 ) ... ( ) ... ( ) ( ... ) ( ) ( 2 1 2 1 2 2 1 1 n n n n b b b i a a a ib a ib a ib a + …

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