kompleks sonlar nazariyasi 2

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1576494051.doc b ) , ( b a = a ib a + = a a b a re = a a im = b a ib a + = a ib a - = a 1 - = i , 1 2 - = i , 3 i i - = , 1 4 = i i i k = + 1 4 1 2 4 - = + k i i i k - = + 3 4 1 4 = k i 1 2 25 4 102 - = = + · i i 1 3 50 4 203 - = = + · i i 1 2 128 4 514 - = = + · i i ) ( ) ( ) ( ) ( bc ab i bd ac id c ib a + + - = + × + = × b a ib a …

2 1 j j + × = i e r r 1 1 1 j i e r z = 2 2 2 j i e r z = = = = - ) ( 2 1 2 1 2 1 2 1 2 1 j j j j i i i e r r e r e r z z [ ] ) sin( ) cos( 2 1 2 1 2 1 j j j j - + - = i r r a i re z = j j in e n r n i re n z = = ÷ ø ö ç è æ ) sin (cos j j n i n n r n z + = 1 = r [ ] j j j j n n n i r sin cos ) sin (cos + = + j i re z = j …

mallar agar α=a+ib va β=c+id kompleks sonlar berilgan bo`lsa: 1. qo`shish va ayirish. α±β=(a+ib)±(c+id)=(a±c)+i(b±d) 2. ko`paytirish va bo`lish agar va o`zaro qo`shma sonlar berilgan bo`lsa: , 1-misol. kompleks sonlarning yig`indisi, ayirmasi, ko`paytmasi va nisbatini toping. yechish. 1. 2. 3. 4. 3. kompleks sonning geometrik tasviri va kompleks tekslik to`g`ri burchakli dekart koordinatalar sistemasi ni tanlab, uning abssissalar o`qiga ning haqiqiy qismi x ni, ordinatalar o`qiga esa mavhum qismining koeffitsienti y ni joylashtirsak, tekislikda nuqtaga ega bo`lamiz. ana shu nuqta kompleks sonning geometrik tasviri deb qabul qilingan. shunday qilib, har bir kompleks songa tekislikda birgina nuqta va aksincha, tekislikdagi har bir nuqta uchun bitta kompleks son mos keladi. o`q – haqiqiy o`q, 0y – mavhum o`q, tekislik esa kompleks tekislik deyiladi. ko`pincha kompleks sonning geometrik tasviri sifatida koordinatalar boshini tekislikdagi nuqta bilan tutashtiruvchi vektor ham qabul qilinadi. bu vektorning moduli yoki uzunligi: 4. kompleks sonning trigonometrik va ko`rsatkichli shakli 1–chizmadan ko`rinadiki: …

berilgan kompleks sonlarni darajaga ko`tarish va ildizdan chiqarish algebraik shaklda berilgan kompleks sonni n-darajaga ko`tarish uchun, uni avval trigonometrik shaklga keltirilib uning modulini shu darajaga ko`tarib, argumentini n ga ko`paytirish kerak: (5.1) ga muavr formulasi deyiladi. 3-misol ni hisoblang yechish. dastlab qavslar ichidagi sonni trigonometrik shaklga keltirib olamiz: . endi (5.1) formulaga asosan, buni darajada ko`tarib soddalashtiramiz: kompleks son berilgan bo`lsa, uning istalgan darajali ildizlarini topish bilan shug`ullanamiz. agar bo`lsa, soni z ning n-darajali ildizi deyilib (5.2) ko`rinishda yoziladi. biz mana shu sonni topish uchun dastlab berilgan z sonni trigonometrik shaklga keltiramiz: kompleks sonlar ustida to`rt amalni bajargan vaqtimizda yana kompleks sonlar hosil bo`lishini ko`rgan edik. kompleks sonning ildizi ham kompleks son bo`ladi,ya`ni (5.3), bunda k=0,1,2,3…, qiymatlarni qabul qilish mumkin. demak, algebraik formada berilgan kompleks sondan ildiz chiqarish uchun, avval uni trigonometrik shaklga keltirib, moduldan shu darajali ildiz chiqariladi, argumenti esa ildiz ko`rsatkichiga bo`linadi. 4. misol. ning qiymatlarini toping. yechish. …

iz chiqarish. kompleks sonning n-darajali ildizi bo`lsa, ya`ni , , , uchun, , (6.4) demak, trigonometrik formada berilgan kompleks sondan ildiz chiqarish uchun, moduldan shu darajali ildiz chiqariladi, argumenti esa ildiz ko`rsatkichiga bo`linadi. 7. kompleks sonning logarifmi kompleks son berilgan bo`lsin. , (7.1) (7.2) 5-misol. ning logarifmini toping. yechish. , , , , _1405780341.unknown _1405780374.unknown _1405780390.unknown _1405780406.unknown _1405780422.unknown _1405780430.unknown _1405780434.unknown _1405780436.unknown _1405780438.unknown _1405780439.unknown _1405780440.unknown _1405780437.unknown _1405780435.unknown _1405780432.unknown _1405780433.unknown _1405780431.unknown _1405780426.unknown _1405780428.unknown _1405780429.unknown _1405780427.unknown _1405780424.unknown _1405780425.unknown _1405780423.unknown _1405780414.unknown _1405780418.unknown _1405780420.unknown _1405780421.unknown _1405780419.unknown _1405780416.unknown _1405780417.unknown _1405780415.unknown _1405780410.unknown _1405780412.unknown _1405780413.unknown _1405780411.unknown _1405780408.unknown _1405780409.unknown _1405780407.unknown _1405780398.unknown _1405780402.unknown _1405780404.unknown _1405780405.unknown _1405780403.unknown _1405780400.unknown _1405780401.unknown _1405780399.unknown _1405780394.unknown _1405780396.unknown _1405780397.unknown _1405780395.unknown _1405780392.unknown _1405780393.unknown _1405780391.unknown _1405780382.unknown _1405780386.unknown _1405780388.unknown _1405780389.unknown _1405780387.unknown _1405780384.unknown _1405780385.unknown _1405780383.unknown _1405780378.unknown _1405780380.unknown _1405780381.unknown _1405780379.unknown _1405780376.unknown _1405780377.unknown _1405780375.unknown _1405780357.unknown _1405780366.unknown _1405780370.unknown _1405780372.unknown _1405780373.unknown _1405780371.unknown _1405780368.unknown _1405780369.unknown _1405780367.unknown _1405780362.unknown _1405780364.unknown _1405780365.unknown _1405780363.unknown _14057

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1576494051.doc b ) , ( b a = a ib a + = a a b a re = a a im = b a ib a + = a ib a - = a 1 - = i , 1 2 - = i , 3 i i - = , 1 4 = i i i k = + 1 4 1 2 4 - = + k i i i k - = + 3 4 1 4 = k i 1 2 25 4 102 - = = + · i i 1 3 50 4 203 - = = + · i i 1 2 128 4 514 - = = + · i i …

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