Ukuphakamisa inombolo eyinkimbinkimbi emandleni emvelo

Kulolu shicilelo, sizocubungula ukuthi inombolo eyinkimbinkimbi inganyuswa kanjani ibe namandla (okuhlanganisa nokusebenzisa ifomula ye-De Moivre). Izinto ezithiyori zihambisana nezibonelo zokuqonda kangcono.

Ukuphakamisa inombolo eyinkimbinkimbi emandleni

Okokuqala, khumbula ukuthi inombolo eyinkimbinkimbi inefomu elijwayelekile: z = a + bi (ifomu le-algebraic).

Manje singaqhubeka ngqo nesixazululo senkinga.

Inombolo yesikwele

Singamela idigri njengomkhiqizo wezinto ezifanayo, bese sithola umkhiqizo wabo (ngenkathi sikhumbula lokho i² =-1).

z² = (a + bi)² = (a + bi)(a + bi)

Isibonelo 1:

z=3+5i

z² = (3 + 5i)² = (3 + 5i)(3 + 5i) = 9 + 15i + 15i + 25i² = -16 + 30i

Ungasebenzisa futhi, okungukuthi isikwele sesamba:

z² = (a + bi)² = a² + 2 ⋅ a ⋅ bi + (bi)² = a² + 2aba – b²

Qaphela: Ngendlela efanayo, uma kunesidingo, amafomula wesikwele somehluko, i-cube yesamba / umehluko, njll.

Iziqu ze-Nth

Phakamisa inombolo eyinkimbinkimbi z ngohlobo n kulula kakhulu uma imelelwa ngefomu le-trigonometric.

Khumbula ukuthi, ngokuvamile, ukuphawula kwenombolo kubonakala kanje: z = |z| ⋅ (cos φ + i ⋅ sin φ).

Ukuze uthole i-exponentiation, ungasebenzisa Ifomula kaDe Moivre (eqanjwe kanjalo ngesazi sezibalo esiyiNgisi u-Abraham de Moivre):

zⁿ = | z |ⁿ ⋅ (cos(nφ) + i ⋅ sin(nφ))

Ifomula itholakala ngokubhala ngefomu le-trigonometric (amamojula aphindaphindeka, futhi izimpikiswano ziyengezwa).

Isibonelo se-2

Phakamisa inombolo eyinkimbinkimbi z = 2 ⋅ (cos 35° + i ⋅ isono 35°) kuze kufike ezingeni lesishiyagalombili.

Isixazululo

z⁸ = 2⁸ ⋅ (cos(8 ⋅ 35°) + i ⋅ isono(8 ⋅ 35°)) = 256 ⋅ (cos 280° + i sin 280°).