Kukwidza nhamba yakaoma kusvika kusimba rezvisikwa

Mubhuku rino, tichaona kuti nhamba yakaoma ingasimudzwa sei kune simba (kusanganisira kushandisa fomula yeDe Moivre). The theoretical material inoperekedzwa nemienzaniso yekunzwisisa zvirinani.

Kukwidza nhamba yakaoma kusvika kune simba

Chekutanga, yeuka kuti nhamba yakaoma ine fomu rakajairwa: z = a + bi (fomu yealgebraic).

Iye zvino tinogona kuenderera zvakananga kune mhinduro yedambudziko.

Square number

Tinogona kumiririra dhigirii sechigadzirwa chezvinhu zvakafanana, tobva tawana chigadzirwa chavo (tichirangarira izvozvo i² =-1).

z² = (a + bi)² = (a + bi)(a + bi)

Muenzaniso 1:

z=3+5i

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

Iwe unogona zvakare kushandisa, kureva sikweya yehuwandu:

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

Cherechedza: Nenzira imwecheteyo, kana zvichidikanwa, mafomula echikwere chemusiyano, cube yehuwandu / musiyano, nezvimwe zvinogona kuwanikwa.

Nth degree

Simudza nhamba yakaoma z mune n zvakanyanya nyore kana inomiririrwa mune trigonometric fomu.

Rangarira kuti, kazhinji, notation yenhamba inoita seizvi: z = |z| ⋅ (cos φ + i ⋅ chivi φ).

Kuti uwedzere, unogona kushandisa Maitiro aDe Moivre (akatumidzwa zita remuChirungu masvomhu Abraham de Moivre):

zⁿ = | z |ⁿ ⋅ (cos(nφ) + i ⋅ chivi(nφ))

Iyo formula inowanikwa nekunyora mune trigonometric fomu (mamodule anowedzerwa, uye nharo dzinowedzerwa).

muenzaniso 2

Simudza nhamba yakaoma z = 2 ⋅ (cos 35° + i ⋅ chivi 35°) kusvika padanho rechisere.

mhinduro

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