Nce tus lej complex rau lub zog ntuj

Hauv kev tshaj tawm no, peb yuav xav txog yuav ua li cas tus lej nyuaj tuaj yeem tsa tau lub zog (xws li siv cov qauv De Moivre). Cov khoom siv theoretical yog nrog rau cov piv txwv kom nkag siab zoo dua.

Nce tus lej complex rau lub zog

Ua ntej, nco ntsoov tias tus lej complex muaj daim ntawv dav dav: z = ib + bi (algebraic form).

Tam sim no peb tuaj yeem mus ncaj qha rau kev daws teeb meem.

Square naj npawb

Peb tuaj yeem sawv cev rau qib raws li cov khoom lag luam ntawm tib yam, thiab tom qab ntawd nrhiav lawv cov khoom (thaum nco txog qhov ntawd i² = -1).

z² = (ib + a)² = (a + bi) (ib + bi)

Piv txwv 1:

z=3+5i

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

Koj tseem tuaj yeem siv, piv txwv li lub square ntawm cov lej:

z² = (ib + a)² = a² + 2 ⋅ a ⋅ ib + (bi)² = a² + 2 ib-b²

Nco ntsoov: Nyob rau hauv tib txoj kev, yog tias tsim nyog, cov qauv rau lub square ntawm qhov sib txawv, lub voos xwmfab ntawm cov lej / qhov sib txawv, thiab lwm yam tuaj yeem tau.

Nth degree

Nce tus lej complex z yam n yooj yim dua yog tias nws tau sawv cev hauv daim ntawv trigonometric.

Nco qab tias, feem ntau, cov cim ntawm tus lej zoo li no: z = |z| ⋅ (cos φ + i ⋅ sin φ).

Rau exponentiation, koj tuaj yeem siv De Moivre's formula (yog li lub npe hu ua English mathematician Abraham de Moivre):

zⁿ = | z |ⁿ ⋅ (cos(nφ) + i ⋅ sin(nφ))

Cov mis yog tau los ntawm kev sau ntawv hauv daim ntawv trigonometric (cov qauv tau muab sib npaug, thiab cov lus sib cav ntxiv).

Piv txwv li 2

Nce tus lej complex z = 2 ⋅ (cos 35° + i ⋅ sin 35°) mus rau qib yim.

tshuaj

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