Te whakanui i te tau matatini ki te mana maori

I roto i tenei whakaputanga, ka whakaarohia me pehea e taea ai te whakanui i te tau matatini ki te mana (tae atu ki te whakamahi i te tauira De Moivre). Kei te taha o te rauemi ariā me nga tauira kia pai ake te maarama.

Te whakanui i te tau matatini ki te mana

Tuatahi, kia mahara he tau matatini te ahua whanui: z = a + bi (ahua taurangi).

Inaianei ka taea e taatau te haere tika ki te otinga o te raru.

Tau tapawha

Ka taea e tatou te tohu i te tohu hei hua o nga ahuatanga rite, katahi ka kimi i a raatau hua (i te maumahara ki tera i² =-1).

z² = (a + bi)² = (a + bi)(a + bi)

Hei tauira 1:

z=3+5i

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

Ka taea hoki e koe te whakamahi, ara te tapawha o te tapeke:

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

Tuhipoka: Waihoki, mehemea e tika ana, ka taea te whiwhi i nga tauira mo te tapawha o te rereketanga, te mataono o te tapeke / rereke, me etahi atu.

Tohu Nth

Whakaarahia he tau matatini z i roto i te ahua n he maamaa ake mena ka whakaatuhia ki te ahua pakoko.

Kia maumahara, i te nuinga o te waa, penei te ahua o te tohu o te tau: z = |z| ⋅ (cos φ + i ⋅ hara φ).

Mo te tauine, ka taea e koe te whakamahi Te tātai a De Moivre (i tapaina i muri i te tohunga pangarau Ingarihi a Abraham de Moivre):

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

Ka whiwhi te tātai mā te tuhi i te puka pākoki (ka whakareatia ngā kōwae, ka tāpirihia ngā tohenga).

tauira 2

Whakaarahia he tau matatini z = 2 ⋅ (cos 35° + i ⋅ hara 35°) ki te tohu tuawaru.

otinga

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