Haɓaka hadadden lamba zuwa ikon halitta

A cikin wannan ɗaba'ar, za mu yi la'akari da yadda za a iya ɗaga hadadden lamba zuwa ƙarfi (ciki har da amfani da dabarar De Moivre). Kayan ka'idar yana tare da misalai don kyakkyawar fahimta.

Ƙara hadaddun lamba zuwa iko

Na farko, ku tuna cewa hadadden lamba tana da sifar gaba ɗaya: z = a + bi (algebraic form).

Yanzu za mu iya ci gaba kai tsaye zuwa maganin matsalar.

Lambar murabba'i

Za mu iya wakiltar digiri a matsayin samfur na abubuwa iri ɗaya, sannan mu sami samfurin su (yayin tunawa da hakan i² = -1).

z² = (a + bi)² = (a + bi) (a + bi)

Misali 1:

z=3+5i

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

Hakanan zaka iya amfani da, wato square na jimlar:

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

lura: Hakazalika, idan ya cancanta, za'a iya samun ƙididdiga don murabba'in bambance-bambance, cube na jimla / bambanci, da dai sauransu.

Nth digiri

Tada hadadden lamba z a cikin kirki n mafi sauki idan an wakilta shi a sigar trigonometric.

Ka tuna cewa, gabaɗaya, bayanin lamba yana kama da haka: z = |z| ⋅ (cos φ + i ⋅ sin φ).

Don ƙarin bayani, zaka iya amfani da shi Ma'anar sunan farko De Moivre (wanda ake kira bayan masanin lissafin Ingilishi Abraham de Moivre):

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

Ana samun dabarar ta hanyar rubuce-rubuce a cikin nau'i na trigonometric (na'urorin suna ninka, kuma ana ƙara gardama).

Misali 2

Tada hadadden lamba z = 2 ⋅ (cos 35° + i ⋅ zunubi 35°) zuwa mataki na takwas.

Magani

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