Esercizi sui numeri complessi

Esercizi sui numeri complessi

Esercizio 1

Z = ^{2 + 5i}/_{1 - 3i} = ^{2 + 5i}/_{1 - 3i} · ^{1 + 3i}/_{1 + 3i} = ^{2 + 6i + 5i - 15}/_{1 + 9} = ^-13/₁₀ + ¹¹ⁱ/₁₀

Esercizio 2

z = ^{i (5 + 6i)2 - 4}/_{1 - i} = i (25 - 36 + 60i) - 4 · ^{(1 + i)}/_{(1 + i)} = -11i - 60 - ^{4 + 4i}/₂ = -11i - 60 - 2 - 2i = -62 - 13i

Esercizio 3

Z = 1 - i p = √(1 + (-1)²) = √2 cosθ = ¹/_√2 sinθ = ^-1/_√2 θ = ⁷/₄π

Z = √2 (cos ⁷/₄π + i sin ⁷/₄π)

Esercizio 4

Z = (2 - 2i)⁵ θ = 2π ^√2/_√2 = ^π/₂ = ³/₂π

Z = (2√2)⁵ (cos ³/₂π + i sen ³/₂π)

Esercizio 1

Z = ^{2 + 5i}⁄_{1 - 3i} = ^{2 + 5i · 1 + 3i}⁄_{1 - 3i · 1 + 3i} = ^{2 + 6i + 5i - 15}⁄_{1 + 9} = ^{- 13 + 11i}⁄₁₀

Z = -¹³⁄₁₀ + ¹¹ⁱ⁄₁₀

Esercizio 2

Z = i(5 + 6i)² - ⁴⁄_{1 - i} = i(25 - 36 + 60i) - 4 · ^{(1 + i)}⁄_{(1 + i)} = -11i - 60 - ^{4 + 4i}⁄₂ = -11i - 60 - 2 - 2i = -62 - 13i

Esercizio 3

Z = 1 - i p = ^{√(1 + (-1)2)} = √2 cosθ = ¹⁄_√2 sinθ = -¹⁄_√2 θ = -⁷⁄₄π

Z = √2 (cos ⁷⁄₄π + i·sin ⁷⁄₄π)

Esercizio 4

Z = (2 - 2i)⁵

X = (2 - 2i) 1 × 1 = √(2² + (-2)²) = 2√2 cosθ = ²⁄_2√2 = ^√2⁄₂ sinθ = -²⁄_2√2 = -^√2⁄₂ θ = -⁷⁄₄π

X = 2√2 (cos ⁷⁄₄π + i sin ⁷⁄₄π)

Z = (2√2)⁵ (cos ³⁵⁄₄π + i sin ³⁵⁄₄π)√₂ · √₂ · √₂ · √₂ · √₂

z = 128 √₂ (6Θ^3π/4 + i sen 3π/4 (1))3 Θ/4 α = Θ^3π/4 + 3π

z = 128 √₂ (6Θ^3π/4α + i sen 3π/4 α)

sen 3π/4 α = - √₂/2

z = 128 √₂ (- √₂/2 + i √₂/2)

z = 128 (1 - 1 + i i)

Esercizio

z³ - |z|² = 0

z = x + i y

1 ε ℂ √_{x² + y²}

x³ + 3 i x² y - 3 x y² - i y³ = x² + y²

x³ - 3 x² y + i (3 x² y - y³) = x² + y²

{x³ - 3 x y² = x² + y²

3 x² y - y³ = 0

x (3 x² - y²) = 0

y = 0

x³ - x² = 0

x² (x - 1) = 0

x = 0   x = 1

x² + x² = 0

2 x² + x³ = 0

y² x + 3 y² x²

x² (2 x + 1) = 0

x = 0

y = ± √₃/2

y₁ = √₃/2

y₂ = - √₃/2

(0, 0) (1, 0) (-½, √₃/2) (-½, -√₃/2)

z₁ = 0, z₂ = 1, z₃ = -½ + i √₃/2, z₄ = -½ - i √₃/2

Secondo metodo

z̅ = ρ (cosθ + i sinθ)

ρ³ (cos3θ + i sin3θ) = ρ²

ρ³ = ρ²

cos3θ = 1

sin3θ = 0

ρ² (ρ - 1) = ρ = 0

ρ = 0

ρ = 1

3θ = 2kπ

θ = 2/3 kπ

3θ = 0

θ = 0

θ = 2/3 π

θ = 4/3 π

θ = 2/3 kπ

k = 0,1,2

θ ≠ 0

ρ = 0 : z = 0

ρ = 1

θ = 0 : 1 (cos0 + i sin0) = 1 z = 1

ρ = 1

θ = 2/3 π : z = (cos 2/3 π + i sin 2/3 π) = -1/2 + i √3/2

ρ = 1

θ = 4/3 π : z = -1/2 - i √3/2

Conclus

z₁ ≅ i z₂ = Δ - i

z₁ z₂ 1 - i 1 + i 1 + i 1 + i

z₁ = i

z₂ = 1 - i

|z₁| = ^|1|⁄_√2 = 1

senΘ = ¹⁄_√2 = ½

Θ = π/2

|z₂| = ^{(√12 + (-1)2)}⁄_√2 = √2

cosΘ = ^½⁄_√2 = ½

sinΘ = - ¹⁄_√2 = -√2/2

Θ = π/4

^z₁⁄_z2 = ¹⁄_√2 [cos(^π⁄₂ - ^π⁄₄π) + i sen(^π⁄₂ - ^π⁄4u)

^z₁⁄_z2 = ¹⁄_√2 [cos(-^π⁄₄π) + i sen(-^π⁄₄π)]

-¹⁄_√2 [-√2/2 + I√2/2] = -½ + ½ i

Radici quarte

ζ⁴ = 1

ζ^a = W

W = z

|w| = 1

W = 1 (cosΘ + i senΘ)

cos Θ = ¹⁄₂ = 1

Θ = 0 + 2κπ

W = 1 (cos(0+2kπ) + i sen(0+2kπ))

ζ^a = w

ζ = ^{√w = W1}⁄₄

ζ = √W = ^√1⁄₂ (cos[^2kπ⁄₄] + i sen[^2kπ⁄₄])

ζ = (cos[^kπ⁄₂] + i sen[^kπ⁄₂])

• K = 0, 1, 2, 3
• K = 0 ζ = 1
• K = 1 ζ = i
• K = 2 ζ = -1
• K = 3 ζ = -i