Esercizi numeri complessi

Esercizio 1

z₁ = 1   Re = 1   Im = 0   ζ₁ = 1

z₂ = -2   Re = -2   Im = 0   ζ₂ = -2

z₃ = 3i   Re = 0   Im = 3   ζ₃ = -3i

z₄ = -4i   Re = 0   Im = -4   ζ₄ = 4i

z₅ = 1 + i   Re = 1   Im = 1   ζ₅ = 1 - i

z₆ = 1 - i   Re = 1   Im = -1   ζ₆ = 1 + i

z₇ = (-√3)/2 ; = 1 - √3/2i   Re = 1/2   Im = -√3/2   ζ₇ = 1/2 + √3/2 i

Esercizio 3

z₁ = 1 + i   z₂ = 1 - i   z₁ · z₂ = (1 + i)(1 - i) = 1 + 1 = 2

z₆ = 1 + 2i   z = 3 - 4i   z₆ · z = (1 + 2i)(3 - 4i) = 3 - 4i + 6i + 8 = 11 + 2i

Esercizio 12

ⁱ/_{1 - i} = ^-i/_{3 - i}   ^{1 + i}/_{1 + i} = ^{i - 1}/_{1 + i}

^{i - i}/₂ = 1/2 + 1/2 i

(1 + i)²(i - 4) = (1 - 4i + 2i)(i - 1) = -2 - 2i

Esercizio 1

z₁ = 1   Re = 1   Im = 0   z̅₁ = 1

z₂ = -2   Re = -2   Im = 0   z̅₂ = -2

z₃ = 3i   Re = 0   Im = 3   z̅₃ = -3i

z₄ = -αi   Re = 0   Im = -α   z̅₄ = αi

z₅ = 1 + i   Re = 1   Im = 1   z̅₅ = 1 - i

z₆ = 1 - i   Re = 1   Im = -1   z̅₆ = 1 + i

z₇ = (-√3i)/2; = 1 - (√3/2)i   Re = 1/2   Im = -√3/2   z̅₇ = 1/2 + (√3/2)i

Esercizio 3

z₁ = 1 + i   z₂ = 1 - i   z₁ · z₂ = (1 + i)(1 - i) = 1 + 1 = 2

z₆ = 1 + 2i   z₇ = 3 - 4i   z₆ · z₇ = (1 + 2i)(3 - 4i) = 3 - 4i + 6i + 8 = 11 + 2i

Esercizio 13

ⁱ/_{1 - i} = ⁱ/_-i, ^{1 + i}/_{1 + i} = ^{i - 1}/_{1 + i} = ^{i - 1}/₂ = 1/2 + 1/2 i

(1 + i)² (i - 1) = (1 - 2i - 1) (i - 1) = -2 - 2i

(2 - i) . (1 + i) = (2 - i) . (1 - i) = 2 - 2i - i - 1 = 1 - 3i

3 + 4i   3 + 4i   3 + 4i   3 + 4i

7 - 3i   3 - 4i   3 - 4i   9 - i   -2 = -9 - 13i = -9 - 13

3 + 4i   3 - 4i   9 + 16   25   25

Esercizio 15

z = √3 + i

|z| = √(√3)² + (1)² = 2

cos θ = √3/2

sin θ = 1/2

θ = π/3

Arg(z) = π/3

z = 2 (cos π/3 + i sin π/3) = 2 e^iπ/3

z = -π i

|z| = √(-π)² = π

cos θ = 0

sin θ = -π/π = -1

θ = 3π/2

z = π (cos 3π/2 + i sin 3π/2) = π i e^{i 3π/2}

z = i - 1   = i - 1   = (i - 1)²   = (-1 + i) - 2i   = - i

i + 1   i - 1   2

z = i   |z| = 1   cos θ = 0   sin θ = 1   θ = π/2   z = e^-i(π/2)

Esercizio 16

cos π + i sin π

z = -1

cos α = -1    sin α = 0    |z| = 1

2 (cos ⁵/₆ π + i sin ⁵/₆ π)

2 (-^√3/₂ + i ¹/₂) = -√3 + i

‾³√₉ (cos ^π/₃ + i sin ^π/₃) = ³√₉ (¹/₂ + ^√3/₂ i) = ³ √₉/₂ + ³ √₉ √₃/₂ i‾

Esercizio 25

z̅² - z̅ z² = 0

z = x + i y    z̅ = x - i y

(z̅ - z̅)(z̅ + z̅) = 0

(x + i y - x + i y) (x + i y + x - i y) = 0

(2 i y) (2 x) = 0

{ 2 x = 0   2y = 0

{ x = 0   y = 0

z = 0

z^2 = \overline{z}^2

(x + iy)^2 = (x - iy)^2

x^2 + 2xyi - y^2 = x^2 - 2xyi - y^2

(x^2 - y^2) + i(2xy) = (x^2 - y^2) + i(-2xy)

2xy = -2xy

xy + xy = 0

2xy = 0

xy = 0

x = 0 V y = 0

z = 0

2z - 3iz = 1 + 4i

2(x + iy) - 3i(x - iy) = 1 + 4i

2x + 2iy - 3ix - 3y = 1 + 4i

(2x - 3y) + i(2y - 3x) = 1 + 4i

2x - 3y = 1   2y - 3x = 4

x = 1/2 + 3/2y   x = 1/2 - 33/10

x = -4/5

2y = 3/2 - 3/2y = 4   y = -11/5

4y - 3x - 3y = 8   -5y = 11

\overline{z} = -\frac{19}{5} - \frac{11}{5}i

\overline{z} = \overline{z}^2

(x + iy) = (x^2 + 2ixy - y^2)(x + iy) = (x^2 - y^2) - 2ixy

x = x^2 - y^2

y = 2xy

z = 1/4 - y^2

x = -1/2

x = -1/2

y^2 = 3/4

y = ±√3/2

Esercizio

3/(z+i) = i/(2-z)

3(2-z) = i(z+i)

6 - 3z = iz - i

z(i+3) = 7 i

z(i+3z) = 7i

(x+3x) - y + 3iy = 7

3x - y = 7

x + 3y = 0

-9y - y = 7

x = -3y

y = -7/10

x = 21/10

z = 21/10 - 7/10i

Oppure

3/(z+i) = i/(2-z)

3(2-z) = i(z+i)

6 - 3z = iz - i

z(i+3) = 7z = 7/(i+3)

7/(i+3) . (i-3)/(i-3) = 7i - 21/-1-37i - 21 = 21 - 7i/10