1. Express 1 - i in Polar Form
Find the magnitude: |1 - i| = √(1² + (-1)²) = √2
Find the argument (angle):
tan(θ) = -1/1 = -1
Since 1 - i is in the fourth quadrant, θ = -π/4
1 - i = √2 * (cos(-π/4) + i * sin(-π/4)) = √2 * e^(-iπ/4)
2. Express (z + i)^4 in Polar Form
Let z + i = r * (cos(θ) + i * sin(θ))
(z + i)^4 = r^4 * (cos(4θ) + i * sin(4θ))
3. Equate
r^4 * (cos(4θ) + i * sin(4θ)) = √2 * e^(-iπ/4)
r^4 * (cos(4θ) + i * sin(4θ)) = √2 * (cos(-π/4) + i * sin(-π/4))
4. Solve for r and θ
Magnitude: r^4 = √2
r = (√2)^(1/4) = 2^(1/8)
Argument: 4θ = -π/4 + 2kπ, where k = 0, 1, 2, 3
θ = -π/16 + (kπ)/2
5. Find the Four Roots
k = 0:
θ = -π/16
z + i = 2^(1/8) * (cos(-π/16) + i * sin(-π/16))
z = 2^(1/8) * (cos(-π/16) + i * sin(-π/16))
k = 1:
θ = -π/16 + π/2 = 7π/16
z + i = 2^(1/8) * (cos(7π/16) + i * sin(7π/16))
z = 2^(1/8) * (cos(7π/16) + i * sin(7π/16)) - i
k = 2:
θ = -π/16 + π = 15π/16
z + i = 2^(1/8) * (cos(15π/16) + i * sin(15π/16))
z = 2^(1/8) * (cos(15π/16) + i * sin(15π/16)) - i
k = 3:
θ = -π/16 + 3π/2 = 23π/16
z + i = 2^(1/8) * (cos(23π/16) + i * sin(23π/16))
z = 2^(1/8) * (cos(23π/16) + i * sin(23π/16)) - i
Therefore, the solutions to the equation (z + i)^4 = 1 - i are:
z = 2^(1/8) * (cos(-π/16) + i * sin(-π/16)) - i
z = 2^(1/8) * (cos(7π/16) + i * sin(7π/16)) - i
z = 2^(1/8) * (cos(15π/16) + i * sin(15π/16)) - i
z = 2^(1/8) * (cos(23π/16) + i * sin(23π/16)) - i
These solutions can be further simplified by converting them from polar form to rectangular form (a + bi).
