Final answer:
The real part of ω, Re(ω), is 0. This conclusion is derived by expressing z in polar form given |z|=1, then substituting into the equation ω=(z-1)/(z+1), and simplifying.
Step-by-step explanation:
Let's solve the mathematical problem completely by finding the real part of ω when |z|=1 and ω=(z−1)/(z+1), given that z≠−1.
Since we know that |z| = 1, z can be represented in polar form as z = cos(θ) + i sin(θ), where θ is the argument of z. Substituting this into the given expression for ω, we get:
ω = (cos(θ) + i sin(θ) - 1) / (cos(θ) + i sin(θ) + 1)
Now, let's multiply the numerator and the denominator by the conjugate of the denominator to remove the imaginary part from the denominator:
ω = [(cos(θ) + i sin(θ) - 1) * (cos(θ) - i sin(θ) + 1)] / [(cos(θ) + i sin(θ) + 1) * (cos(θ) - i sin(θ) + 1)]
Expanding both the numerator and the denominator and simplifying, we yield:
ω = [cos^2(θ) + sin^2(θ) - 1] / [cos^2(θ) + sin^2(θ) + 2cos(θ) + 1]
Given that cos^2(θ) + sin^2(θ) = 1, the expression simplifies to:
ω = 0 / (2cos(θ) + 2)
Therefore, the real part of ω (ω being a complex number in the form a + bi), is found by looking at the real portion of the above fraction. Since the numerator is 0, Re(ω) = 0/anything is 0.
Hence, the real part of ω, Re(ω), is 0.