Aurora has defined a function $f: \mathbb{C} \to \mathbb{C}$ given by $f(z) = (-11 + 4i)z^{-1} + (-11 - 4i)z$. (a) Complex conjugation corresponds to the geometric operation of reflection in the real axis. (b) Explain geometrically or otherwise why $f(z)$ is real whenever $|z| = 1$. Note that the question continues below this essay box. Please scroll down. (c) Find the value of $|(-11 + 4i) + z|^2 - f(z)$ for any $z$ with modulus 1. $|(-11 + 4i) + z|^2 - f(z) = $
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1. Let f(z) = f(x + iy) = x + y + i(x³y – y²). Find (a) f(-1 + 3i) (b) f(3i – 2) 2. Let f(z) = z² + 4zˉz – 5 Re(z) + Im(z). Find (a) f(-3 + 2i) (b) f(2i – 1) 3. Find f(1 + i) for the following functions. (a) f(z) = z + z⁻² + 5 (b) f(z) = 1 / (z² + 1) 4. Find f(2i – 3) for the following functions. (a) f(z) = (z + 3)³(z – 5i)² (b) f(z) = (z + 2 – 3i) / (z + 4 – i) 5. Let f(z) = z²¹ – 5z⁷ + 9z⁴. Use polar coordinates to find (a) f(-1 + i) (b) f(1 + i√3) 6. Express f(z) = zˉ² + (2 – 3i)z in the form u + iv. 7. Express f(z) = (z + 2 – i) / (z – 1 + i) in the form u + iv.
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