Find the Laurent series of the function \[ f(z)=1+\frac{3}{z+2}-\frac{8}{z+3} \] in the region \[ |z|<2. \]
Let \( \Omega \) be a non-empty open connected subset of \( \mathbb{C} \) and \( f: \Omega \to \mathbb{C} \) be a non-constant function. Let the functions \( f^2: \Omega \to \mathbb{C} \) and \( f^3: \Omega \to \mathbb{C} \) be defined by \[ f^2(z) = (f(z))^2 \quad {and} \quad f^3(z) = (f(z))^3, \quad z \in \Omega. \]
Consider the following two statements:
S1: If \( f \) is continuous in \( \Omega \) and \( f^2 \) is analytic in \( \Omega \), then \( f \) is analytic in \( \Omega \).
S2: If \( f^2 \) and \( f^3 \) are analytic in \( \Omega \), then \( f \) is analytic in \( \Omega \). Then, which one of the following is correct?
Let \( U = \{z \in \mathbb{C}: \operatorname{Im}(z) > 0\} \) and \( D = \{z \in \mathbb{C}: |z| < 1\} \), where \( \operatorname{Im}(z) \) denotes the imaginary part of \( z \).
Let \( S \) be the set of all bijective analytic functions \( f: U \to D \) such that \( f(i) = 0 \).
Then, the value of \( \sup_{f \in S} |f(4i)| \) is: