Q1: Let $f: D \rightarrow \mathbb{C}$ be a holomorphic function given by $f(z) = \frac{1}{z} + \frac{1}{z^2}$. Which claim is correct?

A1: $ f(z) = z^{-2} (1+ z) $

A2: $ f(z) = z^{-1} (1+ z) $

A3: $ f(z) = z^{-2} (1+ z^2) $

A4: $ f(z) = z^{2} (1+ z^{-1}) $

Q2: Let $f: D \rightarrow \mathbb{C}$ be a holomorphic function given by $f(z) = \frac{1}{z} + \frac{1}{z^2}$. What is the order of the pole $z_0 = 0$?

A1: $ 2 $

A2: $ 1 $

A3: $ -2 $

A4: $ 0 $

A5: $ -1 $

A6: One needs more information.

Q3: Let $f: D \rightarrow \mathbb{C}$ be a holomorphic function with a pole of order $1$ at $z_0$. What is the correct formula to calculate the residue?

A1: $$ \mathrm{Res}(f, z_0) = \lim_{z \rightarrow z_0} (z-z_0) f(z) $$

A2: $$ \mathrm{Res}(f, z_0) = \lim_{z \rightarrow z_0} \frac{d}{dz} (z-z_0) f(z) $$

A3: $$ \mathrm{Res}(f, z_0) = \frac{1}{2} \lim_{z \rightarrow z_0} (z-z_0) f(z) $$

A4: $$ \mathrm{Res}(f, z_0) = \frac{1}{3!} \lim_{z \rightarrow z_0} \frac{d}{dz} (z-z_0) f(z) $$

Q4: Let $f: D \rightarrow \mathbb{C}$ be a holomorphic function given by two holomorphic functions in the form $f(z) = \frac{h(z)}{g(z)}$ with a pole of oder 1 at $z_0$. This means that $g$ has a zero of order 1 at $z_0$. Which statement is correct?

A1: $$ \mathrm{Res}(f, z_0) = \lim_{z \rightarrow z_0} (z-z_0) \frac{h(z)}{g(z) - g(z_0)} = \frac{h(z_0)}{g^\prime(z_0)}$$

A2: $$ \mathrm{Res}(f, z_0) = 1$$

A3: $$ \mathrm{Res}(f, z_0) = \frac{h(z_0)}{g(z_0)} $$

A4: $$ \mathrm{Res}(f, z_0) = \frac{1}{3!} \lim_{z \rightarrow z_0} \frac{d}{dz} (z-z_0) f(z) = g^\prime(z_0) $$