Q1: Let $f: \mathbb{C} \rightarrow \mathbb{C}$ be a holomorphic function given by $f(x) = \cos(x)$ for all $x \in \mathbb{R}$. What is correct?

A1: For all $z \in \mathbb{C}$, we have $f(z) = \sum_{j=0}^\infty \frac{(-1)^j}{(2j)!} z^{2j}$.

A2: For all $z \in \mathbb{C}$, we have $f(z) = \sum_{j=0}^\infty \frac{(-1)^j}{(2j+1)!} z^{2j+1}$.

A3: There are infinitely many functions $f$ that satisfy the conditions.

A4: There are exactly two functions $f$ that satisfy the conditions.

Q2: Let $f: \mathbb{R}\rightarrow \mathbb{R}$ be a $C^\infty$-function. What is correct?

A1: There is at most one function $g: \mathbb{C} \rightarrow \mathbb{C}$ that is holomorphic and satisfies $g(x) = f(x)$ for all $x \in \mathbb{R}$.

A2: There is at most one function $g: \mathbb{C} \rightarrow \mathbb{C}$ that satisfies $g(x) = f(x)$ for all $x \in \mathbb{R}$.

A3: There are two holomorphic functions $g: \mathbb{C} \rightarrow \mathbb{C}$ that satisfy $g(x) = f(x)$ for all $x \in \mathbb{R}$.