Let $f(z)$ be a power-series (with complex coefficients) centred at $0 \in \mathbb{C}$ and with a radius of convergence 2. Suppose that $f(0) = 0$. Choose the correct statement(s) from below: (A) $f^{-1}(0) = \{0\}$; (B) If $f$ is a non-constant function on $\{|z| < 2\}$, then $f^{-1}(0) = \{0\}$; (C) If $f$ is a non-constant function, then for all $\zeta \in \mathbb{C}$ with sufficiently small $|\zeta|$, the equation $f(z) = \zeta$ has a solution; (D) $$\int_{\gamma} f^{(n)}(z)\,\mathrm{d}z = 0$$ for every $n \geq 1$, where $\gamma$ is a unit circle centred at 0, oriented clockwise, and $f^{(n)}$ is the $n$th derivative of $f(z)$.
Let $f(z)$ be a power-series (with complex coefficients) centred at $0 \in \mathbb{C}$ and with a radius of convergence 2. Suppose that $f(0) = 0$. Choose the correct statement(s) from below:\\
(A) $f^{-1}(0) = \{0\}$;\\
(B) If $f$ is a non-constant function on $\{|z| < 2\}$, then $f^{-1}(0) = \{0\}$;\\
(C) If $f$ is a non-constant function, then for all $\zeta \in \mathbb{C}$ with sufficiently small $|\zeta|$, the equation $f(z) = \zeta$ has a solution;\\
(D)
$$\int_{\gamma} f^{(n)}(z)\,\mathrm{d}z = 0$$
for every $n \geq 1$, where $\gamma$ is a unit circle centred at 0, oriented clockwise, and $f^{(n)}$ is the $n$th derivative of $f(z)$.