Find all functions $f(z)$ that are analytic everywhere in the entire complex plane that satisfy $f(2-i)=4i$

Question

This question is apparently related to Cauchy's Integral Formula and related theorems, but I honestly don't know how to start, other than potentially saying that $f(z)$ is every function such that "insert Gauss' Mean Value Theorem here"

Answer 1

If $f$ is an entire function with $f(z_0) = w_0$ for some (arbitrary) $z_0,\, w_0 \in \mathbb{C}$, then $g = f - w_0$ is an entire function with $g(z_0) = 0$. And that means that there is an entire function $h$ with $g(z) = (z - z_0)\cdot h(z)$ for all $z\in\mathbb{C}$. So we have $f(z) = (z-z_0)\cdot h(z) + w_0$.

Conversely, if $h$ is an arbitrary entire function, then $f$ defined by $f(z) = (z-z_0)\cdot h(z) + w_0$ is an entire function with $f(z_0) = w_0$.

Now specialise to the given $z_0$ and $w_0$.