About $n$-th differentiability

Question

For real function we can use MVT to show that if it is $n$-th differentiable at a point $x_0$ then it can be estimate by an degree $n$ polynomial with error term of $o((x-x_0)^n)$.
Similarly, if we are given an complex function that is $n$-th (complex) differentiable on a point $z_0$ (and not sure about its analycity), then is it true that it can be estimate by an degree $n$ polynomial with error term of $o((z-z_0)^n)$?
In other word, is it true that the function is of such form?
$c_0+c_1(z-z_0)+...+c_n(z-z_0)^n+e(z-z_0)$ where $e\in o(z^n)$

Answer 1

For $n=1$, the result is true by definition of $\mathbb C$-differentiability.

And for $n \ge 2$, the hypothesis of $f$ being $n$ times $\mathbb C$-differentiable at $z_0$ implies by definition that $f$ is $n-1$ times $\mathbb C$-differentiable in a neighborhood of $z_0$. Hence $f$ is holomorphic around $z_0$. Therefore $f$ can be written as a power series. And it can be approximated with a polynomial to any order.