# Using of Newton's method to find a complex root of a polynomial [duplicate]

I have studied the Newton's method for real functions, and these are the conditions to find a function real root on $[a,b]$:

$$f(a) \ f(b)<0$$ $$f'(x)>0 \quad (\text{or} \quad f'(x)<0) \qquad \forall \in [a,b]$$

Convergence theorem:

$$\text{sign}(f' f'') = \text{sign}(x_0 -x^*)$$

Iterations:

$$x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)} \qquad x_n\in \mathbb{R}$$

How can I use Newton's method to find the complex roots of a polynomial (e.g. $z^2+1$) ?

I know a real coefficients polynomial of degree $p$ has $p$ different complex roots (for Fundamental theorem of algebra)

But, how can I verify the convergence of the Newton succession with an initial value $z_0$?

$$z_{n+1}=z_n-\frac{f(z_n)}{f'(z_n)} \qquad z_n\in \mathbb{C}$$

Thanks!