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



$$f(a) \ f(b)&lt;0$$
$$f'(x)&gt;0 \quad (\text{or} \quad f'(x)&lt;0) \qquad \forall \in [a,b] $$



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!
you don't need newton's method. you can use muller's method which can find complex roots, doesn't require a derivate but you need 3 points. 
the method is awesome! basic requirements: knowledge on how to multiply, add, subtract, and divide. knowledge on how to take square roots if you only know that method that you need to have a polar form function, it will just waste your time.
as with square roots, you can find all roots by solving a system of equations:
we have a number called (a+bi)
we know that t^2=(a+bi)
2xy=b
x^2-y^2=a
x^2+y^2=(a^2+b^2)^0.5 solve for equation 2 and 3. then find x and y. obviously, x is real and y is not. y=i x=x
you don't need newton's method. you can use muller's method which can find complex roots, doesn't require a derivate but you need 3 points. 
the method is awesome! basic requirements: knowledge on how to multiply, add, subtract, and divide. knowledge on how to take square roots if you only know that method that you need to have a polar form function, it will just waste your time.
as with square roots, you can find all roots by solving a system of equations:
we have a number called (a+bi)
we know that t^2=(a+bi)
2xy=b
x^2-y^2=a
x^2+y^2=(a^2+b^2)^0.5 solve for equation 2 and 3. then find x and y. obviously, x is real and y is not. y=i x=x