# Newton's Method on Complex Variables

This image contains the question I'm currently working on

I'm stuck on trying on trying to show the result. I started off by writing all z$_n$'s and z$_{n+1}$ in terms of x$_n$ and y$_n$ and then I partially derived the entire equation with respect to x in order to get rid of y$_{n+1}$ to solve for x$_{n+1}$. I end up with a huge mess for partially derived g(x$_n$, y$_n$) and h(x$_n$, y$_n$). I'm going to continue to attempt solving this, but if anyone can provide any tips it would be appreciated. The most prominent issue is that I end up with h$_{xx}$ and g$_{xx}$ and I'm not sure how to get rid of them.

• \begin{align} \dfrac{f}{f'} &= \dfrac{g+ih}{g_x+ih_x} \\ &= \dfrac{g+ih}{g_x+ih_x}\dfrac{g_x-ih_x}{g_x-ih_x} \\ &= \dfrac{gg_x+hh_x+i(hg_x-gh_x)}{g_xg_x+h_xh_x} \\ &= \dfrac{gg_x+hh_x+i(hg_x-gh_x)}{-g_xh_y+h_xg_y} \\ &= \dfrac{gg_x+hh_x}{-g_xh_y+h_xg_y} +i \dfrac{hg_x-gh_x}{-g_xh_y+h_xg_y} \\ &= \dfrac{gh_y-hg_y}{g_xh_y-h_xg_y} -i \dfrac{hg_x-gh_x}{g_xh_y-h_xg_y} \end{align} – Nosrati Oct 9 '17 at 3:38
• Wow, thank you so much. I went in the completely wrong direction – Niko L Oct 9 '17 at 4:02
• You'r welcome!. – Nosrati Oct 9 '17 at 4:07