Given the complex function$$ f_C(z) = z^3 - 1$$ Write the complex function $f_C: \mathbb{C} \rightarrow \mathbb{C}$ as a real vector-valued function $\mathbf{f}: \mathbb{R^2} \rightarrow \mathbb{R^2}$ using rectangular coordinates $z = x_1 +ix_2$. In addition, write down the Jacobian.
So firstly I need to find $F \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} \rightarrow \mathbb{R^2}$. But what $f_C$ tells me to do is compute $(x_1 + ix_2)^3 -1 $ which gives me a value in $\mathbb{C^1}$
I'm also not sure how to go about computing the Jacobian. I don't see how it makes sense to differentiate a complex function $z^3 - 1$ in terms of $x_1$ and $x_2$
You made an error in decomposing $f$ into real and imaginary part, it should be $$ \mathbf{f}\begin{bmatrix} x_1 \\ x_2 \end{bmatrix} = \begin{bmatrix} x_1^3 -3x_1x_2^2 -1 \\ 3x_1^2x_2 -x_2^3 \end{bmatrix} $$ Then the Jacobian derivative matrix is $$ J_F = \begin{bmatrix} 3x_1^2-3x_2^2&-6x_1x_2\\6x_1x_2&3x_1^2-3x_2^2\end{bmatrix} = 3 \begin{bmatrix} x_1^2-x_2^2&-2x_1x_2\\2x_1x_2&x_1^2-x_2^2\end{bmatrix} $$ This is also the matrix representation of the complex number $3z^2=3[(x_1^2-x_2^2)+2i\,x_1x_2]$.