I haven't been able to decide whether the image $X$ of the holomorphic function $f: \mathbb{C} \rightarrow \mathbb{C}^2$ given by $f(z) = (z^2 - 1, z^3 - z)$ is analytic or not. Here is what I have tried so far:
(1) Finding for each $x \in X$ a single holomorphic function $g_x$ on a neighborhood $V$ of $x$ such that the zero set of $g_x$ is equal to $X \cap V$. The function $g(z_1,z_2) = (\frac{z_2}{z_1})^2 - z_1 - 1$ is holomorphic on $V = \{ z = (z_1, z_2) \in \mathbb{C} \; | \; z_1 \neq 0 \} $ and hence on $X \setminus \{ (0, 2i), (0, -2i) \} $. Further, it is easy to show that $g$ is zero on $X \setminus \{ (0,2i),(0,-2i) \}$. Thus, for each $x \in X \setminus \{ (0, 2i), (0, -2i) \}$, we have a function holomorphic on the neighborhood $V$ of $x$ such that $X \cap V = \{ z \; | \; g(z) = 0 \}$. However, I have not been able to think of functions holomorphic on neighborhoods of $(0, 2i)$ or $(0, -2i)$ that satisfy the requirements to make $X$ an analytic set.
(2) My difficulty in finding proper functions at the two points previously mentioned has lead me to think that I might be able to show that $X$ isn't analytic. Can I show that if a function holomorphic on a neighborhood $U$ of $(0,2i)$ had a zero set of $U \cap X$ then there would be a contradiction?
(3) The only other thing I can think of is to show that $(z^2 - 1, z^3 - z)$ can be a local coordinate system near each $x \in X$. However, I do not know how to do this.
Am I on the right track with any of these ideas? Thanks!