A function $f$, defined on $D(0,R) \subset \mathbb{R}^2$ and with complex values, expands as a power series on $D(0,R)$ if there exists a complex sequence $(a_n)$ such that $$\forall (x,y) \in D(0,R), \quad f(x,y) = \sum_{n=0}^{+\infty} a_n (x + \mathrm{i}y)^n$$ We admit the following result: a function $h$ from $D(0,R)$ to $\mathbb{C}$ expands as a power series on $D(0,R)$ if and only if $h$ is of class $\mathcal{C}^1$ on $D(0,R)$ and for all $(x,y) \in D(0,R)$, $\frac{\partial h}{\partial y}(x,y) = \mathrm{i}\frac{\partial h}{\partial x}(x,y)$. Show that if $f$ does not vanish on $D(0,R)$ then $1/f$ expands as a power series on $D(0,R)$.
A function $f$, defined on $D(0,R) \subset \mathbb{R}^2$ and with complex values, expands as a power series on $D(0,R)$ if there exists a complex sequence $(a_n)$ such that
$$\forall (x,y) \in D(0,R), \quad f(x,y) = \sum_{n=0}^{+\infty} a_n (x + \mathrm{i}y)^n$$
We admit the following result: a function $h$ from $D(0,R)$ to $\mathbb{C}$ expands as a power series on $D(0,R)$ if and only if $h$ is of class $\mathcal{C}^1$ on $D(0,R)$ and for all $(x,y) \in D(0,R)$, $\frac{\partial h}{\partial y}(x,y) = \mathrm{i}\frac{\partial h}{\partial x}(x,y)$. Show that if $f$ does not vanish on $D(0,R)$ then $1/f$ expands as a power series on $D(0,R)$.