We say that a function $f$, defined on $D(0,R) \subset \mathbb{R}^2$ and with complex values, expands in a power series on $D(0,R)$ if there exists a complex sequence $(a_n)$ such that $$\forall (x,y) \in D(0,R), \quad f(x,y) = \sum_{n=0}^{+\infty} a_n (x + \mathrm{i} y)^n$$ Throughout this part, $f$ denotes a function that expands in a power series on $D(0,R)$. Show that $f$ is of class $\mathcal{C}^1$ on $D(0,R)$ and that its partial derivatives expand in power series on $D(0,R)$. What can we deduce about the function $f$?
We say that a function $f$, defined on $D(0,R) \subset \mathbb{R}^2$ and with complex values, expands in a power series on $D(0,R)$ if there exists a complex sequence $(a_n)$ such that
$$\forall (x,y) \in D(0,R), \quad f(x,y) = \sum_{n=0}^{+\infty} a_n (x + \mathrm{i} y)^n$$
Throughout this part, $f$ denotes a function that expands in a power series on $D(0,R)$.
Show that $f$ is of class $\mathcal{C}^1$ on $D(0,R)$ and that its partial derivatives expand in power series on $D(0,R)$. What can we deduce about the function $f$?