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 $$\forall (x,y) \in D(0,R), \quad f(x,y) = \sum_{n=0}^{+\infty} a_n (x + \mathrm{i} y)^n$$ We denote by $u$ and $v$ the real and imaginary parts of $f$, so that, for any $(x,y) \in D(0,R)$, $$u(x,y) \in \mathbb{R}, \quad v(x,y) \in \mathbb{R}, \quad f(x,y) = u(x,y) + \mathrm{i} v(x,y).$$ Show that $u$ and $v$ are harmonic functions on $D(0,R)$.
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
$$\forall (x,y) \in D(0,R), \quad f(x,y) = \sum_{n=0}^{+\infty} a_n (x + \mathrm{i} y)^n$$
We denote by $u$ and $v$ the real and imaginary parts of $f$, so that, for any $(x,y) \in D(0,R)$,
$$u(x,y) \in \mathbb{R}, \quad v(x,y) \in \mathbb{R}, \quad f(x,y) = u(x,y) + \mathrm{i} v(x,y).$$
Show that $u$ and $v$ are harmonic functions on $D(0,R)$.