Let $f$ be a function that expands as a power series on $D(0,R)$, i.e., there exists a complex sequence $(a_n)$ such that
$$\forall (x,y) \in D(0,R), \quad f(x,y) = \sum_{n=0}^{+\infty} a_n (x + \mathrm{i}y)^n$$
Show that for all $r \in [0, R[$, we have $f(0) = \frac{1}{2\pi} \int_0^{2\pi} f(r\cos(t), r\sin(t))\, \mathrm{d}t$.