@TedShifrin I have also an other question.
I am looking at the proof of the theorem: If $\phi \in C^0(\partial{B_R})$ then there exists a unique solution of $\left\{\begin{matrix}
\Delta u=0 & \text{ in } B_R \\
u|_{\partial{B_R}}=\phi &
\end{matrix}\right. (\star)$ and $u(x)=\frac{R^2-|x|^2}{w_n R} \int_{\partial{B_R}} \frac{\phi(\xi) dS}}{|x-\xi|^n}$.

At the proof: if $x_0 \in \partial{B_R}$ then it must hold $\lim_{x \to x_0} u(x)=\phi(x_0)$.

$P(x, \xi)=\frac{R^2-|x|^2}{w_n R |x-\xi|^n}$

$|u(x)-\phi(x_0)|=\left| \int_{\partial{B_R}} P(x, \xi) (\phi(\xi)-\phi(x_0))\right| dS= \dots$

Let $F$ be a subfield of the field $K$ and $S$ a non-empty subset (not necessarily, subfield) of $K$, finite or infinite. Let $F(S)$ be the subset of $K$ that is defined as follows:
An element $u\in K$ is in $F(S)$ iff there are finitely many elements of $S$, say $s_1, \dots , s_n$, and polynomials with $n$ variables $f,g\in F[x_1, \dots , x_n]$, with $g(s_1, \dots , s_n)\neq 0$, so that $u=f(s_1, \dots , s_n)/g(s_1, \dots , s_n)$.

Let $E$ a subfield of $K$ and $F\cup S\subseteq E$.
The elements of $F(S)$ are of the form $\frac{f(s_1, \dots , s_n)}{g(s_1, \dots , s_n)}$.