According to https://en.wikipedia.org/wiki/Schwarz_integral_formula, which I will paraphrase here, if $f$ is a holomorphic on the closed unit disk $\{ \lvert z \rvert \leq 1 \}$, then
$f(z) = \frac{1}{2\pi i} \oint_{\lvert \zeta \rvert = 1} \frac{\zeta+z}{\zeta-z}\operatorname{Re}(f(\zeta)) \frac{d\zeta}{\zeta} + i \operatorname{Im}(f(0))$,
for all $\lvert z \rvert < 1$.
First, when they say that $f$ is holomorphic on the closed disk, does it mean that it is holomorphic on an open set containing the closed disk?
Second, there has to be a way to recover the real part of $f$ on the boundary circle from this formula. But here I ran into some problems. Since $f$ is continuous up to the boundary, it follows that letting $z$ approach a boundary point should work. But then one should not interchange that limit with the integral, because it leads to the whole integral term (including the factor of $1/(2\pi i)$) being pure imaginary.
What is the right careful way to recover the real part of $f$ on the boundary circle? A reference will do.
I would like to implement this formula numerically, so if anyone has tips, and warnings etc, then I would appreciate it. Particularly, how can one recover the real part of $f$ numerically? The kernel becomes singular as $z$ approaches a boundary point.
Edit 1: I found notes by Paul Garrett http://www-users.math.umn.edu/~garrett/m/complex/notes_2014-15/08c_harmonic.pdf containing a very simple proof for the Poisson kernel formula, which is one of the questions I asked about. It immediately follows from the mean-value property of harmonic functions, after a change of variable by a Möbius transformation of the open unit disk. So the remaining question is a numerical one. How does one numerically evaluate the Schwarz integral formula when $z$ is close to the boundary circle?
Edit 2: as an idea, can one use this idea in reverse? Starting from a point $z$ close to the boundary, apply a Möbius transformation to send $z$ to the origin, then apply the mean-value property? I mean, of course theoretically speaking, one can do that, but my question is, will this behave better numerically?