Let $f$ be a holomorphic function in a region containing the upper-plane and the $x$-axis.
Suppose $f$ converges to $0$ uniformly, when $|z|\to\infty$.
Q:
1) For any $y>0$, how to show that $$f(z)=\frac{1}{\pi i}\int_{-\infty}^{\infty}\frac{\xi-x}{(\xi-x)^2+y^2}f(\xi)d\xi。$$
2) How to use $u(\xi,0)$ to represent $f(z)$, where $f(z)=u(z)+i v(z)$.
PS:
For the 1) question, it seems that we consider the integral $f(\xi)(\frac{1}{\xi-z}+\frac{1}{\xi -\bar{z}})$ on the upper semi-circle with large radius, and then let the radius goes to infinity.
But, it should be $f(z)+f(\bar{z})$, I am confused.
For the question 1)
Let $C_r=[-r,r] \cup \{ |z| = r, \Im(z) > 0\}$ the upper-semicircle of radius $r$. The Cauchy integral formula gives for $|z| < r, \Im(z) >0$ : $$f(z) = \frac{1}{2i \pi} \int_{C_r} \frac{f(s)}{s-z}ds$$ The assumption is that $|f(s)| \le g(|s|)$ which means $$\left|\int_{|s| = r, \Im(s) >0} \frac{f(s)}{s-z}ds\right| = \left|\int_{t=0}^\pi \frac{f(re^{it})}{re^{it}-z}d(re^{it})\right| \le \int_0^\pi \ldots$$
For the question 2) I would look at the Schwartz integral formula, starting with the case $f$ real on the real axis.