= Schwarz integral formula
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The Schwarz integral formula reconstructs a <holomorphic function> from boundary values of its <real part>. For a real boundary function $f$ on the <complex upper half-plane> satisfying $\int_{\mathbb R}|f(s)|/(1+s^2)\,ds<\infty$, the regularized form is
$$
W(w)=\frac1{\pi i}\int_{\mathbb R}f(s)\left[\frac1{s-w}-\frac{s}{1+s^2}\right]ds.
$$
Its <real part> is the <Poisson integral> of $f$. The subtraction makes the integral converge and changes only the imaginary normalization. <Integration by parts>, when the boundary terms vanish, gives $W'(w)=(\pi i)^{-1}\int f'(s)/(s-w)\,ds$; jumps of $f$ are included as <Dirac delta function> terms in its <distributional derivative>.
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