= Solution
For $|z|<r$, the <Schwarz integral on the unit disk>, rescaled to radius $r$, is
$$
g(z)=\frac1{2\pi}\int_0^{2\pi}u(re^{i\theta})\frac{re^{i\theta}+z}{re^{i\theta}-z}\,d\theta.
$$
On each smaller closed disk $|z|\leq s<r$, its kernel and all its derivatives in $z$ are uniformly bounded. Since the boundary values of the real <harmonic function> $u$ are continuous on this circle, <differentiation under the integral sign> proves that $g$ is holomorphic. Taking the real part gives the <Poisson integral> of those boundary values. It agrees with $u$ by uniqueness of the <Dirichlet problem> on the radius-$r$ disk. Hence \b[$g$ is holomorphic and $\operatorname{Re}g=u$ on $|z|<r$]. Its imaginary part is the <harmonic conjugate> normalized by $\operatorname{Im}g(0)=0$.
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