I was studying logarithmic potential $f(z)=\int_E \log(\frac{1}{ \mid{z-a} \mid })d\mu(a)$ (Tsuji's Potential theory in modern function theory ). I am trying to prove $f(z)$ is harmonic outside of $E$, where $E$ is a closed and bounded Euclidean subset. I don't know how to proceed.
2026-03-29 20:46:33.1774817193
Logarithmic Potential
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Because $E$ is compact, if $z_0\notin E$ then there is $r>0$ such that the ball $K:=B(z_0,r)$ with center $z_0$ and radius $r$ is contained in $E^c$. Now $\mu$ is a (positive) measure supported by $E$ with finite total mass. The function $z\mapsto \log|z-a|$ is smooth on $K$, uniformly in $a\in E$. So differentiating (repeatedly) under the integral is no problem. The harmonicity of $f$ on the interior of $K$ now follows because $\Delta_z \log|z-a|\equiv0$ there.