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What is the symbol behind the integration means in logarithmic potential ? Let's say $$x_{\mu}(z)=\int_D \ln|z-\xi|d\mu(\xi)$$ What does $\xi$ means ? I have been studying harmonic moment and I come across this term.

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$\xi$ (the Greek letter "xi") is just the integration variable. You might as well rename it $w$ if you like...

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    Then how to integrate the integral above if we integrate w.r.t. $\xi$ ? Treat the integrand as constant ?2012-12-10
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    No, it is not constant, it is the function $\xi \mapsto \ln |z-\xi|$. You can only explicitly integrate this if you know what measure $\mu$ is.2012-12-11
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    but isn't the function is $z \rightarrow ln|z-\xi|$ ? as the LHS bracket is z2012-12-11
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    $z$ is a parameter in the integral, so the result of the integration depends on $z$, and is the logarithmic potential of $\mu$ evaluated at $z$.2012-12-11