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I am looking to determine the value of the integral $$ \int_{\partial K(0,1)} \log|z-w| dA(w) = ? $$ where $dA$ denotes the 2-dimensional Lebesgue measure in $\mathbb{C}$.

What I have done so far:

1) When $|z|>1$ we have $$ \int_{\partial K(0,1)} \log|z-w| dA(w) = 2\pi\log|z| + \int_{\partial K(0,1)} \log\Big|1-w/|z|\Big| dA(w) = 2\pi\log|z| $$ where the last integral is $0$, since the integrand is the real part of a holomorphic branch of the logarithm.

2) I don't see how I should apply the same trick when $|z| \leq 1$, so I am pretty lost here :/

Any help is appreciated, thanks!

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    Btw, I expect (I hope) the value of the integral when $|z|\leq 1$ is $0$. – 2017-01-18
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    Cauchy's integral formula is everything you need: https://en.wikipedia.org/wiki/Poisson_kernel – 2017-01-18

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