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For the proof of the Poisson-Jensen formula, I want to show that the following equality holds: $$\log(|z-a|) = \int_{0}^{2\pi} \log(|Re^{i\theta} - a|) P(z, Re^{i\theta}) \frac{\mathrm{d} \theta}{2\pi},$$ where $a$ is a point on the circle of radius $R$ and $P(z,w) = \frac{|w|^2 - |z|^2}{|w-z|^2} = \text{Re}\left\{ \frac{w + z}{w-z} \right\}$ is the Poisson kernel. Normally this would be easy, however $\log{|Re^{i \theta} - a|}$ has a singularity, so the Poisson formula doesn't apply immediately.

Thanks in advance.

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1 Answer

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It always applies. To prove, first prove when $a$ is not on the circle, then consider the limit, when $a$ tends to this circle. The singularity is like $\log$ so the integral is convergent and the limit procedure is justified.

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