Possible to integrate $\ln |x-y| dy$ on a circle?

Question

I am wondering what is the solution to $$\int_{\partial D}\ln |x-y| dy$$

when $D$ is the unit disc in $\mathbb{R^2}$ and $x \in \partial D$. Is this even possible analytically?

Answer 1

Let $y=x e^{i\theta}$, then the integral becomes

$$ \int_{[0,2\pi]}\ln|x(1-e^{i\theta})| d\theta =2\pi |x|+ \int_{[0,2\pi]}\ln| 1-e^{i\theta} | d\theta =2\pi+ \int_{[0,2\pi]}\ln|2-2\cos\theta| d\theta$$ $$= 2\pi+ \int_{[0,2\pi]}\ln(4\sin^2(\theta/2)) d\theta = 2\pi+8\pi+ \int_{[0,2\pi]}\ln(\sin^2(\theta/2)) d\theta$$

$$= 10\pi+ 2 \int_{[0,2\pi]}\ln(\sin (\theta/2)) d\theta=10\pi+ 4 \int_{[0,\pi]}\ln(\sin \theta) d\theta=10\pi -4\pi\ln 2$$