Logarithmic potentials of plane sets

Compute the single-layer logarithmic potential $u(x,y)=\displaystyle \oint_{C} \kappa \ln \dfrac {1}{r}\, ds$, where $\kappa =$ constant density, $r=\sqrt{(\xi -x)^{2}+(\eta -y)^{2}}$, and the contour $C$ is the circle $\xi^{2}+\eta^{2}=R^{2}$.

Using the polar coordinates $\rho$ and $\varphi$, compute the single-layer logarithmic potentials $I_{1}=\displaystyle \int_{0}^{2\pi } \cos m\psi \ln \dfrac {1}{r}\, d\psi$ and $I_{2}=\displaystyle \int_{0}^{2\pi } \sin m\psi \ln \dfrac {1}{r}\, d\psi$, where $r$ is the distance between the point $(\rho ,\varphi )$ and the moving point $(1,\psi )$, and $m$ is a natural number.

Study the variation of the function $P(x)$, where $P(x)=\mu \int_{a}^{b}\left|y-x\right| \, dy$, as $x$ varies over the interval $(-\infty ,+\infty )$.

Find the second derivative of

where the function $\mu (y)$ is continuous.

Using Poisson's integral

find the logarithmic potential of a homogeneous disk, i.e., the integral $\mu \iint_{\sigma } \ln r \, d\sigma$, where $r$ is the distance between a point $M$ and a point on the area of a disk $\sigma$ of radius $R$.

The logarithmic potential of a homogeneous segment $(-1,1)$ is called the integral $P=\int_{-1}^{+1}\ln r \, d\xi$, where $r$ is the distance between the points $(x,y)$ and $(\xi ,0)$. Find the equation of the curve on which $P(x,y)=P(1,0)$.

The logarithmic potential of an area $\sigma$ at the point $(x,y)$ is called the integral

Find the logarithmic potential of the disk $x^{2}+y^{2}=a^{2}$, if the density $\mu$ at the point $(x,y)$ is a given function of the radius vector from the origin: $\mu =f(r)$, where $r=\sqrt{x^{2}+y^{2}}$.

Find the logarithmic potential of a single layer

where $\Gamma$ is the circle $\xi^{2}+\eta^{2} = 1$, oriented counterclockwise, $r = \sqrt{(\xi -x)^{2}+(\eta -y)^{2}}$, if:

Compute the logarithmic potential of a double layer

where $\Gamma$ is the circle $\xi^{2}+\eta^{2} = 1$, oriented counterclockwise, $\mathbf{r} = (\xi -x ; \eta -y), \quad r = \sqrt{(\xi -x)^{2}+(\eta -y)^{2}}, \mathbf{n}$ is the outward normal to $\Gamma$, if:

Find the logarithmic potential, if $\Omega = \left\{ x^{2}+y^{2} \leqslant a^{2}\right\}$, and, moreover: