# On 2D Inverse Problems/Riemann mapping theorem

Let ${\displaystyle f:U\rightarrow D}$ be a Riemann map for the region U, the the kernel of the Dirichlet-to-Neumann map for the region U can be expressed in terms of the kernel for the disc.
Exercise (*). Proof that ${\displaystyle K_{U}(\phi ,\theta )=|f'(\phi )|K_{D}(f(\phi ),f(\theta ))|f'(\theta )|}$ off the diagonal.
The Cayley transform ${\displaystyle {\frac {1-z}{1+z}}:C^{+}\rightarrow D}$ maps the complex right half-plane onto the unit disc.
Exercise (**). Derive the formula for the kernel of the DN map for the unit disc D: ${\displaystyle K_{D}(\phi ,\theta )={\frac {-1}{\pi (1-cos(\phi -\theta ))}}}$ w/the kernel formula for the half-plane and the radial derivative of the Poisson kernel,solving Dirichlet boundary problem on the disc:
${\displaystyle K_{P}(z,\theta )={\frac {1-|z|^{2}}{2\pi |1-ze^{-i\theta }|}}.}$
${\displaystyle \Lambda _{kl}={\begin{cases}{\frac {2n(n+1)}{3}},{\mbox{ }}k=l,\\{\frac {-1}{1-\cos {\frac {2\pi (k-l)}{2n+1}}}},{\mbox{ }}k\neq l,\end{cases}}}$ where n is a natural number and k,l = 1,2, ... 2n+1.
Exercise (**). Prove that the eigenvalues of the matrix are natural numbers(!) ${\displaystyle 2n,4n-2,6n-6,\ldots ,n(n+1)}$ w/multiplicity two and 0 w/multiplicity one.