On 2D Inverse Problems/Cauchy matrices

Let $x_{k}$ be an ordered set of n complex numbers. The corresponding Cauchy matrix is the matrix $C_{x}=\{{\frac {1}{x_{k}+x_{l}}}\}$.

Principal submatrices of a Cauchy matrix are Cauchy matrices. 
The determinant of a Cauchy matrix is given by the following formula:

$\det(C_{x})={\frac {\prod _{1\leq k<l\leq n}(x_{l}-x_{k})^{2}}{\prod _{1\leq ,k,l\leq n}(x_{k}+x_{l})}}.$

It follows, that if $x_{k}$'s are distinct positive numbers, then the Cauchy matrix 's positive definite.

Exercise (*). Prove that for any positive numbers $x_{k}$ there is a Stieltjes continued fraction, interpolating the constant unit function at these numbers, $\beta _{x}(x_{k})=1$.

(Hint.) Use the solution of the Pick-Nevanlinna interpolation problem w/the appropriate Cauchy matrix.

The latter exercise has the following functional equation corollary for the discrete and continuous Dirichlet-to-Neumann maps.

Exercise (**). Prove that for any positive definite matrix M there is a Stieltjes continued fraction, such that $\beta _{M}(M)={\sqrt {M}}$.

The next chapter is devoted to the applications of the functional equation.