# On 2D Inverse Problems/Stieltjes continued fractions

Let be a set of *n* positive numbers. The Stieltjes continued fraction is an expression of the form, see [KK] & also [JT],

or its reciprocal

The function defines a rational *n*-to-*1* map of the right half of the complex plane onto itself,

since

**Exercise(***).** Use the mapping properties of a Stieltjes continued fractions to prove that it's a rational functions w/a zero or a pole at the origin w/simple, symmetric, interlacing zeros and poles lying on the imaginary axes and that the above properties characterize the continued fractions.

**Exercise(**).** Prove that the continued fractions have the following representation:

where

are non-negative real numbers and the converse of the statement is true.

The function is determined by the pre-image set (of size *n*, counting multiplicities) of the point {*z = 1*}, since

and a complex polynomial is determined by its roots up to a multiplicative constant by the *fundamental theorem of algebra*.

Let be the elementary symmetric functions of the set . That is,

Then, the coefficients of the continued fraction are the pivots in the Gauss-Jordan elimination algorithm of the following *n* by *n* square Hurwitz matrix:

and, therefore, can be expressed as the ratios of monomials of the determinants of the blocks of .

**Exercise (**).** Prove that

**Exercise (*).** Use the previous exercise to prove that

**Exercise (**).** Let *A* be a diagonal matrix with the alternating in sign diagonal entries:

and *D* the *(0,1)*-matrix

Prove that

That is

**Exercise (*).** Find the constant in the equation above.