# On 2D Inverse Problems/Stieltjes continued fractions

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Let be a sequence ofnpositive numbers. TheStieltjes continued fractionis 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 Stieltjes continued fractions to prove that their interlacing, simple and symmetric zeros and poles lie at the origin and the imaginary axes and that the properties and rationality characterize the continued fractions.

**Exercise(**).**Prove that the continued fractions 've the representation , 're non-negative real numbers, and the fractions 're characterized by it.

The function is determined by the pre-image of unity (i.e.npoints, counting multiplicities), since

and a complex polynomial is determined by its roots up to a multiplicative constant by thefundamental 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 theGauss-Jordan eliminationalgorithm of the following squareHurwitz matrix:

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

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

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