# On the homogeneous Hilbert problem for effecting Kinnersley-Chitre transformations

## Abstract

For each real number z_{o}, the multiplicative group of all 2 x 2 unimodular matrix functions v(τ) of a complex variable τ which are holomorphic at z_{o} and are real for real τ is a realization of the Geroch group; the union of these realizations is denoted by K(R). The HHP (homogeneous Hilbert problem) of Hauser and Ernst is given an advantageous reformulation which employs K(R) and which dispenses with the use of contours. The use of and the potentials which occur in the HHP are discussed. A complex extension K(C) of K(R) is introduced by dropping the reality conditions, n-parameter families of members of K(C) are discussed and examples of analytic continuation in the complex parameter space are given.Some basic solutions of the HHP are reviewed; in particular an alternative form and derivation of Cosgrove's solution for the double Harrison family are given.

## Keywords

Fundamental Solution Branch Point Analytic Continuation Complex Extension Translational Mapping## Preview

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## Footnotes

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_{±}is the transpose of his X±, and our G is his (G^{T})^{−l}.To say that G satisfies a Hölder condition on L means there exist 0 < μ > l and A > 0 such that ‖ G(s′)-G(s)‖ < A|s′-s μ for all s,s′ on L where ‖ M‖ denotes the norm of M.Google Scholar - 12.This means that there exists no other holomorphic function f
^{a}_{b}(τ) of τ such that v^{a}_{b}(τ) = f^{a}_{b}(τ) for all τ in at least one neighborhood of z_{o}and such that dom v^{a}_{b}is a proper subset of dom f^{a}_{b}.Google Scholar - 13.The domain of v as defined here may not be connected, but it always has a connected component which covers z
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