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Quadratic differentials ( A ( z - a ) ( z - b ) / ( z - c ) 2 ) d z 2 and algebraic Cauchy transform

Mohamed Jalel Atia, Faouzi Thabet (2016)

Czechoslovak Mathematical Journal

We discuss the representability almost everywhere (a.e.) in of an irreducible algebraic function as the Cauchy transform of a signed measure supported on a finite number of compact semi-analytic curves and a finite number of isolated points. This brings us to the study of trajectories of the particular family of quadratic differentials A ( z - a ) ( z - b ) ( z - c ) - 2 d z 2 . More precisely, we give a necessary and sufficient condition on the complex numbers a and b for these quadratic differentials to have finite critical trajectories....

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