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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 $ℂ$ 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....

An example of nonsymmetric semi-classical form of class $s = 1$ ; generalization of a case of Jacobi sequence.

Atia, Mohamed Jalel — 2000

International Journal of Mathematics and Mathematical Sciences

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

An example of nonsymmetric semi-classical form of class s = 1 ; generalization of a case of Jacobi sequence.

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

An example of nonsymmetric semi-classical form of class $s = 1$ ; generalization of a case of Jacobi sequence.