## Quadrature Identities and the Schwarz Function |

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### Contents

QUADRATURE IDENTITY PROBLEMS AND THE SCHWARZ FUNCTION | 8 |

DOMAIN FUNCTIONS AND THE QUADRATURE IDENTITY PROBLEM | 23 |

CONSTRUCTION OF DOMAINS WHICH ADMIT QUADRATURE IDENTITIES | 46 |

Copyright | |

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### Common terms and phrases

2TTi admit quadrature identities admits a quadrature Aharonov and Shapiro assumption automorphic function Bergman's reproducing kernel boundary relation bounded domain canonical domain chapter circular domain Combining complex plane conclude condition 1.1 conformal image constant Conversely assume domain functions domain which admits doubly connected domains equivalent f G L finite connectivity function h G 3D grad v(z Green's function harmonic function harmonic measures Hence integrable analytic functions kernel function L-kernel Lemma Let D C C meromorphic function metric space necessary and sufficient never zero number of poles obtain outer boundary plane that admit preceding equality prove quadrature domains quadrature identity problem regular analytic function result Riemann mapping theorem satisfies the condition Schiffer Schwarz function S(z shell domain simply connected domains Stokes formula Theorem 13 tion ture identity unit disc univalent function