## The Schwarz function and its applications |

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### Contents

Chapter Page 1 Prologue | 1 |

Conjugate Coordinates in the Plane | 5 |

Elementary Geometrical Facts | 7 |

Copyright | |

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algebra analytic arc analytic continuation analytic curve analytic function anti-analytic arc with Schwarz assume bilinear bisector boundary Chapter clinant coefficients complex potential complex variables conformal mapping conjugate coordinates contraction map convergence curvilinear angle defined derivative designate differential equations disc dz dz eigenvalues Example ff(z)dxdy fixed point follows formal power series formula func functional composition functional equation geometry harmonic function Hence horn angle identity integral interior intersect inverse inversive geometry iteration Lemma Let f(z linear mathematics matrix meromorphic Mobius nine-point circle obtain orthocenter orthogonal polynomial Proof real axis regular analytic root of unity satisfies schlicht Schroeder Function Schwarz Function S(z Schwarzian reflection simple simply connected simply connected region single-valued singularities spiral straight line Suppose symmetric with respect tangent theorem theory tion transformation triangle unique unit circle write x-axis zn+i