## The Schwarz function and its applications |

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### Contents

CHAPTER Prologue Conjugate Coordinates in the Plane Elementary Geometrical Facts The NinePoint Circle | 1 |

The Schwarz Function for an Analytic Arc Geometrical Interpretation of the Schwarz Fimction Schwarzian Reﬂection | 2 |

Properties in the Large of the Schwarz Fimction | 10 |

Copyright | |

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### Common terms and phrases

algebra analytic arc analytic continuation analytic curve analytic function anti-analytic assume bilinear bisector boundary Cauchy Chapter clinant coefficients complex number complex plane complex variables conformal invariant conformal mapping conjugate coordinates contraction map convergence curvilinear angle deﬁned deﬁnition derivative designate differential equations disc eigenvalues ellipse Example ﬁgure ﬁnd ﬁrst ﬁxed point ﬂow follows formal power series formula func function f Function for f functional composition functional equation geometry H. A. Schwarz harmonic function Hence horn angle identity inﬁnite integral interior intersect invariant curves invariant under f inverse inversive geometry iteration LEMMA linear Math mathematics matrix meromorphic neighborhood nine-point circle obtain orthocenter orthogonal polynomial Proof real axis reﬂection satisﬁes schlicht Schroeder Function Schwarz Function 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