## Conformal mapping: lectures given at Oklahoma A. and M. College, Dept. of Mathematics, summer session, 1951 |

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### Contents

HARMONIC FUNCTIONS 1 Definition | 8 |

Harnacks Principle | 10 |

Schwartz Theorem | 11 |

Symmetry Principle | 12 |

Dirichlets Principle lh 7 Subharmonic Functions | 15 |

Perrons Method | 16 |

Barrier Functions | 18 |

Regions regular for the Dirichlet Problem | 20 |

Extremal properties of p z and qz | 43 |

Slit mappings for arbitrary Q hi 8 Minimal slit regions | 51 |

An extremal property of pq | 54 |

Null sets of class D | 56 |

An extremal property of p+q | 60 |

Other canonical mappings | 64 |

EXTREMAL METRIC 1 Statement of the problem | 67 |

Explicit calculation of in some simple cases | 71 |

Greens Function | 23 |

C0NF0RMAL MAPPING 1 The Riemann Mapping Theorem | 29 |

Multiply Connected Regions and Approximating Regions32 | 32 |

Topological Structure of Qn Differentials | 33 |

U The Greens function and the generalized Greens Function | 36 |

Slit mappings of Qn | 38 |

U The comparison ani composition laws Ik 5 Historical note | 78 |

Extremal distance | 80 |

Proof that nE1E2 jy 8U 9 Further examples | 88 |

Reduced extremal distance | 92 |

Conclusion | 96 |