## Analytic Function Theory, Volume 1Emphasizes the conceptual and historical continuity of analytic function theory. This book covers canonical topics including elliptic functions, entire and meromorphic functions, as well as conformal mapping. It also features chapters on majorization and on functions holomorphic in a half-plane. |

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### Contents

NUMBER SYSTEMS | 1 |

THE COMPLEX PLANE | 18 |

FRACTIONS POWERS AND ROOTS | 46 |

HOLOMORPHIC FUNCTIONS | 68 |

POWER SERIES | 102 |

COMPLEX INTEGRATION | 160 |

REPRESENTATION THEOREMS | 196 |

THE CALCULUS OF RESIDUES | 241 |

APPENDIX A Some Properties of Point Sets | 277 |

On the Theory of Integration | 288 |

297 | |

### Common terms and phrases

absolutely convergent algebra analytic angle asserted bounded BV[a Cauchy Cauchy-Riemann equations Cauchy's coefficients complex numbers complex plane continuous function convergent series converges uniformly corresponding defined definition denote derivative disk domain double series elements entire function equation EXERCISE exists finite number follows formula given half-plane HB[D Hence holomorphic function imaginary implies inequality infinite infinity integral interior intersection interval inverse Lemma limit point line segment logarithm mapping meromorphic function multiplication neighborhood obtain partial sums point z0 poles polynomial positive integer power series Problem properties prove radius of convergence rational function real axis real numbers rectifiable curve residue result Riemann roots scroc Section sequence series converges simple closed polygon singular point solution subset Suppose tangent tends to zero that/(z theorem theory triangle uniform convergence unique unit circle vector Verify z-plane

### Popular passages

Page 298 - Complex integration and Cauchy's theorem. (Cambridge Tracts in Mathematics and Mathematical Physics, No. 15.) Cambridge, University Press, 1914.