## Functions of One Complex Variable IThis book is intended as a textbook for a first course in the theory of functions of one complex variable for students who are mathematically mature enough to understand and execute E - 8 arguments. The actual pre requisites for reading this book are quite minimal; not much more than a stiff course in basic calculus and a few facts about partial derivatives. The topics from advanced calculus that are used (e.g., Leibniz's rule for differ entiating under the integral sign) are proved in detail. Complex Variables is a subject which has something for all mathematicians. In addition to having applications to other parts of analysis, it can rightly claim to be an ancestor of many areas of mathematics (e.g., homotopy theory, manifolds). This view of Complex Analysis as "An Introduction to Mathe matics" has influenced the writing and selection of subject matter for this book. The other guiding principle followed is that all definitions, theorems, etc. |

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It is a great book. Clear writing and good exercise collection but a bit hard to follow.

### Contents

The Complex Number System | 1 |

6 The extended plane and its spherical representation | 8 |

Elementary Properties and Examples of Analytic Functions | 30 |

Complex Integration | 58 |

Entire Functions | 77 |

Singularities | 103 |

The Maximum Modulus Theorem | 128 |

Compactness and Convergence in | 142 |

Analytic Continuation and Riemann Surfaces | 210 |

Harmonic Functions | 252 |

Harmonic Functions Redux | 253 |

Potential Theory in the Plane | 276 |

The Range of an Analytic Function | 292 |

Calculus for Complex Valued Functions | 303 |

References | 311 |

List of Symbols | 317 |

### Common terms and phrases

analytic continuation analytic function analytic manifold analytic on G Cauchy sequence Cauchy's Theorem circle closed curve closed rectifiable curve compact set compact subset complete analytic function complex numbers component contains continuous function converges absolutely converges uniformly Corollary covering space curve in G Definition differentiable entire function equation example Exercise fact finite number fixed follows formula function element function f function on G g is analytic gives harmonic function Hence homeomorphism homotopic If/is implies integer Lemma Let f Let G lim sup limit point logarithm meromorphic function metric space Mobius transformation Monodromy Theorem one-one open set open subset path point in G polygon polynomial power series expansion properties Proposition prove radius of convergence rational function reader real number region G removable singularity result Runge's Theorem satisfies simply connected subset of G topological space uniformly continuous z e G zero