## Lecture Notes on Complex AnalysisThis book is based on lectures presented over many years to second and third year mathematics students in the Mathematics Departments at Bedford College, London, and King's College, London, as part of the BSc. and MSci. program. Its aim is to provide a gentle yet rigorous first course on complex analysis.Metric space aspects of the complex plane are discussed in detail, making this text an excellent introduction to metric space theory. The complex exponential and trigonometric functions are defined from first principles and great care is taken to derive their familiar properties. In particular, the appearance of π, in this context, is carefully explained.The central results of the subject, such as Cauchy's Theorem and its immediate corollaries, as well as the theory of singularities and the Residue Theorem are carefully treated while avoiding overly complicated generality. Throughout, the theory is illustrated by examples.A number of relevant results from real analysis are collected, complete with proofs, in an appendix.The approach in this book attempts to soften the impact for the student who may feel less than completely comfortable with the logical but often overly concise presentation of mathematical analysis elsewhere. |

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### Contents

Complex Numbers | 1 |

Sequences and Series | 17 |

Metric Space Properties of the Complex Plane | 29 |

Analytic Functions | 59 |

The Complex Exponential and Trigonometric Functions | 79 |

The Complex Logarithm | 97 |

Complex Integration | 111 |

Cauchys Theorem | 127 |

Singularities and Meromorphic Functions | 167 |

Theory of Residues | 175 |

The Argument Principle | 185 |

Maximum Modulus Principle | 195 |

Mobius Transformations | 207 |

Harmonic Functions | 219 |

Appendix A Some Results from Real Analysis | 231 |

Bibliography | 241 |

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absolutely convergent analytic function annulus argument Argz Cauchy sequence Cauchy-Riemann equations Cauchy's Integral Formula Cauchy's Theorem circle closed contour complex numbers complex plane conclude constant continuous convergent subsequence converges absolutely Corollary defined Definition denote differentiable at z0 disc of convergence domain example f is analytic f(zo finite Furthermore G H(D given Hence hypothesis Identity Theorem implies inequality inverse isolated singularity Laurent expansion limit point line segment locally uniformly logarithm Logz Maximum Modulus Principle Mobius transformation non-empty Note open disc open sets partial derivatives particular point z0 pole polygon power series proof is complete Proposition punctured disc real and imaginary real number Remark series converges absolutely star-domain star-like subset Suppose that f Taylor series triangle upper bound Value Theorem vanish