## Complex AnalysisWith this second volume, we enter the intriguing world of complex analysis. From the first theorems on, the elegance and sweep of the results is evident. The starting point is the simple idea of extending a function initially given for real values of the argument to one that is defined when the argument is complex. From there, one proceeds to the main properties of holomorphic functions, whose proofs are generally short and quite illuminating: the Cauchy theorems, residues, analytic continuation, the argument principle. With this background, the reader is ready to learn a wealth of additional material connecting the subject with other areas of mathematics: the Fourier transform treated by contour integration, the zeta function and the prime number theorem, and an introduction to elliptic functions culminating in their application to combinatorics and number theory. Thoroughly developing a subject with many ramifications, while striking a careful balance between conceptual insights and the technical underpinnings of rigorous analysis, The Princeton Lectures in Analysis represents a sustained effort to introduce the core areas of mathematical analysis while also illustrating the organic unity between them. Numerous examples and applications throughout its four planned volumes, of which |

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### Contents

1 | |

Chapter 2 Cauchys Theorem and Its Applications | 32 |

Chapter 3 Meromorphic Functions and the Logarithm | 71 |

Chapter 4 The Fourier Transform | 111 |

Chapter 5 Entire Functions | 134 |

Chapter 6 The Gamma and Zeta Functions | 159 |

Chapter 7 The Zeta Function and Prime Number Theorem | 181 |

Chapter 8 Conformal Mappings | 205 |

Chapter 10 Applications of Theta Functions | 283 |

Asymptotics | 318 |

Simple Connectivity and Jordan Curve Theorem | 344 |

Notes and References | 365 |

369 | |

Symbol Glossary | 373 |

375 | |

Chapter 9 An Introduction to Elliptic Functions | 261 |