## An introduction to complex analysisRecent decades have seen profound changes in the way we understand complex analysis. This new work presents a much-needed modern treatment of the subject, incorporating the latest developments and providing a rigorous yet accessible introduction to the concepts and proofs of this fundamental branch of mathematics. With its thorough review of the prerequisites and well-balanced mix of theory and practice, this book will appeal both to readers interested in pursuing advanced topics as well as those wishing to explore the many applications of complex analysis to engineering and the physical sciences. * Reviews the necessary calculus, bringing readers quickly up to speed on the material * Illustrates the theory, techniques, and reasoning through the use of short proofs and many examples * Demystifies complex versus real differentiability for functions from the plane to the plane * Develops Cauchy's Theorem, presenting the powerful and easy-to-use winding-number version * Contains over 100 sophisticated graphics to provide helpful examples and reinforce important concepts |

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### Contents

Preliminaries | 1 |

Tools | 83 |

E Uniqueness of the Power Series Representation | 174 |

Copyright | |

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### Common terms and phrases

angles antiderivative boundary point boundary values bounded calculus Cauchy Cauchy-Riemann equations closed path coefficients compact complex number compute conformal equivalence conformal mapping Consider constant contains contour cosh cosine defined definition denote derivative domain equipotentials Evaluate Example exists exponential Figure finite number follows function F function given graph harmonic conjugate harmonic function heat flow Help on Selected holomorphic function Identify integral equals integrand interval inverse Laurent series Lemma Let f level curves line segments linear mappings logarithm meromorphic Mobius map nonzero notation one-to-one open connected set open set picture plane pole of order polynomials power series problem Proof Proposition prove pullback radius of convergence rational function real axis real numbers real-valued removable singularity Residue Theorem result Riemann Section sequence series representation sine sketch Solution subset Taylor series temperature triangle unit circle unit disk upper half-plane vector vertical write zero