## COMPLEX VARIABLES: THEORY AND APPLICATIONSThe second edition of this comprehensive and accessible text continues to offer students a challenging and enjoyable study of complex variables that is infused with perfect balanced coverage of mathematical theory and applied topics. The author explains fundamental concepts and techniques with precision and introduces the students to complex variable theory through conceptual develop-ment of analysis that enables them to develop a thorough understanding of the topics discussed. Geometric interpretation of the results, wherever necessary, has been inducted for making the analysis more accessible. The level of the text assumes that the reader is acquainted with elementary real analysis. Beginning with the revision of the algebra of complex variables, the book moves on to deal with analytic functions, elementary functions, complex integration, sequences, series and infinite products, series expansions, singularities and residues. The application-oriented chapters on sums and integrals, conformal mappings, Laplace transform, and some special topics, provide a practical-use perspective. Enriched with many numerical examples and exercises designed to test the student's comprehension of the topics covered, this book is written for a one-semester course in complex variables for students in the science and engineering disciplines. |

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### Contents

Analytic Functions 3978 | 39 |

Elementary Functions 7997 | 79 |

Complex Integration 98152 | 98 |

Sequences Series and Products 153187 | 153 |

Series Expansions 188240 | 188 |

Singularities and Residues 241294 | 241 |

Sums and Deﬁnite Integrals 295334 | 295 |

Conformal Mappings 335375 | 335 |

Laplace Transform | 376 |

Special Topics | 406 |

Objectivetyre Questions | 467 |

487 | |

### Common terms and phrases

absolutely convergent analytic function bilinear transformation boundary bounded branch C-R equations Cauchy integral formula coefﬁcients complex numbers complex plane constant continuous function converges uniformly Corollary cosh curve deﬁned deﬁnition differentiable domain Q entire function evaluate Example exists exponential order f is analytic f(zo ﬁnd ﬁnite number ﬁrst ﬁxed points function deﬁned function f(z given harmonic function Hence implies inequality inﬁnite product interior inverse iv(x Laplace transform Let f Let f(z limit point line segment maps multivalued function nonzero Note number of zeros path point 20 pole of order polynomial positively oriented power series principal value Problem Set Proposition prove radius of convergence real axis real number region Remark residue theorem result roots satisﬁes sequence series expansion simple closed contour simple poles singular point sinh sufﬁcient Suggestion Taylor series uniform convergence unit disc upper half-plane w-plane zero of order

### Popular passages

Page 8 - R2, it is simply the fact that the length of one side of a triangle is less than or equal to the sum of the lengths of the other two sides. We will not give the solution to Example 2.2 as we generalize this example in Example 2.3. As F...