## Complex Variables with ApplicationsComplex numbers can be viewed in several ways: as an element in a field, as a point in the plane, and as a two-dimensional vector. Examined properly, each perspective provides crucial insight into the interrelations between the complex number system and its parent, the real number system. The authors explore these relationships by adopting both generalization and specialization methods to move from real variables to complex variables, and vice versa, while simultaneously examining their analytic and geometric characteristics, using geometry to illustrate analytic concepts and employing analysis to unravel geometric notions. The engaging exposition is replete with discussions, remarks, questions, and exercises, motivating not only understanding on the part of the reader, but also developing the tools needed to think critically about mathematical problems. This focus involves a careful examination of the methods and assumptions underlying various alternative routes that lead to the same destination. The material includes numerous examples and applications relevant to engineering students, along with some techniques to evaluate various types of integrals. The book may serve as a text for an undergraduate course in complex variables designed for scientists and engineers or for mathematics majors interested in further pursuing the general theory of complex analysis. The only prerequistite is a basic knowledge of advanced calculus. The presentation is also ideally suited for self-study. |

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### Contents

1 | |

5 | |

15 | |

Topological and Analytic Preliminaries | 25 |

22 Sequences | 32 |

23 Compactness | 39 |

24 Stereographic Projection | 44 |

25 Continuity | 48 |

74 Cauchys Theorem | 226 |

Applications of Cauchys Theorem | 243 |

82 Cauchys Inequality and Applications | 263 |

83 Maximum Modulus Theorem | 275 |

Laurent Series and the Residue Theorem | 285 |

92 Classification of Singularities | 293 |

93 Evaluation of Real Integrals | 308 |

94 Argument Principle | 331 |

Bilinear Transformations and Mappings | 61 |

32 Linear Fractional Transformations | 66 |

33 Other Mappings | 85 |

Elementary Functions | 91 |

42 Mapping Properties | 100 |

43 The Logarithmic Function | 108 |

44 Complex Exponents | 114 |

Analytic Functions | 121 |

52 Analyticity | 130 |

53 Harmonic Functions | 141 |

Power Series | 153 |

62 Uniform Convergence | 164 |

63 Maclaurin and Taylor Series | 173 |

64 Operations on Power Series | 186 |

Complex Integration and Cauchys Theorem | 195 |

72 Parameterizations | 207 |

73 Line Integrals | 217 |

Harmonic Functions | 348 |

102 Poisson Integral Formula | 358 |

103 Positive Harmonic Functions | 371 |

Conformal Mapping and the Riemann Mapping Theorem | 379 |

112 Normal Families | 390 |

113 Riemann Mapping Theorem | 395 |

114 The Class S | 405 |

Entire and Meromorphic Functions | 411 |

122 Weierstrass Product Theorem | 422 |

123 MittagLeffler Theorem | 437 |

Analytic Continuation | 445 |

132 Special Functions | 458 |

473 | |

475 | |

479 | |

Hints for Selected Questions and Exercises | 485 |