## Complex Analysis (Google eBook)This unusually lively textbook on complex variables introduces the theory of analytic functions, explores its diverse applications and shows the reader how to harness its powerful techniques. Complex Analysis offers new and interesting motivations for classical results and introduces related topics that do not appear in this form in other texts. Stressing motivation and technique, and containing a large number of problems and their solutions, this volume may be used as a text both in classrooms and for self-study. Topics covered include: The complex numbers; functions of a complex variable; analytic functions; line integrals and entire functions; properties of entire functions and of analytic functions; simply connected domains; isolated singularities; the residue theorem and applications; contour integral techniques; conformal mapping and the riemann mapping theorem; maximum-modulus theorems for unbounded domains; harmonic functions; forms of analytic functions; analytic continuation; the gamma and zeta functions; application to other areas of mathematics. For this second edition, the authors have revised some of the existing material and have provided new exercises and solution |

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### Contents

XXXIX | 139 |

XL | 146 |

XLI | 149 |

XLII | 152 |

XLIII | 156 |

XLIV | 157 |

XLV | 163 |

XLVI | 175 |

X | 25 |

XI | 27 |

XII | 31 |

XIII | 34 |

XIV | 39 |

XV | 43 |

XVI | 50 |

XVII | 54 |

XVIII | 56 |

XIX | 62 |

XX | 64 |

XXI | 67 |

XXII | 71 |

XXIII | 73 |

XXIV | 78 |

XXV | 80 |

XXVI | 85 |

XXVII | 90 |

XXVIII | 92 |

XXIX | 99 |

XXX | 101 |

XXXI | 103 |

XXXII | 107 |

XXXIII | 113 |

XXXIV | 115 |

XXXV | 121 |

XXXVI | 127 |

XXXVII | 130 |

### Common terms and phrases

2m Jc Algebra analytic function annulus apply Argz assume automorphism boundary bounded C-analytic Cauchy Cauchy-Riemann equations Chapter Closed Curve Theorem coefficients compact complex numbers complex plane conformal mapping consider constant contained contour Corollary defined Definition differentiable entire function equal evaluate example Exercise exists f is analytic f is entire fact finite formula function f given harmonic function Hence hypothesis imaginary axis infinitely inside integral isolated singularity Lemma line segment linear Mapping Theorem maximum Maximum-Modulus Theorem modulus Morera's Theorem nonconstant nonzero Note open set polynomial power series Proposition Prove radius of convergence real axis real numbers real-valued rectangle Residue Theorem Riemann Mapping Theorem right half-plane sequence Show simple pole simply connected simply connected domain solution Suppose f Theorem Suppose Uniqueness Theorem unit circle unit disc upper half-plane values zero

### Popular passages

Page 2 - A complex number is any number that can be written in the form a + bi, where a and b are real numbers, and i...

Page 1 - In the geometric representation, he wrote, one finds the "intuitive meaning of complex numbers completely established and more is not needed to admit these quantities into the domain of arithmetic.

Page 2 - С is the set of ordered pairs of real numbers (a, b) with addition and multiplication defined by (a, b) + (c, d) = (a + c, b + d} (a, fo)(c, if) = (ac - bd, ad + foe).

Page 1 - Euler, too, felt that complex numbers "exist only in the imagination" and considered complex roots of an equation useful only in showing that the equation actually has no solutions.