## Complex AnalysisThe present book is meant as a text for a course on complex analysis at the advanced undergraduate level, or first-year graduate level. The first half, more or less, can be used for a one-semester course addressed to undergraduates. The second half can be used for a second semester, at either level. Somewhat more material has been included than can be covered at leisure in one or two terms, to give opportunities for the instructor to exercise individual taste, and to lead the course in whatever directions strikes the instructor's fancy at the time as well as extra read ing material for students on their own. A large number of routine exer cises are included for the more standard portions, and a few harder exercises of striking theoretical interest are also included, but may be omitted in courses addressed to less advanced students. In some sense, I think the classical German prewar texts were the best (Hurwitz-Courant, Knopp, Bieberbach, etc. ) and I would recommend to anyone to look through them. More recent texts have emphasized connections with real analysis, which is important, but at the cost of exhibiting succinctly and clearly what is peculiar about complex analysis: the power series expansion, the uniqueness of analytic continuation, and the calculus of residues. |

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### Contents

IV | 3 |

V | 8 |

VI | 12 |

VII | 17 |

VIII | 21 |

IX | 27 |

X | 31 |

XI | 33 |

LXI | 278 |

LXII | 288 |

LXIII | 293 |

LXIV | 295 |

LXVI | 299 |

LXVII | 305 |

LXVIII | 308 |

LXX | 310 |

XII | 37 |

XIII | 47 |

XIV | 60 |

XV | 64 |

XVI | 66 |

XVII | 68 |

XVIII | 72 |

XIX | 76 |

XX | 83 |

XXI | 86 |

XXII | 92 |

XXIII | 94 |

XXIV | 104 |

XXV | 110 |

XXVI | 115 |

XXVII | 119 |

XXVIII | 125 |

XXIX | 133 |

XXX | 134 |

XXXI | 138 |

XXXII | 147 |

XXXIII | 149 |

XXXIV | 156 |

XXXV | 161 |

XXXVI | 165 |

XXXVIII | 166 |

XXXIX | 168 |

XL | 173 |

XLI | 186 |

XLII | 193 |

XLIII | 196 |

XLIV | 199 |

XLV | 201 |

XLVI | 210 |

XLVIII | 212 |

XLIX | 214 |

L | 217 |

LI | 222 |

LII | 233 |

LIII | 243 |

LV | 248 |

LVI | 250 |

LVII | 254 |

LVIII | 261 |

LIX | 273 |

LX | 275 |

LXXI | 313 |

LXXII | 316 |

LXXIII | 324 |

LXXV | 333 |

LXXVI | 337 |

LXXVII | 339 |

LXXVIII | 341 |

LXXX | 342 |

LXXXI | 348 |

LXXXII | 356 |

LXXXIV | 358 |

LXXXV | 360 |

LXXXVI | 362 |

LXXXVII | 367 |

LXXXVIII | 374 |

XCI | 378 |

XCII | 384 |

XCIII | 389 |

XCIV | 393 |

XCVI | 397 |

XCVII | 402 |

XCVIII | 405 |

XCIX | 410 |

CI | 411 |

CII | 415 |

CIV | 418 |

CV | 420 |

CVI | 422 |

CVII | 424 |

CVIII | 426 |

CIX | 433 |

CX | 435 |

CXI | 442 |

CXIII | 443 |

CXIV | 448 |

CXV | 451 |

CXVI | 455 |

CXVIII | 459 |

CXIX | 463 |

CXX | 467 |

CXXI | 469 |

CXXII | 474 |

479 | |

481 | |

483 | |

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### Common terms and phrases

Algebraic analytic continuation analytic function analytic isomorphism annulus apply assume automorphism bounded calculus Cauchy's theorem Chapter circle of radius closed disc closed path coefficients compact set complex numbers concludes the proof connected open set contained continuous function converges absolutely converges uniformly curve define deleting derivative differentiable disc of radius entire function equation estimate Example Exercise exists Figure finite number follows fractional linear map function f give given harmonic function Hence holomorphic function interval inverse isomorphism Jensen's formula Lemma Let f Let z0 maximum modulus principle meromorphic function neighborhood obtain open disc open set point z0 pole polynomial positive integer power series expansion proves the theorem radius of convergence Re(s Re(z real axis real numbers rectangle residue Riemann sequence series converges Show simply connected subset Suppose Theorem 1.2 unit circle unit disc upper half plane winding number write zeta function