Geared toward advanced undergraduates and graduate students, this substantially revised and updated edition of a popular text offers a concise treatment that provides careful and complete explanations as well as numerous problems and solutions. Topics include elementary theory, general Cauchy theorem and applications, analytic functions, and prime number theorem. 2004 edition.
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analytic function analytic logarithm analytic on D(O analytic on Q argument principle assume that f Cauchy-Riemann equations Cauchy’s theorem closed curve closed path complex numbers constant on Q continuous argument continuous logarithm continuous on D(O D(zQ deﬁned Deﬁnition Let essential singularity f and g f is analytic f is constant f on Q f(Zo ﬁnite ﬁrst ﬁxed follows function f function on Q harmonic functions hence identity theorem isolated singularity Let f limit point linear fractional transformation logarithm of f maximum principle minimum principles open mapping theorem open set Q point in Q Poisson integral formula pole of order preserves angles Problem Proof Q is connected radius of convergence real-differentiable region Q removable singularity satisﬁes sequence Show that f singular point subset of Q Suppose f Theorem Let Theorem Suppose zero of order zeros of f