Titlebook: Complex Analysis; Serge Lang Textbook 1993Latest edition Springer-Verlag GmbH Germany, part of Springer Nature 1993 Complex analysis.Merom

疏忽

Conformal MappingsIn this chapter we consider a more global aspect of analytic functions, describing geometrically what their effect is on various regions. Especially important are the analytic isomorphisms and automorphisms of various regions, of which we consider many examples.

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The Riemann Mapping TheoremIn this chapter we give the general proof of the Riemann mapping theorem. We also prove a general result about the boundary behavior.

Conspiracy

Applications of the Maximum Modulus Principle and Jensen’s FormulaWe return to the maximum principle in a systematic way, and give several ways to apply it, in various contexts.

Assault

Elliptic FunctionsIn this chapter we give the classical example of entire and meromorphic functions of order 2. The theory illustrates most of the theorems proved so far in the book. A self-contained “analytic” continuation of the topics discussed in this chapter can be found in Chapters 3, 4, and 18 of my book on . [La 73].

invulnerable

The Gamma and Zeta FunctionsWe now come to a situation where the natural way to define a function is not through a power series but through an integral depending on a parameter. We shall give a natural condition when we can differentiate under the integral sign, and we can then use Goursat’s theorem to conclude that the holomorphic function so defined is analytic.

乐章

The Prime Number TheoremAt the turn of the century, Hadamard and de la Vallee Poussin independently gave a proof of the prime number theorem, exploiting the theory of entire functions which had been developed by Hadamard. Here we shall give D. J. Newman’s proof, which is much shorter. I have also benefited from Korevaar’s exposition.

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角斗士

https://doi.org/10.1007/978-3-540-26542-9rincipal ways will be by means of power series. Thus we shall see that the series.converges for all . to define a function which is equal to .. Similarly, we shall extend the values of sin . and cos . by their usual series to complex valued functions of a complex variable, and we shall see that they

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