Titlebook: Complex Analysis in one Variable; Raghavan Narasimhan Book 19851st edition Springer Science+Business Media New York 1985 Complex analysis.

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https://doi.org/10.1007/978-1-4302-2498-3This chapter is devoted to various theorems which can be proved using Runge’s theorem : the existence of functions with prescribed zeros or poles, a “cohomological” version of Cauchy’s theorem, and related theorems. The last section concerns itself with .. (Ω) as a ring (or ℂ-algebra).

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Transaction Management in Spring,In this chapter, we shall prove that any simply connected open set in ℂ, which is not all of ℂ, is analytically isomorphic to the unit disc .= {z∊ℂ∣∣z∣<1}. The proof will also enable us to characterize simple connectedness in several ways.

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EJB, Spring Remoting, and Web Services,We saw, in Chapter 6, that if Ω is open in ℂ and f., …. , f. ∈ ℋ (Ω) and have no common zeros in Ω, then there exist g. ... , g. ∈ ℋ (Ω) such that ∑ g.f. ≡1.

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Transaction Management in Spring,In this chapter, we introduce, and study, subharmonic functions and use them to solve the Dirichlet problem for harmonic functions (on reasonable domains). We shall indicate some other applications of these functions at the end of the chapter.

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Elementary Theory of Holomorphic Functions,In this chapter, we shall develop the classical theory of holomorphic functions. The Looman-Menchoff theorem, proved in § 6, is less standard than the rest of the material.

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The Riemann Mapping Theorem and Simple Connectedness in the Plane,In this chapter, we shall prove that any simply connected open set in ℂ, which is not all of ℂ, is analytically isomorphic to the unit disc .= {z∊ℂ∣∣z∣<1}. The proof will also enable us to characterize simple connectedness in several ways.