Abstract
This chapter is much simpler than the previous one. It can be studied independently and even much earlier. The chapter is devoted to the machinery of series, one of the most important technical tools of analysis. We consider different types of convergence of sequences and series of functions, and discuss conditions under which certain useful properties of functions persist after passing to a limit.
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- 1.
In the exceptional case when \(\mathop{\varlimsup}\nolimits_{n\rightarrow\infty}\sqrt[n]{|c_{n}|}=\infty\), we take \(R=0\) and the disk \(K\) degenerates to the single point \(z_{0}\).
- 2.
U. Dini (1845–1918) – Italian mathematician best known for his work in the theory of functions.
- 3.
F.W. Bessel (1784–1846) – German astronomer.
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E. Cesàro (1859–1906) – Italian mathematician who studied analysis and geometry.
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G.H. Hardy (1877–1947) – British mathematician who worked mainly in number theory and theory of functions.
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A. Tauber (b. 1866, year of death unknown) – Austrian mathematician who worked mainly in number theory and theory of functions.
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If you have not completely mastered the general concepts of Chap. 9, you may assume without any loss of content in the following that the functions discussed always map ℝ into ℝ or ℂ into ℂ, or \(\mathbb{R}^{m}\) into \(\mathbb{R}^{n}\).
- 8.
M.H. Stone (1903–1989) – American mathematician who worked mainly in topology and functional analysis.
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© 2016 Springer-Verlag Berlin Heidelberg
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Zorich, V.A. (2016). Uniform Convergence and the Basic Operations of Analysis on Series and Families of Functions. In: Mathematical Analysis II. Universitext. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-48993-2_8
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DOI: https://doi.org/10.1007/978-3-662-48993-2_8
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