The Higher Calculus--A History of Real and Complex Analysis from Euler to Wei

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作者：Umberto Bottazini
文件大小：35.40MB
文件类型：扫描版PDF
语言：英语
出版：Springer-Verlag，1986年
页数：170
书签：无

内容简介：
This is a translation of the author's 1981 original [Boringhieri,Turin, 1981]; it has also been substantially revised.

The book begins with a brief account of 18th-century views on the fundamental concepts of analysis, especially the notion of function and the use of imaginary quantities. The author discusses Lagrange's attempt to make analysis a branch of algebra, and then Fourier's approach to mathematical physics, including the special role played in his work by boundary conditions and his use of the method of separation of variables for solving partial differential equations. The next chapter is concerned with the achievements of Gauss, Bolzano, Cauchy and Abel in rigorising real-variable analysis.

The author then turns to functions of a complex variable, discussing the various attempts to prove the fundamental theorems of algebra, the geometrical representation of complex numbers and Cauchy's creation of complex function theory. In the same chapter he also describes Cauchy's definition of the integral and his methods of establishing existence theorems for the solutions of differential equations.

We next find an account of early work on the convergence of Fourier series (Cauchy, Poisson, Dirichlet) and of the emergence of the concept of uniform convergence. A chapter on Riemann's work covers his investigations on integration and trigonometric series and also his approach to the foundations of complex function theory, including Riemann surfaces and the use of Dirichlet's principle. The final chapter is concerned not only with the construction of the real numbers by Dedekind, Cantor and others, and the beginnings of set theory, but also with Weierstrass's approach to complex analysis, his infinite-product representation for entire function, Mittag-Leffler's theorem on meromorphic functions, and Picard's theorem. An interesting feature of this chapter is an account of Casorati's discussions on the foundations of analysis with Kronecker and Weierstrass in 1864.

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