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2013 | OriginalPaper | Buchkapitel

Chapter 9 Several Complex Variables

verfasst von : Umberto Bottazzini, Jeremy Gray

Erschienen in: Hidden Harmony—Geometric Fantasies

Verlag: Springer New York

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Abstract

Throughout the nineteenth century and well into the twentieth the study of complex functions of several variables posed a challenge to the experts in the function theory of a single variable. As we have seen in Chap. 6, the prospect of creating a theory of Abelian functions was one that Weierstrass continually had in mind; it was the ultimate goal of all his work. And yet a marked distinction between the theories of one and several variables persists to the present day. Almost all universities offer a mainstream course in single variable complex function theory; few, if any, present the theory of several variables as other than a specialist option. We shall see that this distinction is in the nature of the functions studied. Because this dichotomy survives in the modern syllabus, we have divided this chapter into three sections. The first is a survey of the claim that the theories of one and several variables diverge markedly. We give an indication of what was discovered about the complex function theory of several variables, but generally slight the proofs so that the section can be read by a broad audience. The second section looks at the history of the principal results about Abelian functions and theta functions which, for a long time, were the only examples known of complex functions of several variables. In the final section we reconsider the general theory, look at some of the techniques used, and seek sharpen the discussion of the opening section. The latter sections naturally place greater demands on the reader.

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Vorheriges Kapitel Chapter 8 Advanced Topics in the Theory of Functions

Nächstes Kapitel Chapter 10 The Textbook Tradition

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We have taken this example from (Osgood 1914) which has generally been a very helpful guide to the material in this chapter.

There were investigations of multiple complex integrals by Maximilien Marie in the 1850s, but these seem to have been rather formal, superficial, and occasionally wrong as shown in Poincaré (1887a, 440–441). In the end they were without much influence, perhaps because, to quote (Coolidge 1924, 77) Marie “possessed the knack of quarrelling with his contemporaries almost to the point of genius”.

So it is particularly interesting to note Krantz’s comment (Krantz 1982, 16) that, by comparison with their comparatively modest role in the single-variable theory, when it comes to constructing functions in several complex variables the Cauchy–Riemann equations are central.

The function \(p\left (u,z\right )\) was called an algebroid function, by Poincaré in his then-unpublished thesis (1879, lii).

This account follows (Osgood 1914, 50–53).

This takes care of the examples such as \(\frac{x} {y}\) above.

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Gronwall, in his (1917), observed that Osgood had noted in (1914) that Cousin’s argument is sketchy when it comes to specifying what branches are involved and gave Osgood’s modifications to make the matter precise.

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Osgood commented (1914, 38) that “this theorem in its present form was first given by Fabry C.R. 134 (1902) pp. 1190, and rediscovered by Hartogs.” He also noted that a certain A. Meyer had already given a geometric interpretation of such a function in 1883: the graph of logs as a function of logr is continuous and concave downwards.

This marks the first appearance of the Levi form. See Range (2012) for an introduction to the convexity issues that arise.

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Titel: Chapter 9 Several Complex Variables
verfasst von: Umberto Bottazzini
Jeremy Gray
Verlag: Springer New York
Buch: Hidden Harmony—Geometric Fantasies
Print ISBN: 978-1-4614-5724-4

Electronic ISBN: 978-1-4614-5725-1

Copyright-Jahr: 2013
DOI: https://doi.org/10.1007/978-1-4614-5725-1_10

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