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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Notes
- 1.
- 2.
We have taken this example from (Osgood 1914) which has generally been a very helpful guide to the material in this chapter.
- 3.
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”.
- 4.
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.
- 5.
The function \(p\left (u,z\right )\) was called an algebroid function, by Poincaré in his then-unpublished thesis (1879, lii).
- 6.
This account follows (Osgood 1914, 50–53).
- 7.
This takes care of the examples such as \(\frac{x} {y}\) above.
- 8.
One of the most gifted Italian mathematicians of his generation, in his short life—he died in 1917 at the front during WWI—in addition to function theory Levi gave substantial contributions to various fields of mathematics, including the theory of partial differential equations and the calculus of variations. See Picone’s introduction in (Levi, Opere 1, V–XV).
- 9.
The generalised Laurent theorem is explained in (Osgood 1924, 68–70).
- 10.
- 11.
Otto Blumenthal was the first of Hilbert’s students. He taught at the University of Aachen for many years, and became an editor of Mathematische Annalen, but when the Nazis came to power he was forced to resign because one of his grandparents was a Jew, although he himself was a practising Protestant. In 1939 he obtained a teaching position in Delft, but in 1943 he was captured, and he died in Theresienstadt in November 1944, age 68.
- 12.
For an analysis of Cousin’s work and the subsequent developments in sheaf theory, see Chorlay (2010).
- 13.
See Maurey, B. and J.-P. Tacchi, La genèse du théorème de recouvrement de Borel, http://www.math.jussieu.fr/~maurey/articles/GenRev.pdf, who cite “le recueil de l’association des anciens élèves de l’École Normale Supérieure” for 1934. It seems that Cousin was born in Paris on 18 March 1867 and studied mathematics at the École Normale where he wrote his doctorate in 1886. He then taught at the Lycée in Caen until 1894, when he became a maître de conférences at Grenoble, and then in 1902 he became a professor at Bordeaux. His last publication was published in Acta mathematica in 1910, he was Head of the Faculté des sciences at Bordeaux from 1924 to 1930, and he died in 1933.
- 14.
- 15.
- 16.
Friedrich Hartogs became a Privatdozent in 1909, and an extraordinary professor at Munich in 1912, but he had to wait until 1927 to be appointed full professor there. After the Nazi party came to power in 1933 Hartogs, who was a Jew, suffered increasingly difficult times until he was forced to retire in 1935. After the Kristallnacht of 9–10 November 1938 he was imprisoned at Dachau for several weeks. Further humiliations followed his release, including a enforced divorce from his Aryan wife to protect her, and in the end he committed suicide in 1943. For biographical information on Pringsheim, Hartogs and other Jewish mathematicians at Munich University, see Bauer (1997).
- 17.
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.
- 18.
This marks the first appearance of the Levi form. See Range (2012) for an introduction to the convexity issues that arise.
- 19.
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Bottazzini, U., Gray, J. (2013). Chapter 9 Several Complex Variables. In: Hidden Harmony—Geometric Fantasies. Sources and Studies in the History of Mathematics and Physical Sciences. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-5725-1_10
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