Abstract
Weierstrass is remembered for many specific discoveries in complex function theory, and here we consider two of them: his insight into the distinction between poles and essential singularities and the idea of a natural boundary of a complex function and the connection to nowhere differentiable real functions. We then look more briefly at his systematic presentation of the theory of elliptic functions, and content ourselves with a mention of his representation theorem, and his final account of a theory of Abelian functions.
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Notes
- 1.
And even more accurately as the Casorati-Sokhotskii-Weierstrass theorem, after the Russian mathematician Sokhotskii who came to the same ideas at about the same time, see Bottazzini and Gray (2013, § 7.7.3).
- 2.
Riemann too, as Dedekind reported when editing Riemann’s papers, found his examples of pathological functions looking at limit case of \(\theta \)-functions. But, as Klein wrote in his history of mathematics in the 19th Century (1926, I, 286), Riemann’s attitude was quite different from Weierstrass’s: “Riemann excluded natural boundaries from his considerations. Weierstrass, on the contrary, was led precisely by his systematic manner of thought to look closer at the behaviour of an analytic function in the neighbourhood its natural boundaries”.
- 3.
See Du Bois-Reymond (1875) and Weierstrass (1872), the version in Weierstrass’s Mathematische Werke, which differs from the account given by du Bois-Reymond only in the correction of an simple error.
- 4.
Weierstrass is our source for this information about Riemann, and interestingly he was of the opinion that differentiability fails everywhere for Riemann’s function but is “somewhat difficult to demonstrate”; see Weierstrass (1872, 72).
- 5.
For more detail, see Schubring (2012, 570–575) and Gispert (1983, 49).
- 6.
The Formeln und Lehrsätze (or, Formulae and Results) of 1885.
- 7.
The order is reversed in the Formeln und Lehrsätze.
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Gray, J. (2015). Weierstrass’s Foundational Results. In: The Real and the Complex: A History of Analysis in the 19th Century. Springer Undergraduate Mathematics Series. Springer, Cham. https://doi.org/10.1007/978-3-319-23715-2_20
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