Elliptic functions according to Eisenstein and Kronecker.
type: Item
instance of: scholarly article
title: Elliptic functions according to Eisenstein and Kronecker.
zbMATH Open document ID: 0955.11001
published in: Classics in Mathematics
publication date: 1999-01-07T00:00:00Z
review text: As \textit{N. K. Takare} wrote in the review of the first edition of this very rich and stimulating book (1976; Zbl 0318.33004): ``Professor André Weil tries to resurrect Eisenstein's long-forgotten approach to the study of elliptic functions and, doing so, adds much value to this interesting part of special function theory. His opinion is that Eisenstein's, together with Kronecker's works, can be profitably applied to some current problems in the theory of elliptic functions and the modular group, and extended beyond, in particular to the arithmetical study of Eisenstein series for Hilbert modular group''. The book is divided, roughly, into three parts. In the first one, the author presents the theory of elliptic functions as developed by Eisenstein as early as 1847 in Crelles Journal. This theory includes the definition, by means of ``Eisenstein summation process'', of the function \(E_2(z)\), which is \(\wp(z)\) up to an additive constant, and the determination of its differential equation, fifteen years before Weierstrass. The connection beetween the function \(E_2\) and the cotangent, as well as the fact that Eisenstein uses only elementary computations (avoiding Cauchy's theory), turn out to be very illuminating. Eisenstein obtains, by his approach, most results of the theory of elliptic functions. André Weil completes the theory by proving the relations between theta series and infinite products, what Eisenstein refrained from doing by ``lack of time'' (Eisenstein died in 1852 aged 29). The second part of the book is devoted to Kronecker's memoirs on elliptic functions, which were published by the Berlin Academy in 1883, 1885, 1886, 1889 and 1990, mainly to those concerned with double series of the form \[ G(s,\chi)= \sum_{w\in W}\chi(w) |w|^{-2s}, \] where \(W\) is a lattice in the complex plane, and \(\chi\) a character of the additive group \(W\). One of the results presented by the author is Kronecker's limit formula, which gives the expansion of \(G(s,1)…
Mathematics Subject Classification ID: 11-03
Mathematics Subject Classification ID: 33E05
Mathematics Subject Classification ID: 01A55
Mathematics Subject Classification ID: 11-02
Mathematics Subject Classification ID: 33-02
Mathematics Subject Classification ID: 11F03
Mathematics Subject Classification ID: 30F35
zbMATH DE Number: 1236956
zbMATH Keywords: elliptic functions
zbMATH Keywords: Eisenstein
zbMATH Keywords: Kronecker
zbMATH Keywords: modular group
reviewed by: Daniel Duverney
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1236956