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Chapter 4 Complex Functions and Elliptic Integrals

verfasst von : Umberto Bottazzini, Jeremy Gray

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Abstract

In this chapter we consider how elliptic function theory and complex variable theory were finally drawn together in the 1830s and 1840s. As the recognition of the importance of the work of Abel and Jacobi grew, mathematicians came to feel that it was unsatisfactory to base the theory of elliptic functions on the inversion of many-valued integrals. One alternative would have been to adopt and develop Cauchy’s theory of complex integrals. By and large this was not done, and it is interesting to examine why. The study of elliptic integrals was felt by many to be fraught with ambiguity because of the square root in the integrand. Moreover, Cauchy’s system of definitions, based on his newly defined concepts of limit, continuity, differentiability, and integrability, was incompatible with talk of many-valued functions—Cauchy did not define continuity for a many-valued function, and indeed a many-valued function cannot be continuous according to Cauchy’s use of the term. Although a doubly periodic function is a meromorphic function defined on the whole of the complex plane, an elliptic integral makes better sense on something like a Riemann surface (a torus in this case). Thus the many-valued nature of an elliptic integral posed a challenge to mathematicians throughout the 1830s and 1840s. So the perceived problem with the foundations did not meet with a ready answer in the newly emerging theory of complex functions. Matters were to be worse with hyperelliptic integrals, because the corresponding inverse functions could not be treated as multiply-periodic functions in the plane.

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Vorheriges Kapitel Chapter 3 Cauchy’s “Modern Analysis”

Nächstes Kapitel Chapter 5 Riemann’s Geometric Function Theory

This is not the same as the object defined by Eq. (1.51).

For the theorem, see Jacobi Werke 1, p. 506, and for the quote p. 499. The identities below occur on pp. 510–511.

Quoted in Koenigsberger (1904b, 186–187).

It has been discussed, briefly, in Houzel (1978) and in more detail in Gray (1992).

See Cauchy (1827e, 336).

Cayley (1845b, 148). This is stated in slightly different terms also in Cayley (1845c, 175).

The matter has been investigated by Lützen in his magisterial biography of Liouville (Lützen 1990), upon which our account is based.

A student of Bordoni and Brioschi at the University of Pavia, he was appointed Extraordinarius for algebra and analytic geometry there in 1859, becoming Ordinarius in 1862. The following year he switched to the chair of calculus that he occupied for almost 30 years until the end of his life. For an account of his life and work, see Bertini’s obituary in Casorati, Opere (1, 3–30).

For the chronology, see Koenigsberger (1879).

For this incident, see Ore (1957, 150–151); on Cauchy’s notoriety, see Lützen (1990, 138).

Abel’s monument more lasting as bronze (Legendre 1832). The original is Horace, Odes 3.30, “Exegi monumentum aere perennius”, “I have erected a monument more lasting than bronze”, where he prophesied an eternal life for his poems and fame for himself.

For a full account, upon which ours is largely based, see Ore (1957, 246–261). The discovery of the remaining eight pages is described in Del Centina (2002, 2003).

The papers (Broch 1840, 1842; Jürgensen 1839, 1842; Minding 1842; Rosenhain 1844–1845) are described in Brill and Noether (1894, 226–234) and following them (Hancock 1897, 258–260).

Following (Kleiman 2004, 398).

For a mathematical discussion, see Kleiman (2004, 404–405), who points out that the proof is an exercise in the use of the symmetric functions of the x _j, which depend on the coefficients of the polynomial Q.

For a similar conclusion, see Cooke (1989, 412).

The magnitude of Jacobi’s achievement may be judged by Eisenstein’s failure to follow him here, a failure the older man judged harshly. See Hancock (1897, 278).

See Göpel (1847) and Rosenhain (1851); described in Brill and Noether (1894, 236–239).

See, for example, Brill and Noether (1894, 234–239) and Houzel (1978).

Principia, Book I, Section 6. For a discussion in modern terms, see Arnold (1990, 83–105).

See Euler (1760, 1762b, 1763). Euler showed it was integrable and gave the “general solution as an equation between two elliptic integrals with separated variables” (Wilson 1994, 1054). For Lagrange, see his (1766–1769) and (1815, 101–114). Legendre (1811–1817, 3, §384) observed that the variables must be allowed to go complex if complete branches of the solutions are to be obtained.

See Appell (1879) and Whittaker (1904, 4th ed. 1937, 73), where the motion is expressed in terms of the Jacobian elliptic function sn.

In this case the polhode is a circle, and the angular momentum vector rotates around the body axis; the phenomenon of precession.

See Klein and Sommerfeld (1898, 2, 473) and Jacobi (1850, 307).

See Jacobi (1850, 293–294).

Jacobi to Hermite, 6 August 1845, see Jacobi (1846a).

Published posthumously as Jacobi (1866).

See Ges. Werke. Supplementband, 207.

Binet (1811). This theorem was generalised in Dupin (1813).

Jacobi pointed out that it was in this way that Lagrange had shown that the problem of attraction to two fixed centres led to a proof of the fundamental theorem for elliptic functions (corresponding to the case n = 2 here); see Lagrange (1766–1769).

They have an ambiguous sign factor that Neumann kept track of.

Neumann’s notation is ambiguous and this expression is our interpretation of what he meant.

See Gauss Werke 10.1, 560–561 or also Dunnington (2004, 481).

See Gauss Werke 10.1, 25.

A summary will be found in Berndt and Evans (1981).

Gauss’s unpublished remarks are analysed by Schlesinger (1912, 122–3). See also Hermite, Oeuvres 1, 486.

They can be found in Schoeneberg (1974, 220).

See Cauchy (1840b). Cauchy published this paper twice, first in the Comptes rendus of the Académie and then in the Journal de mathématiques.

See Kronecker (1880). This proof was flawed, and in Kronecker (1889) he gave a better proof. See also the account in Landau (1958, 203–207).

See, e.g. Mordell (1918) and Landau (1958).

See Lagrange (1775–1777).

Good introductions to this material are Cox (1989) and Goldstein et al. (2007).

See Heegner (1952) for a treatment of h = 1 and for entirely rigorous accounts (Baker 1966) and (Stark 1967); for h = 2 see Baker (1971) and Stark (1971); for h = 3 see Gross and Zagier (1983). In each case, Gauss’s conjecture was affirmed. For an informative and historical account of these developments, see Goldfeld (1985) and Zagier (1984).

See Dirichlet (1852, 241), in his Math. Werke 2, quote on p. 241. Dirichlet’s paper on the class number formula is his (1839, 1840). A lucid and informative account of Dirichlet’s proof of the class number formula will be found in Scharlau and Opolka (1985, 121–143).

See Dirichlet (1837b). There are many good accounts in the literature, e.g. Apostol (1976), Davenport (1967), Serre (1973).

Landau’s function-theoretic proof that the L series do not vanish when s = 1 is much later (Landau 1910). He defined \(\zeta _{m}(s) = \Pi _{\omega }L_{\omega }(s)\). The function L ₁(s) has a simple pole at s = 1. If any L _ω(s) were zero, ζ _m(s) would be holomorphic at s = 1, but this contradicts other conclusions that can be reached using the theory of Dirichlet series.

See Gauss Werke (10.1, 571) or also Dunnington (2004, 484).

See Gauss Werke (10.1, 572).

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Titel: Chapter 4 Complex Functions and Elliptic Integrals
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_5

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