Abstract
One of the great excitements in the mathematical world of the 1830s and 1840s was the discovery by Abel and Jacobi independently of elliptic functions. These are complex-valued functions of a complex variable, and as such they were new and proved an important stimulus for the growth of a theory of complex functions. Equally importantly, they had properties akin to the trigonometric functions—they were, however, not singly but doubly periodic, as will be explained below—and this made them attractive to study. And, as Legendre had already shown, they were likely to have many applications in mechanics.
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Notes
- 1.
Procès-verbaux des séances de l’Académie 9 (1921), 373; Poisson Rapport (1831).
- 2.
Among those passed over for the prize on this occasion was Évariste Galois .
- 3.
Quoted in Koenigsberger (1904b, 54) and Krazer (1909, 55n).
- 4.
On Abel ’s short life see Stubhaug (2000). The earlier Ore (1957) is more reliable on the mathematics, which is described in greater detail and depth in Houzel (2004).
- 5.
In due course Abel ’s proof that the quintic equation is not solvable by radicals was one of the great breakthroughs opening the door to Galois ’s work.
- 6.
Quoted in Ore (1957, 65).
- 7.
See e.g. Krazer (1909, 56).
- 8.
The actual number is \(n^2 \) if n is odd and \(2n^2 \) if n is even. A sign enters the expression for the general solution that must be positive when n is odd but may be positive or negative when n is even.
- 9.
As commentators noted, despite its title it concentrated on pure mathematics, to the point that wits called it the Journal für die reine unangewandte Mathematik, the Journal for purely unapplied mathematics. It became, and remains, one of the leading journals for mathematics in the world.
- 10.
He seems not to have known at this stage of the comparable work of Ruffini from 1799.
- 11.
Both remarks are in Bjerknes (1885, 92).
- 12.
See Abel (1902, 45–46).
- 13.
See Abel (1881, 2, 261). The circumference of the lemniscate is divisible into m equal parts by ruler and compass alone (italics Abel ’s) if m is of the form \(2^n\) or a prime of the form \(2^n + 1\), or if m is a product of numbers of these two kinds. See Houzel (2004).
- 14.
Compare the equation for \(\sin (nx)\) given \(\sin (x)\).
- 15.
Abel also observed that these results followed from those in Legendre ’s Exercises.
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Gray, J. (2015). Abel. 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_7
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