Abstract
At the time when Legendre published his treatise [128], Abel, a young Norwegian mathematician, was beginning to publish a huge generalization of the addition theorems for elliptic functions. This generalization dealt with all the integrals of the form: 9.1 y being an arbitrary algebraic function of the variable x. Those integrals were later called “abelian transcendents” by Jacobi in [107]. In the sequel, we use the simpler name of abelian integrals.
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Notes
- 1.
That is, the square root of a polynomial of degree at most four.
References
N.H. Abel, Mémoire sur une propriété générale d’une classe très étendue de fonctions transcendantes, Presented at the French Académie des Sciences in Paris on the 30 October 1826. Republished in Œuvres complètes de Niels Henrik Abel, vol. I, ed. by L. Sylow, S. Lie (Grondahl and Son, Christiania, 1881), pp. 145–211
C. Jacobi, Considerationes generales de transcendentibus abelianis. J. Reine Angew. Math. 9, 394–403 (1832)
S.L. Kleiman, What is Abel’s theorem anyway? in [127], pp. 395–440
A.-M. Legendre, Traité des fonctions elliptiques, Tome Premier (Huzard-Courcier, Paris, 1825)
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Popescu-Pampu, P. (2016). Abel and the New Transcendental Functions. In: What is the Genus?. Lecture Notes in Mathematics(), vol 2162. Springer, Cham. https://doi.org/10.1007/978-3-319-42312-8_9
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DOI: https://doi.org/10.1007/978-3-319-42312-8_9
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