On the periodicity of trigonometric functions generalized to quotient rings of R[x]
Claude Gauthier
Open Mathematics, Tome 4 (2006), p. 395-412 / Harvested from The Polish Digital Mathematics Library
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We apply a method of Euler to algebraic extensions of sets of numbers with compound additive inverse which can be seen as quotient rings of R[x]. This allows us to evaluate a generalization of Riemann’s zeta function in terms of the period of a function which generalizes the function sin z. It follows that the functions generalizing the trigonometric functions on these sets of numbers are not periodic.

Publié le : 2006-01-01
EUDML-ID : urn:eudml:doc:269604 
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Claude Gauthier. On the periodicity of trigonometric functions generalized to quotient rings of R[x]. Open Mathematics, Tome 4 (2006) pp. 395-412. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-006-0020-y/

[1] H. Cartan: Théorie élémentaire des fonctions analytiques d’une ou plusieurs variables complexes, Hermann, Paris, 1961. | Zbl 0094.04401

[2] P. Deguire and C. Gauthier: “Sur la dérivation dans certains anneaux quotients de R[x]”, Ann. Sci. Math. Québec, Vol. 24, (2000), pp. 19–31. | Zbl 1094.13547

[3] L. Euler: “De summis serierum reciprocarum”, Comment. Acad. Sci. Petropolit., Vol. 7(1734/35), (1740), pp. 123–134; Opera omnia, Ser. 1, Vol. 14, Leipzig-Berlin, 1924, pp. 73–86.

[4] C. Gauthier: “Quelques propriétés algébriques des ensembles de nombres à inverse additif composé”, Ann. Sci. Math. Québec, Vol. 26, (2002), pp. 47–59.

[5] I.J. Good: “A simple generalization of analytic function theory”, Expo. Math., Vol. 6 (1988), pp. 289–311. | Zbl 0662.30045

[6] E. Grosswald: Topics from the Theory of Numbers, Birkhäuser, Boston, 1984.

[7] M.E. Muldoon and A.A. Ungar: “Beyond sin and cos”, Math. Mag., Vol. 69 (1996), pp. 2–14. http://dx.doi.org/10.2307/2691389 | Zbl 0860.33003

[8] H. Silverman: Complex Variables, Houghton Mifflin, Boston, 1975.

[9] G. Valiron: Théorie des fonctions, Masson, Paris, 1948.