On the periodicity of trigonometric functions generalized to quotient rings of R[x]

• Autorzy: Claude Gauthier
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

Compound inverse; generalized trigonometric functions; zeta function; 30G35; 11M41

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2006
• Tom: 4
• Numer: 3
• Strony: 395-412

Daty

wydano

2006-09-01

online

2006-09-01

Twórcy

autor

Claude Gauthier

• Université de Moncton,

Bibliografia

• [1] H. Cartan: Théorie élémentaire des fonctions analytiques d’une ou plusieurs variables complexes, Hermann, Paris, 1961.
• [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.
• [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.
• [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
• [8] H. Silverman: Complex Variables, Houghton Mifflin, Boston, 1975.
• [9] G. Valiron: Théorie des fonctions, Masson, Paris, 1948.

Identyfikatory

DOI

10.2478/s11533-006-0020-y