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

100%

Gauthier C.

Open Mathematics

2006

tom 4

nr 3

395-412

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.

Ellipses numbers and geometric measure representations

44%

Richter W.-D.

Journal of Applied Analysis

2011

tom Vol. 17, nr 2

165-179

Ellipses will be considered as subsets of suitably defined Minkowski planes in such a way that, additionally to the well-known area content property A(r) = Π (a,b) r 2, the number Π (a,b) = abΠ reflects a generalized circumference property U (a,b)(r) = 2Π (a,b) r of the ellipses E (a,b)(r) with main axes of lengths 2ra and 2rb, respectively. In this sense, the number Π (a,b) is an ellipse number w.r.t. the Minkowski functional r of the reference set E (a,b)(1). This approach is closely connected with a generalization of the method of indivisibles and avoids elliptical integrals. Further, several properties of both a generalized arc-length measure and the ellipses numbers will be discussed, e.g. disintegration of the Lebesgue measure and an elliptically contoured Gaussian measure indivisiblen representation, wherein the ellipses numbers occur in a natural way as norming constants.