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The class number one problem for the non-abelian normal CM-fields of degree 16

100%

Louboutin S.

Acta Arithmetica

1997

tom 82

nr 2

173-196

Remarkable identities

100%

Górowski J. , Żabowski J.

Journal of Mathematics and Applications

2014

tom Vol. 37

39--50

In the paper a number of identities involving even powers of the values of functions tangent, cotangent, secans and cosecans are proved. Namely, the following relations are shown: [wzory] where m, n are positive integers, f is one of the functions: tangent, cotangent, secans or cosecans and wf(x),vf(x),~wf(x) are some polynomials from Q[x]. One of the remarkable identities is the following: [wzór] Some of these identities are used to find, by elementary means, the sums of the series of the form [wzór] , where n is a fixed positive integer. One can also notice that Bernoulli numbers appear in the leading coeficients of the polynomials wf(x),vf(x) and ~ wf(x).

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

88%

Gauthier C.

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.

On a class of integrals appearing in subcritical thermodynamics of superconductors

75%

Gonczarek R. , Czerwonko J.

2000

tom Vol. 48, no 4

403-406

A formula for the family of integrals I[n] = [...], n [belongs to] N, is proved.