New integral representations of the polylogarithm function

2006 ◽  
Vol 463 (2080) ◽  
pp. 897-905 ◽  
Author(s):  
Djurdje Cvijović
Keyword(s):  
Bernoulli Polynomials ◽  
Integral Representations ◽  
The Other ◽  
Important Special Case ◽  
Polylogarithm Function ◽  
Physical Problems ◽  
The Riemann Zeta Function ◽  
Dilogarithm Function ◽  
Riemann Zeta ◽  
Special Case

Maximon has recently given an excellent summary of the properties of the Euler dilogarithm function and the frequently used generalizations of the dilogarithm, the most important among them being the polylogarithm function Li s ( z ). The polylogarithm function appears in several fields of mathematics and in many physical problems. We, by making use of elementary arguments, deduce several new integral representations of the polylogarithm Li s ( z ) for any complex z for which | z |<1. Two are valid for all complex s , whenever Re  s >1. The other two involve the Bernoulli polynomials and are valid in the important special case where the parameter s is a positive integer. Our earlier established results on the integral representations for the Riemann zeta function ζ (2 n +1), n ∈ N , follow directly as corollaries of these representations.

scholarly journals A Probabilistic Proof for Representations of the Riemann Zeta Function

Mathematics ◽  
2019 ◽  
Vol 7 (4) ◽  
pp. 369
Author(s):  
Jiamei Liu ◽  
Yuxia Huang ◽  
Chuancun Yin
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Probabilistic Approach ◽  
Recurrence Formula ◽  
Integral Representations ◽  
The Riemann Zeta Function ◽  
Probabilistic Proof ◽  
Riemann Zeta ◽  
Positive Integers

In this paper, we present a different proof of the well known recurrence formula for the Riemann zeta function at positive even integers, the integral representations of the Riemann zeta function at positive integers and at fractional points by means of a probabilistic approach.

Bending and Stretching of Certain Types of Heterogeneous Aeolotropic Elastic Plates

1961 ◽  
Vol 28 (3) ◽  
pp. 402-408 ◽  
Author(s):  
E. Reissner ◽  
Y. Stavsky
Keyword(s):  
The Other ◽  
Explicit Solutions ◽  
Elastic Plates ◽  
Important Special Case ◽  
Elastic Symmetry ◽  
Transverse Bending ◽  
Special Case ◽  
Equal Thickness

The class of plates with which this paper is concerned includes as an important special case plates consisting of two orthotropic sheets of equal thickness which are laminated in such a way that the axes of elastic symmetry enclose an angle +θ with the x, y-axes in one sheet and an angle −θ in the other sheet. For plates of this type there occurs a coupling phenomenon between in-plane stretching and transverse bending which does not occur in the theory of homogeneous plates and which has not been considered in earlier work for such plates. The general results of the present paper are illustrated by means of explicit solutions for two specific plate problems.

scholarly journals Families of log Legendre Chi function integrals

2021 ◽  
pp. 21-21
Author(s):  
Anthony Sofo
Keyword(s):  
Zeta Function ◽  
Explicit Form ◽  
Riemann Zeta Function ◽  
Special Functions ◽  
Polylogarithm Function ◽  
The Riemann Zeta Function ◽  
Riemann Zeta

In this paper we investigate the representation of integrals involving the product of the Legendre Chi function, polylogarithm function and log function. We will show that in many cases these integrals take an explicit form involving the Riemann zeta function, the Dirichlet Eta function, Dirichlet lambda function and many other special functions. Some examples illustrating the theorems will be detailed.

A RELATION BETWEEN THE ZEROS OF TWO DIFFERENT L-FUNCTIONS WHICH HAVE AN EULER PRODUCT AND FUNCTIONAL EQUATION

2005 ◽  
Vol 01 (03) ◽  
pp. 401-429
Author(s):  
MASATOSHI SUZUKI
Keyword(s):  
Functional Equation ◽  
Zeta Function ◽  
Riemann Zeta Function ◽  
Modular Forms ◽  
Functional Equations ◽  
Euler Product ◽  
The Riemann Zeta Function ◽  
Euler Products ◽  
Riemann Zeta ◽  
Special Case

As automorphic L-functions or Artin L-functions, several classes of L-functions have Euler products and functional equations. In this paper we study the zeros of L-functions which have Euler products and functional equations. We show that there exists a relation between the zeros of the Riemann zeta-function and the zeros of such L-functions. As a special case of our results, we find relations between the zeros of the Riemann zeta-function and the zeros of automorphic L-functions attached to elliptic modular forms or the zeros of Rankin–Selberg L-functions attached to two elliptic modular forms.

scholarly journals Devising methods for planning a multifactorial multilevel experiment with high dimensionality

2021 ◽  
Vol 5 (4 (113)) ◽  
pp. 64-72
Author(s):  
Lev Raskin ◽  
Oksana Sira
Keyword(s):  
Original Problem ◽  
Accurate Solution ◽  
The Other ◽  
High Dimensionality ◽  
Optimal Plan ◽  
Approximate Methods ◽  
Important Special Case ◽  
Decomposition Procedure ◽  
Relevant Task ◽  
Special Case

This paper considers the task of planning a multifactorial multilevel experiment for problems with high dimensionality. Planning an experiment is a combinatorial task. At the same time, the catastrophically rapid growth in the number of possible variants of experiment plans with an increase in the dimensionality of the problem excludes the possibility of solving it using accurate algorithms. On the other hand, approximate methods of finding the optimal plan have fundamental drawbacks. Of these, the main one is the lack of the capability to assess the proximity of the resulting solution to the optimal one. In these circumstances, searching for methods to obtain an accurate solution to the problem remains a relevant task. Two different approaches to obtaining the optimal plan for a multifactorial multilevel experiment have been considered. The first of these is based on the idea of decomposition. In this case, the initial problem with high dimensionality is reduced to a sequence of problems of smaller dimensionality, solving each of which is possible by using precise algorithms. The decomposition procedure, which is usually implemented empirically, in the considered problem of planning the experiment is solved by employing a strictly formally justified technique. The exact solutions to the problems obtained during the decomposition are combined into the desired solution to the original problem. The second approach directly leads to an accurate solution to the task of planning a multifactorial multilevel experiment for an important special case where the costs of implementing the experiment plan are proportional to the total number of single-level transitions performed by all factors. At the same time, it has been proven that the proposed procedure for forming a route that implements the experiment plan minimizes the total number of one-level changes in the values of factors. Examples of problem solving are given

scholarly journals On an Exact Relation between ζ″(2) and the Meijer G -Functions

Mathematics ◽  
2019 ◽  
Vol 7 (4) ◽  
pp. 371
Author(s):  
Luis Acedo
Keyword(s):  
Taylor Series ◽  
Riemann Zeta Function ◽  
Recurrence Relations ◽  
Integral Representations ◽  
Standard Approach ◽  
Exact Relation ◽  
Complex Half Plane ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Dominant Part

In this paper we consider some integral representations for the evaluation of the coefficients of the Taylor series for the Riemann zeta function about a point in the complex half-plane ℜ ( s ) > 1 . Using the standard approach based upon the Euler-MacLaurin summation, we can write these coefficients as Γ ( n + 1 ) plus a relatively smaller contribution, ξ n . The dominant part yields the well-known Riemann’s zeta pole at s = 1 . We discuss some recurrence relations that can be proved from this standard approach in order to evaluate ζ ″ ( 2 ) in terms of the Euler and Glaisher-Kinkelin constants and the Meijer G -functions.

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