scholarly journals A Quadruple Definite Integral Expressed in Terms of the Lerch Function

Symmetry ◽  
2021 ◽  
Vol 13 (9) ◽  
pp. 1638
Author(s):  
Robert Reynolds ◽  
Allan Stauffer
Keyword(s):  
Special Functions ◽  
Definite Integral ◽  
Polynomial Functions ◽  
Fundamental Constants ◽  
Zero Distribution ◽  
Special Cases ◽  
Almost All

A quadruple integral involving the logarithmic, exponential and polynomial functions is derived in terms of the Lerch function. Special cases of this integral are evaluated in terms of special functions and fundamental constants. Almost all Lerch functions have an asymmetrical zero-distribution. The majority of the results in this work are new.

scholarly journals A Note on a Triple Integral

Symmetry ◽  
2021 ◽  
Vol 13 (11) ◽  
pp. 2056
Author(s):  
Robert Reynolds ◽  
Allan Stauffer
Keyword(s):  
Closed Form ◽  
Closed Form Expression ◽  
Polynomial Functions ◽  
Fundamental Constants ◽  
Form Expression ◽  
Zero Distribution ◽  
Special Cases ◽  
Almost All

A closed form expression for a triple integral not previously considered is derived, in terms of the Lerch function. Almost all Lerch functions have an asymmetrical zero-distribution. The kernel of the integral involves the product of the logarithmic, exponential, quotient radical, and polynomial functions. Special cases are derived in terms of fundamental constants; results are summarized in a table. All results in this work are new.

scholarly journals A Quadruple Integral Involving Chebyshev Polynomials Tn(x): Derivation and Evaluation

Symmetry ◽  
2022 ◽  
Vol 14 (1) ◽  
pp. 100
Author(s):  
Robert Reynolds ◽  
Allan Stauffer
Keyword(s):  
Zeta Function ◽  
Chebyshev Polynomial ◽  
Chebyshev Polynomials ◽  
Zeta Functions ◽  
Fundamental Constants ◽  
Zero Distribution ◽  
Special Cases ◽  
Lerch Zeta Function ◽  
Almost All

The aim of the current document is to evaluate a quadruple integral involving the Chebyshev polynomial of the first kind Tn(x) and derive in terms of the Hurwitz-Lerch zeta function. Special cases are evaluated in terms of fundamental constants. The zero distribution of almost all Hurwitz-Lerch zeta functions is asymmetrical. All the results in this work are new.

scholarly journals Definite Integral Involving Rational Functions of Powers and Exponentials Expressed in Terms of the Lerch Function

2021 ◽  
Vol 26 (3) ◽  
pp. 58
Author(s):  
Robert Reynolds ◽  
Allan Stauffer
Keyword(s):  
Exponential Function ◽  
Special Functions ◽  
Rational Functions ◽  
Definite Integral ◽  
Fundamental Constants ◽  
Definite Integrals ◽  
Special Cases ◽  
Quotient Of Functions ◽  
Integral Formulae

This paper gives new integrals related to a class of special functions. This paper also showcases the derivation of definite integrals involving the quotient of functions with powers and the exponential function expressed in terms of the Lerch function and special cases involving fundamental constants. The goal of this paper is to expand upon current tables of definite integrals with the aim of assisting researchers in need of new integral formulae.

scholarly journals Note on a Stieltjes Transform in terms of the Lerch Function

2021 ◽  
Vol 14 (3) ◽  
pp. 723-736
Author(s):  
Robert Reynolds ◽  
Allan Stauffer
Keyword(s):  
Closed Form ◽  
Special Functions ◽  
Complex Numbers ◽  
Definite Integral ◽  
Fundamental Constants ◽  
Logarithmic Function ◽  
Stieltjes Transform ◽  
Closed Form Solutions ◽  
General Complex

In this work the authors derive the Stieltjes transform of the logarithmic function in terms of the Lerch function. This transform is used to derive closed form solutions involving fundamental constants and special functions. Specifically we derive the definite integral given by\[\int_{0}^{\infty} \frac{(1-b x)^m \log ^k(c (1-b x))+(b x+1)^m \log ^k(c (b x+1))}{a+x^2}dx\]where $a,b,c,m$ and $k$ are general complex numbers subject to the restrictions given in connection with the formulas.

scholarly journals The Logarithmic Transform of a Polynomial Function Expressed in Terms of the Lerch Function

Mathematics ◽  
2021 ◽  
Vol 9 (15) ◽  
pp. 1754
Author(s):  
Robert Reynolds ◽  
Allan Stauffer
Keyword(s):  
Polynomial Function ◽  
Special Functions ◽  
Polynomial Functions ◽  
Fundamental Constants ◽  
Definite Integrals

This is a collection of definite integrals involving the logarithmic and polynomial functions in terms of special functions and fundamental constants. All the results in this work are new.

scholarly journals A Quadruple Integral Involving Product of the Struve Hv(βt) and Parabolic Cylinder Du(αx) Functions

Symmetry ◽  
2021 ◽  
Vol 14 (1) ◽  
pp. 9
Author(s):  
Robert Reynolds ◽  
Allan Stauffer
Keyword(s):  
Zeta Function ◽  
Special Functions ◽  
Zeta Functions ◽  
Parabolic Cylinder ◽  
Fundamental Constants ◽  
Lerch Zeta Function ◽  
Parabolic Cylinder Functions ◽  
Infinite Integral ◽  
Almost All

The objective of the present paper is to obtain a quadruple infinite integral. This integral involves the product of the Struve and parabolic cylinder functions and expresses it in terms of the Hurwitz–Lerch Zeta function. Almost all Hurwitz-Lerch Zeta functions have an asymmetrical zero distributionSpecial cases in terms fundamental constants and other special functions are produced. All the results in the work are new.

scholarly journals ON PROBABILTY DISTRIBUTION IN ASSOCIATION WITH A CERTAIN GENERALIZED HYPERGEOMETRIC FUNCTION

2015 ◽  
Vol 14 (3) ◽  
pp. 5569-5577
Author(s):  
Yashwant Singh ◽  
Nanda Kulkarni
Keyword(s):  
Density Function ◽  
Probability Function ◽  
Special Function ◽  
Special Functions ◽  
Density Functions ◽  
Chi Square ◽  
Recurrence Relationship ◽  
Special Cases ◽  
Almost All ◽  
Special Case

In the present paper, a probability function has been introduced in terms of the -function and its properties are studied. It is shown that the classical non-central distributions such as, non-central chi-square, non-central Student- , non-central and almost all classical central continuous distributions can be obtained as special cases of this general density function. This general density function is introduced with the hope that any density function, which can be represented in terms of any known special function as well as the density of the ratio of any two independent stochastic variables whose density functions can be represented in terms of any known special functions, is contained in as a special case. The properties of , discussed in this paper, include the characteristic function, moments, recurrence relationship among moments and the distribution function.

scholarly journals The Double Laplace Transform Expressed in terms of the Lerch Transcendent

2021 ◽  
Vol 14 (3) ◽  
pp. 618-637
Author(s):  
Robert Reynolds ◽  
Allan Stauffer
Keyword(s):  
Laplace Transform ◽  
Special Functions ◽  
Fundamental Constants ◽  
Definite Integrals ◽  
Special Values ◽  
Double Laplace Transform ◽  
Lerch Transcendent

In this manuscript, the authors derive a formula for the double Laplace transform expressed in terms of the Lerch Transcendent. The log term mixes the variables so that the integral is not separable except for special values of k. The method of proof follows the method used by us to evaluate single integrals. This transform is then used to derive definite integrals in terms of fundamental constants, elementary and special functions. A summary of the results is produced in the form of a table of definite integrals for easy referencing by readers.

scholarly journals Fractional trapezium type inequalities for twice differentiable preinvex functions and their applications

2020 ◽  
Vol 10 (2) ◽  
pp. 226-236
Author(s):  
Artion Kashuri ◽  
Rozana Liko
Keyword(s):  
Integral Operator ◽  
Integral Inequalities ◽  
Fractional Integrals ◽  
Recent Past ◽  
Trapezium Type ◽  
Special Means ◽  
Special Cases ◽  
Type Integral ◽  
Preinvex Functions ◽  
Almost All

Trapezoidal inequalities for functions of divers natures are useful in numerical computations. The authors have proved an identity for a generalized integral operator via twice differentiable preinvex function. By applying the established identity, the generalized trapezoidal type integral inequalities have been discovered. It is pointed out that the results of this research provide integral inequalities for almost all fractional integrals discovered in recent past decades. Various special cases have been identified. Some applications of presented results to special means have been analyzed. The ideas and techniques of this paper may stimulate further research.

scholarly journals On the General Consensus Protocol in Multiagent Networks with Double-Integrator Dynamics and Coupling Time Delay

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Tao Dong ◽  
Xiaofeng Liao
Keyword(s):  
Time Delay ◽  
Matrix Theory ◽  
Sufficient Conditions ◽  
Consensus Algorithm ◽  
Directed Network ◽  
Consensus Protocol ◽  
Special Cases ◽  
Coupling Time ◽  
Almost All ◽  
Double Integrator

This paper considers the problem of the convergence of the consensus algorithm for multiple agents in a directed network where each agent is governed by double-integrator dynamics and coupling time delay. The advantage of this protocol is that almost all the existing linear local interaction consensus protocols can be considered as special cases of the present paper. By combining algebraic graph theory and matrix theory and studying the distribution of the eigenvalues of the associated characteristic equation, some necessary and sufficient conditions are derived for reaching the second-order consensus. Finally, an illustrative example is also given to support the theoretical results.

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