scholarly journals Mellin Transform of Logarithm and Quotient Function with Reducible Quartic Polynomial in Terms of the Lerch Function

Axioms ◽  
2021 ◽  
Vol 10 (3) ◽  
pp. 236
Author(s):  
Robert Reynolds ◽  
Allan Stauffer
Keyword(s):  
Mellin Transform ◽  
Contact Problems ◽  
Fundamental Constants ◽  
Definite Integrals ◽  
Special Cases ◽  
Quartic Polynomial

A class of definite integrals involving a quotient function with a reducible polynomial, logarithm and nested logarithm functions are derived with a possible connection to contact problems for a wedge. The derivations are expressed in terms of the Lerch function. Special cases are also derived in terms fundamental constants. The majority of 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 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 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.

Derivation of Green-type, transitional, and uniform asymptotic expansions from differential equations II. Whittaker functions W k,m for large k , and for large | K 2 – m 2 |

1964 ◽  
Vol 281 (1384) ◽  
pp. 111-129 ◽  
Keyword(s):  
Differential Equations ◽  
Asymptotic Expansions ◽  
Mellin Transform ◽  
Bessel Functions ◽  
Parabolic Cylinder ◽  
Whittaker Functions ◽  
Transform Method ◽  
Special Cases ◽  
Uniform Expansions ◽  
Uniform Asymptotic Expansions

The method for deriving Green-type asymptotic expansions from differential equations, introduced in I and illustrated therein by detailed calculations on modified Bessel functions, is applied to Whittaker functions W k,m , first for large k , and then for large |k 2 —m 2 |. Following the general theory of I, combination of this procedure with the Mellin transform method yields asymptotic expansions valid in transitional regions, and general uniform expansions. Weber parabolic cylinder and Poiseuille functions are examined as important special cases.

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 Distribution of LRC for testing sphericity of a complex multivariate Gaussian model

1985 ◽  
Vol 8 (3) ◽  
pp. 555-562 ◽  
Author(s):  
D. K. Nagar ◽  
S. K. Jain ◽  
A. K. Gupta
Keyword(s):  
Covariance Matrix ◽  
Hypergeometric Functions ◽  
Mellin Transform ◽  
Gaussian Model ◽  
Null Distribution ◽  
Multivariate Normal ◽  
Likelihood Ratio Criterion ◽  
Ratio Criterion ◽  
Special Cases ◽  
Multivariate Gaussian Model

In this paper, exact null distribution of the likelihood ratio criterion for testing sphericity structure in a complex multivariate normal covariance matrix is obtained in computable series form. The method of inverse Mellin transform and contour integration has been used. Certain special cases are given explicitly in terms of the hypergeometric functions.

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.

The Incremental Variational Principles for Frictional Contact Problems of Linear Elasticity

1993 ◽  
Vol 60 (4) ◽  
pp. 982-985 ◽  
Author(s):  
G. Zboinski
Keyword(s):  
Variational Principles ◽  
Contact Problems ◽  
Complementary Energy ◽  
Energy Functionals ◽  
Variational Functional ◽  
Elastic Bodies ◽  
Static Contact ◽  
Special Cases ◽  
The Common ◽  
Type Potential

Four types of the most frequently used variational functional are employed in order to form the inequality principles of the kineto-static contact problem of two elastic bodies in the common relative motion. As the general case, the principle based on the Hu- Washizu functional is proposed. The principles formed with the Reissner type, potential energy, and complementary energy functionals are derived as the special cases.

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.

Quadruple Integral Involving the Logarithm and Product of Bessel Functions Expressed in Terms of the Lerch Function

Axioms ◽  
2021 ◽  
Vol 10 (4) ◽  
pp. 324
Author(s):  
Robert Reynolds ◽  
Allan Stauffer
Keyword(s):  
Integral Representation ◽  
Bessel Functions ◽  
Fundamental Constants ◽  
Special Cases

In this paper, we have derived and evaluated a quadruple integral whose kernel involves the logarithm and product of Bessel functions of the first kind. A new quadruple integral representation of Catalan’s G and Apéry’s ζ(3) constants are produced. Some special cases of the result in terms of fundamental constants are evaluated. All the results in this work are new.

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