scholarly journals The Mellin Transform of Logarithmic and Rational Quotient Function in terms of the Lerch Function

2021 ◽  
Vol 2021 ◽  
pp. 1-13
Author(s):  
Robert Reynolds ◽  
Allan Stauffer
Keyword(s):  
Current Literature ◽  
Mellin Transform ◽  
Integral Transforms ◽  
Complex Numbers ◽  
Definite Integrals ◽  
Famous Book ◽  
General Complex

Upon reading the famous book on integral transforms volume II by Erdeyli et al., we encounter a formula which we use to derive a Mellin transform given by ∫ 0 ∞ x m − 1 log k a x / β 2 + x 2 γ + x d x , where the parameters a , k , β , and γ are general complex numbers. This Mellin transform will be derived in terms of the Lerch function and is not listed in current literature to the best of our knowledge. We will use this transform to create a table of definite integrals which can be used to extend similar tables in current books featuring such formulae.

scholarly journals Definite Integral of Algebraic, Exponential and Hyperbolic Functions Expressed in Terms of Special Functions

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1425
Author(s):  
Robert Reynolds ◽  
Allan Stauffer
Keyword(s):  
Closed Form ◽  
Special Function ◽  
Special Functions ◽  
Complex Numbers ◽  
Definite Integral ◽  
Hyperbolic Functions ◽  
Closed Form Solutions ◽  
Definite Integrals ◽  
Arbitrary Complex ◽  
Famous Book

While browsing through the famous book of Bierens de Haan, we came across a table with some very interesting integrals. These integrals also appeared in the book of Gradshteyn and Ryzhik. Derivation of these integrals are not listed in the current literature to best of our knowledge. The derivation of such integrals in the book of Gradshteyn and Ryzhik in terms of closed form solutions is pertinent. We evaluate several of these definite integrals of the form ∫0∞(a+y)k−(a−y)keby−1dy, ∫0∞(a+y)k−(a−y)keby+1dy, ∫0∞(a+y)k−(a−y)ksinh(by)dy and ∫0∞(a+y)k+(a−y)kcosh(by)dy in terms of a special function where k, a and b are arbitrary complex numbers.

Integral transforms based upon fractional integration

1963 ◽  
Vol 59 (1) ◽  
pp. 63-71 ◽  
Author(s):  
Charles Fox
Keyword(s):  
Functional Equation ◽  
Mellin Transform ◽  
Fourier Transforms ◽  
Fractional Integration ◽  
Integral Transforms ◽  
Image Position

AbstractThe theory of Fourier transformscan be developed from the functional equation K(s) K(1 – s) = 1, where K(s) is the Mellin transform of the kernel k(x).In this paper I show that reciprocities can be obtained which are analogous to the Fourier transforms above but which develop from the much more general functional equationThe reciprocities are obtained by using fractional integration. In addition to the reciprocities we have analogues of the Parseval theorem and of the discontinuous integrals usually associated with Fourier transforms.In order to simplify the analysis I confine myself to the case n = 1 and to L2 space.

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 LOCAL RIEMANN HYPOTHESIS FOR COMPLEX NUMBERS

2009 ◽  
Vol 05 (07) ◽  
pp. 1221-1230 ◽  
Author(s):  
RIKARD OLOFSSON
Keyword(s):  
Riemann Hypothesis ◽  
Special Class ◽  
Mellin Transform ◽  
Natural Generalization ◽  
Hermite Functions ◽  
Complex Numbers

In this paper, a special class of local ζ-functions is studied. The main theorem states that the functions have all zeros on the line ℜ(s) = 1/2. This is a natural generalization of the result of Bump and Ng stating that the zeros of the Mellin transform of Hermite functions have ℜ(s) = 1/2.

scholarly journals Inequalities for some classical integral transforms

2016 ◽  
Vol 47 (3) ◽  
pp. 351-356
Author(s):  
Piyush Kumar Bhandari ◽  
Sushil Kumar Bissu
Keyword(s):  
Mellin Transform ◽  
Integral Transforms ◽  
Hankel Transform ◽  
Schwarz Inequality ◽  
Fourier Cosine ◽  
Sine Transform ◽  
Fourier Sine Transform ◽  
Fourier Cosine Transform ◽  
Classical Integral ◽  
New Inequalities

By using a form of the Cauchy-Bunyakovsky-Schwarz inequality, we establish new inequalities for some classical integral transforms such as Laplace transform,Fourier transform, Fourier cosine transform, Fourier sine transform, Mellin transform and Hankel transform.

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 New Inversion Techniques for Some Integral Transforms via the Generalized Product Theorem of the Mellin Transform

2012 ◽  
Vol 2012 ◽  
pp. 1-10 ◽  
Author(s):  
Alireza Ansari ◽  
Mohammadreza Ahmadi Darani
Keyword(s):  
Integral Equations ◽  
Singular Integral ◽  
Mellin Transform ◽  
Singular Integral Equations ◽  
Integral Transforms ◽  
Product Theorem ◽  
Inversion Techniques

We introduce the generalized product theorem for the Mellin transform, and we solve certain classes of singular integral equations with kernels coincided with conditions of this theorem. Also, new inversion techniques for the Wright, Mittag-Leffler, Stieltjes, and Widder potential transforms are obtained.

scholarly journals On certain methods of solving a class of integral equations of Fredholm type

1992 ◽  
Vol 52 (1) ◽  
pp. 1-10 ◽  
Author(s):  
H. M. Srivastava ◽  
R. K. Raina
Keyword(s):  
Integral Equation ◽  
Integral Equations ◽  
Mellin Transform ◽  
Integral Transforms ◽  
Fredholm Type ◽  
Reduction Techniques ◽  
Hypergeometric Integral ◽  
Type Integral ◽  
Interesting Generalization

AbstractThe authors begin by presenting a brief survey of the various useful methods of solving certain integral equations of Fredholm type. In particular, they apply the reduction techniques with a view to inverting a class of generalized hypergeometric integral transforms. This is observed to lead to an interesting generalization of the work of E. R. Love [9]. The Mellin transform technique for solving a general Fredholm type integral equation with the familiar H-function in the kernel is also considered.

scholarly journals An extension of the method of brackets. Part 2

2020 ◽  
Vol 18 (1) ◽  
pp. 983-995
Author(s):  
Ivan Gonzalez ◽  
Lin Jiu ◽  
Victor H. Moll
Keyword(s):  
Mellin Transform ◽  
Feynman Diagrams ◽  
Half Line ◽  
Definite Integrals ◽  
Small Set

Abstract The method of brackets, developed in the context of evaluation of integrals coming from Feynman diagrams, is a procedure to evaluate definite integrals over the half-line. This method consists of a small number of operational rules devoted to convert the integral into a bracket series. A second small set of rules evaluates this bracket series and produces the result as a regular series. The work presented here combines this method with the classical Mellin transform to extend the class of integrands where the method of brackets can be applied. A selected number of examples are used to illustrate this procedure.

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