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 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 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 Definite Integrals Involving Combinations of Powers and Logarithmic Functions of Complicated Arguments Expressed in Terms of the Hurwitz Zeta Function

2021 ◽  
Vol 2021 ◽  
pp. 1-6
Author(s):  
Robert Reynolds ◽  
Allan Stauffer
Keyword(s):  
Zeta Function ◽  
Special Functions ◽  
Zeta Functions ◽  
Hurwitz Zeta Function ◽  
Fundamental Constants ◽  
Closed Formula ◽  
Definite Integrals ◽  
Logarithmic Functions

In this manuscript, the authors derive closed formula for definite integrals of combinations of powers and logarithmic functions of complicated arguments and express these integrals in terms of the Hurwitz zeta functions. These derivations are then expressed 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 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 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.

Matrix representation of formal polynomials over max-plus algebra

2020 ◽  
pp. 2150216
Author(s):  
Cailu Wang ◽  
Yuegang Tao
Keyword(s):  
Matrix Method ◽  
Polynomial Algorithm ◽  
Polynomial Function ◽  
Matrix Representation ◽  
Sufficient Condition ◽  
Polynomial Functions ◽  
Canonical Forms ◽  
Necessary And Sufficient Condition ◽  
The Matrix ◽  
Necessary And Sufficient

This paper proposes the matrix representation of formal polynomials over max-plus algebra and obtains the maximum and minimum canonical forms of a polynomial function by standardizing this representation into a canonical form. A necessary and sufficient condition for two formal polynomials corresponding to the same polynomial function is derived. Such a matrix method is constructive and intuitive, and leads to a polynomial algorithm for factorization of polynomial functions. Some illustrative examples are presented to demonstrate the results.

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.

scholarly journals A note on local polynomial functions on commutative semigroups

1982 ◽  
Vol 33 (1) ◽  
pp. 50-53
Author(s):  
P. A. Grossman
Keyword(s):  
Positive Integer ◽  
Universal Algebra ◽  
Polynomial Function ◽  
Polynomial Functions ◽  
Commutative Semigroups ◽  
Local Polynomial ◽  
A Chain

AbstractGiven a universal algebra A, one can define for each positive integer n the set of functions on A which can be “interpolated” at any n elements of A by a polynomial function on A. These sets form a chain with respect to inclusion. It is known for several varieties that many of these sets coincide for all algebras A in the variety. We show here that, in contrast with these results, the coincident sets in the chain can to a large extent be specified arbitrarily by suitably choosing A from the variety of commutative semigroups.

scholarly journals The Extrapolar SWIFT model (version 1.0): fast stratospheric ozone chemistry for global climate models

2018 ◽  
Vol 11 (2) ◽  
pp. 753-769 ◽  
Author(s):  
Daniel Kreyling ◽  
Ingo Wohltmann ◽  
Ralph Lehmann ◽  
Markus Rex
Keyword(s):  
Polynomial Function ◽  
Stratospheric Ozone ◽  
Ozone Layer ◽  
Rate Of Change ◽  
Polynomial Functions ◽  
Mixing Ratio ◽  
Fourth Degree ◽  
Stratospheric Chemistry ◽  
Wide Range ◽  
Atlas Model

Abstract. The Extrapolar SWIFT model is a fast ozone chemistry scheme for interactive calculation of the extrapolar stratospheric ozone layer in coupled general circulation models (GCMs). In contrast to the widely used prescribed ozone, the SWIFT ozone layer interacts with the model dynamics and can respond to atmospheric variability or climatological trends. The Extrapolar SWIFT model employs a repro-modelling approach, in which algebraic functions are used to approximate the numerical output of a full stratospheric chemistry and transport model (ATLAS). The full model solves a coupled chemical differential equation system with 55 initial and boundary conditions (mixing ratio of various chemical species and atmospheric parameters). Hence the rate of change of ozone over 24 h is a function of 55 variables. Using covariances between these variables, we can find linear combinations in order to reduce the parameter space to the following nine basic variables: latitude, pressure altitude, temperature, overhead ozone column and the mixing ratio of ozone and of the ozone-depleting families (Cly, Bry, NOy and HOy). We will show that these nine variables are sufficient to characterize the rate of change of ozone. An automated procedure fits a polynomial function of fourth degree to the rate of change of ozone obtained from several simulations with the ATLAS model. One polynomial function is determined per month, which yields the rate of change of ozone over 24 h. A key aspect for the robustness of the Extrapolar SWIFT model is to include a wide range of stratospheric variability in the numerical output of the ATLAS model, also covering atmospheric states that will occur in a future climate (e.g. temperature and meridional circulation changes or reduction of stratospheric chlorine loading). For validation purposes, the Extrapolar SWIFT model has been integrated into the ATLAS model, replacing the full stratospheric chemistry scheme. Simulations with SWIFT in ATLAS have proven that the systematic error is small and does not accumulate during the course of a simulation. In the context of a 10-year simulation, the ozone layer simulated by SWIFT shows a stable annual cycle, with inter-annual variations comparable to the ATLAS model. The application of Extrapolar SWIFT requires the evaluation of polynomial functions with 30–100 terms. Computers can currently calculate such polynomial functions at thousands of model grid points in seconds. SWIFT provides the desired numerical efficiency and computes the ozone layer 104 times faster than the chemistry scheme in the ATLAS CTM.

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