scholarly journals A Note on Some Definite Integrals of Arthur Erdélyi and George Watson

Mathematics ◽  
2021 ◽  
Vol 9 (6) ◽  
pp. 674
Author(s):  
Robert Reynolds ◽  
Allan Stauffer
Keyword(s):  
Analytic Continuation ◽  
Current Literature ◽  
Integral Method ◽  
Special Functions ◽  
Contour Integral ◽  
Complex Conjugate ◽  
Definite Integrals ◽  
Special Cases ◽  
Bose Einstein ◽  
Fourier Kernel

This manuscript concerns two definite integrals that could be connected to the Bose-Einstein and the Fermi-Dirac functions in the integrands, separately, with numerators slightly modified with a difference in two expressions that contain the Fourier kernel multiplied by a polynomial and its complex conjugate. In this work, we use our contour integral method to derive these definite integrals, which are given by ∫0∞ie−imx(log(a)−ix)k−eimx(log(a)+ix)k2eαx−1dx and ∫0∞ie−imx(log(a)−ix)k−eimx(log(a)+ix)k2eαx+1dx in terms of the Lerch function. We use these two definite integrals to derive formulae by Erdéyli and Watson. We derive special cases of these integrals in terms of special functions not found in current literature. Special functions have the property of analytic continuation, which widens the range of computation of the variables involved.

scholarly journals A Note on a Fourier Sine Transform

Symmetry ◽  
2021 ◽  
Vol 13 (10) ◽  
pp. 1828
Author(s):  
Robert Reynolds ◽  
Allan Stauffer
Keyword(s):  
Integral Method ◽  
Contour Integral ◽  
Summary Table ◽  
Zero Distribution ◽  
Definite Integrals ◽  
Sine Transform ◽  
Fourier Sine ◽  
Fourier Sine Transform ◽  
General Power ◽  
Almost All

This is a compilation of definite integrals of the product of the hyperbolic cosecant function and polynomial raised to a general power. In this work, we used our contour integral method to derive a Fourier sine transform in terms of the Lerch function. Almost all Lerch functions have an asymmetrical zero-distribution. A summary table of the results are produced for easy reading. A vast majority of the results 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 Infinite Sum of the Incomplete Gamma Function Expressed in Terms of the Hurwitz Zeta Function

Mathematics ◽  
2021 ◽  
Vol 9 (16) ◽  
pp. 1952
Author(s):  
Robert Reynolds ◽  
Allan Stauffer
Keyword(s):  
Zeta Function ◽  
Gamma Function ◽  
Integral Method ◽  
Special Functions ◽  
Incomplete Gamma Function ◽  
Hurwitz Zeta Function ◽  
Contour Integral ◽  
Fundamental Constants ◽  
Infinite Sum

We apply our simultaneous contour integral method to an infinite sum in Prudnikov et al. and use it to derive the infinite sum of the Incomplete gamma function in terms of the Hurwitz zeta function. We then evaluate this formula to derive new series 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.

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