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15 pages, 301 KiB

Open AccessArticle

Definite Integral of Logarithmic Functions in Terms of the Incomplete Gamma Function

by Robert Reynolds and Allan Stauffer

Mathematics 2021, 9(13), 1506; https://doi.org/10.3390/math9131506 - 28 Jun 2021

Viewed by 1743

In this article we derive some entries and errata for the book of Gradshteyn and Ryzhik which were originally published by Bierens de Haan. We summarize our results using tables for easy reading and referencing. Full article

10 pages, 275 KiB

Open AccessArticle

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

by Robert Reynolds and Allan Stauffer

Mathematics 2020, 8(9), 1425; https://doi.org/10.3390/math8091425 - 25 Aug 2020

Viewed by 2148

Abstract

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

\int_{0}^{\infty} \frac{{(a + y)}^{k} - {(a - y)}^{k}}{e^{b y} - 1} d y

\int_{0}^{\infty} \frac{{(a + y)}^{k} - {(a - y)}^{k}}{e^{b y} + 1} d y

\int_{0}^{\infty} \frac{{(a + y)}^{k} - {(a - y)}^{k}}{sinh (b y)} d y

and

\int_{0}^{\infty} \frac{{(a + y)}^{k} + {(a - y)}^{k}}{cosh (b y)} d y

in terms of a special function where k, a and b are arbitrary complex numbers. Full article

6 pages, 237 KiB

Open AccessFeature PaperArticle

Derivation of Logarithmic and Logarithmic Hyperbolic Tangent Integrals Expressed in Terms of Special Functions

by Robert Reynolds and Allan Stauffer

Mathematics 2020, 8(5), 687; https://doi.org/10.3390/math8050687 - 1 May 2020

Cited by 8 | Viewed by 5716

Abstract

The derivation of integrals in the table of Gradshteyn and Ryzhik in terms of closed form solutions is always of interest. We evaluate several of these definite integrals of the form [...] Read more.

The derivation of integrals in the table of Gradshteyn and Ryzhik in terms of closed form solutions is always of interest. We evaluate several of these definite integrals of the form

\int_{0}^{\infty} log (1 \pm e^{- α y}) R (k, a, y) d y

in terms of a special function, where

R (k, a, y)

is a general function and k, a and

α

are arbitrary complex numbers, where

R e (α) > 0

. Full article

5 pages, 226 KiB

Open AccessArticle

A Definite Integral Involving the Logarithmic Function in Terms of the Lerch Function

by Robert Reynolds and Allan Stauffer

Mathematics 2019, 7(12), 1148; https://doi.org/10.3390/math7121148 - 24 Nov 2019

Cited by 8 | Viewed by 66338

Abstract

We present a method using contour integration to evaluate the definite integral of the form

\int_{0}^{\infty} {log}^{k} (a y) R (y) d y

in terms of special functions, where [...] Read more.

We present a method using contour integration to evaluate the definite integral of the form

\int_{0}^{\infty} {log}^{k} (a y) R (y) d y

in terms of special functions, where

R (y) = \frac{y^{m}}{1 + α y^{n}}

and

k, m, a, α

and n are arbitrary complex numbers. We use this method for evaluation as well as to derive some interesting related material and check entries in tables of integrals. Full article

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