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

Mathematics ◽  
2019 ◽  
Vol 7 (12) ◽  
pp. 1148 ◽  
Author(s):  
Robert Reynolds ◽  
Allan Stauffer
Keyword(s):  
Special Functions ◽  
Contour Integration ◽  
Complex Numbers ◽  
Definite Integral ◽  
Logarithmic Function ◽  
Related Material ◽  
Arbitrary Complex

We present a method using contour integration to evaluate the definite integral of the form ∫ 0 ∞ log k ( a y ) R ( y ) d y in terms of special functions, where R ( y ) = 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.

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 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 Definite Integral of Arctangent and Polylogarithmic Functions Expressed as a Series

Mathematics ◽  
2019 ◽  
Vol 7 (11) ◽  
pp. 1099 ◽  
Author(s):  
Robert Reynolds ◽  
Allan Stauffer
Keyword(s):  
Special Functions ◽  
Contour Integration ◽  
Definite Integral ◽  
Catalan’S Constant ◽  
Infinite Sums ◽  
Mathematical Constants

We present a method using contour integration to evaluate the definite integral of arctangent reciprocal logarithmic integrals in terms of infinite sums. In a similar manner, we evaluate the definite integral involving the polylogarithmic function L i k ( y ) in terms of special functions. In various cases, these generalizations give the value of known mathematical constants such as Catalan’s constant G, Aprey’s constant ζ ( 3 ) , the Glaisher–Kinkelin constant A, l o g ( 2 ) , and π .

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

Mathematics ◽  
2020 ◽  
Vol 8 (5) ◽  
pp. 687 ◽  
Author(s):  
Robert Reynolds ◽  
Allan Stauffer
Keyword(s):  
Closed Form ◽  
Special Function ◽  
Special Functions ◽  
Complex Numbers ◽  
General Function ◽  
Hyperbolic Tangent ◽  
Closed Form Solutions ◽  
Definite Integrals ◽  
Arbitrary Complex

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 ∫ 0 ∞ log ( 1 ± 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 .

On methods of summability based on integral functions

1959 ◽  
Vol 55 (1) ◽  
pp. 23-30 ◽  
Author(s):  
D. Borwein
Keyword(s):  
Integral Function ◽  
Radius Of Convergence ◽  
Complex Numbers ◽  
Arbitrary Complex ◽  
Image Position

Suppose throughout thatand thatis an integral function. Suppose also that l, sn(n = 0,1,…) are arbitrary complex numbers and denote by ρ(ps) the radius of convergence of the series

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 Integral of Power and Algebraic Functions in terms of the Lerch Function

2021 ◽  
Vol 14 (3) ◽  
pp. 980-988
Author(s):  
Robert Reynolds ◽  
Allan Stauffer
Keyword(s):  
Present Note ◽  
Special Functions ◽  
Integral Formula ◽  
Definite Integral ◽  
Algebraic Functions ◽  
Definite Integrals ◽  
Cauchy’S Integral Formula ◽  
Logarithmic Integral

Bierens de haan (1867) evaluated a definite integral involving the cotangent function and this result was also listed in Gradshteyn and Ryzhik (2007). The objective of this present note is to use this integral along with Cauchy's integral formula to derive a definite logarithmic integral in terms of the Lerch function. We will use this integral formula to produce a table of known and new results in terms of special functions and thereby expanding the list of definite integrals in both text books.

Tauberian Theorems for Borel-Type Methods of Summability

1974 ◽  
Vol 17 (2) ◽  
pp. 167-173 ◽  
Author(s):  
D. Borwein ◽  
E. Smet
Keyword(s):  
Complex Variable ◽  
Negative Integer ◽  
Complex Numbers ◽  
Tauberian Theorems ◽  
Arbitrary Complex ◽  
Principal Value ◽  
Image Position ◽  
Borel Type

Suppose throughout that s, an (n=0,1, 2,…) are arbitrary complex numbers, that α>0 and β is real and that N is a non-negative integer such that αN+β≧l. Letwhere z=x+iy is a complex variable and the power zr is assumed to have its principal value.

Spectral Problem for a Polynomial Pencil of the Sturm-Liouville Equations

2020 ◽  
pp. 163-181
Author(s):  
Anar Adiloğlu-Nabiev
Keyword(s):  
Differential Equation ◽  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Spectral Problem ◽  
Boundary Value ◽  
Asymptotic Formulas ◽  
Complex Numbers ◽  
Arbitrary Complex ◽  
Sturm Liouville ◽  
Complex Valued

A boundary value problem for the second order differential equation -y′′+∑_{m=0}N−1λ^{m}q_{m}(x)y=λ2Ny with two boundary conditions a_{i1}y(0)+a_{i2}y′(0)+a_{i3}y(π)+a_{i4}y′(π)=0, i=1,2 is considered. Here n>1, λ is a complex parameter, q0(x),q1(x),...,q_{n-1}(x) are summable complex-valued functions, a_{ik} (i=1,2; k=1,2,3,4) are arbitrary complex numbers. It is proved that the system of eigenfunctions and associated eigenfunctions is complete in the space and using elementary asymptotical metods asymptotic formulas for the eigenvalues are obtained.

Sign in / Sign up

Export Citation Format

Share Document