Related papers: Some exponential and trigonometric integrals invol…
We explore a well-known integral representation of the logarithmic function, and demonstrate its usefulness in obtaining compact, easily-computable exact formulas for quantities that involve expectations and higher moments of the logarithm…
From the integration of non-symmetrical hyperboles, a one-parameter generalization of the logarithmic function is obtained. Inverting this function, one obtains the generalized exponential function. We show that functions characterizing…
We show that the higher derivatives of the Riemann zeta function may be expressed in terms of integrals involving the digamma function. Related integrals for the Stieltjes constants are also shown. We also present a formula for the…
A solution is proposed for the problem of composition of ordinary generating functions. A new class of functions that provides a composition of ordinary generating functions is introduced; main theorems are presented; compositae are written…
In this note, we are interested in the *-version of various special functions. Noting that many special functions are defined by integrals involving the exponential functions, we define *-special functions by similar integral formula…
This paper investigates the generalized beta-logarithmic matrix function (GBLMF),which combines the extended beta matrix function and the logarithmic mean. The study establishes essential properties of this function, including functional…
Series containing the digamma function arise when calculating the parametric derivatives of the hypergeometric functions and play a role in evaluation of Feynman diagrams. As these series are typically non-hypergeometric, a few instances…
Motivated by existing results, we present some completely monotonic functions involving the polygamma functions.
We study three different $q$-analogues of the harmonic numbers. As applications, we present some generating functions involving number theoretical functions and give the $q$-generalization of Gosper's exponential generating function of…
In this note we derive some interesting definite integrals involving Malmsten logarithm forms, reciprocal logarithm forms and K\"{o}lbig type integrals in terms of special functions.
We present two new classes of orthogonal functions, log orthogonal functions (LOFs) and generalized log orthogonal functions (GLOFs), which are constructed by applying a $\log$ mapping to Laguerre polynomials. We develop basic approximation…
In the paper, the authors establish integral representations of some functions related to the remainder of Burnside's formula for the gamma function and find the (logarithmically) complete monotonicity of these and related functions. These…
We give closed-form expressions for the Dirichlet beta function at even positive integers and for the Dirichlet lambda function at odd positive integers, based on the function J(s) defined via convergent integral. We also show fundamental…
We present the evaluation of some definite integrals in the classical table by I. S. Gradshteyn and I. M. Ryzhik where the integrand is a combination of powers, exponentials and logarithms.
In this paper we study the a.e. exponential strong summability problem for the rectangular partial sums of double trigonometric Fourier series of the functions from $L\log L$ .
Relations among integrals of logarithms, polylogarithms and Euler sums are presented. A unifying element being the introduction of Nielsen's generalized polylogarithms.
The need to evaluate Logarithmic integrals is ubiquitous in essentially all quantitative areas including mathematical sciences, physical sciences. Some recent developments in Physics namely Feynman diagrams deals with the evaluation of…
The elliptic gamma function is a generalization of the Euler gamma function. Its trigonometric and rational degenerations are the Jackson q-gamma function and the Euler gamma function. We prove multiplication formulas for the elliptic gamma…
The focus of our investigation will be integrals of form $\int_0^1 \log^a(1-x) \log^b x \log^c(1+x) /f(x) dx$, where $f$ can be either $x,1-x$ or $1+x$. We show that these integrals possess a plethora of linear relations, and give…
Two distinct systems of commutative complex numbers in n dimensions are described, of polar and planar types. Exponential forms of n-complex numbers are given in each case, which depend on geometric variables. Azimuthal angles, which are…