Journal ArticleDOI
Integral representation of some functions related to the Gamma function
TLDR
In this paper, it was shown that the functions Θ(Θ(x) = [Gamma (x + 1)]^{1/x} (1 + 1/x)^x /x) and Θ (Θ (x, Θ) = Θ((Θ + 1)/x)) are Stieltjes transforms.Citations
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Four logarithmically completely monotonic functions involving gamma function
TL;DR: In this article, two classes of functions, involving a parameter and the classical Euler gamma function, were verified to be logarithmically completely monotonic in or ; some inequalities involving the Euler Gamma function were deduced and compared with those originating from certain problems of traffic flow, relating to the well known Stirling's formula.
Journal ArticleDOI
Complete monotonicity of a function involving the divided difference of psi functions
TL;DR: In this paper, necessary and sufficient conditions are presented for a function involving the divided difference of the psi function to be completely monotonic and for the ratio of two gamma functions to be logarithmically complete.
Journal ArticleDOI
Complete Monotonicity of Functions Connected with the Exponential Function and Derivatives
Chun-Fu Wei,Bai-Ni Guo +1 more
TL;DR: In this paper, complete monotonicity results concerning the polylogarithm of the functions are discovered. But they are not complete monotonic in the sense that the ratios between consecutive derivatives of these two functions are decreasing on the same order.
Journal ArticleDOI
Refinements, Extensions and Generalizations of the Second Kershaw's Double Inequality
TL;DR: In this article, the second Kershaw double inequality concerning the γρερδεratio of two gamma functions is refined, extended, and generalized elegantly.
Journal ArticleDOI
Function Spaces as Dirichlet Spaces (About a Paper by Maz'ya and Nagel)
Niels Jacob,René L. Schilling +1 more
TL;DR: In this paper, it was shown that these Sobolev norms are equivalent to Dirichlet norms, and they were extended to a wider class of anisotropic norms which arise in the theory of Markov processes.
References
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Book
Table of Integrals, Series, and Products
TL;DR: Combinations involving trigonometric and hyperbolic functions and power 5 Indefinite Integrals of Special Functions 6 Definite Integral Integral Functions 7.Associated Legendre Functions 8 Special Functions 9 Hypergeometric Functions 10 Vector Field Theory 11 Algebraic Inequalities 12 Integral Inequality 13 Matrices and related results 14 Determinants 15 Norms 16 Ordinary differential equations 17 Fourier, Laplace, and Mellin Transforms 18 The z-transform
A table of integrals
TL;DR: Basic Forms x n dx = 1 n + 1 x n+1 (1) 1 x dx = ln |x| (2) udv = uv − vdu (3) 1 ax + bdx = 1 a ln|ax + b| (4) Integrals of Rational Functions