Journal ArticleDOI
Integral representation of some functions related to the Gamma function
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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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Logarithmically complete monotonicity of ratios of q-gamma functions
Jing-Feng Tian,Zhenhang Yang +1 more
TL;DR: In this article, the necessary and sufficient conditions for the ratio (W q ; u, v / W q ; r, s ) to be logarithmically completely monotonic on ( − ρ, ∞ ), where ρ = min { u, v, r, s }.
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Complete monotonicity of a function involving a ratio of gamma functions and applications
Feng Qi,Bai-Ni Guo +1 more
TL;DR: In this article, necessary and sufficient conditions are presented for a function involving a ratio of gamma functions to be logarithmically completely monotonic, and several inequalities involving the volume of the unit ball in R n are derived, which refine, generalize and extend some known inequalities.
Journal ArticleDOI
Logarithmically completely monotonic rational functions
TL;DR: In this article , the class of logarithmically completely monotonic (LCM) functions is studied and conditions are proposed to ensure a rational function is LCM, a result that enables the known space of linear continuous-time externally positive systems to be enlarged and an efficient and optimal pole-placement procedure for the tracking controller synthesis problem to be developed.
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A Brief Overview and Survey of the Scientific Work by Feng Qi
TL;DR: In the paper, the authors present a brief overview and survey of the scientific work by Chinese mathematician Feng Qi and his coauthors.
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