scholarly journals Complete monotonicity, completely monotonic degree, integral representations, and an inequality related to the exponential, trigamma, and modified Bessel functions

2014 ◽  
Vol 2 (3) ◽  
Author(s):  
Feng Qi ◽  
Shu-Hong Wang
Keyword(s):  
Bessel Functions ◽  
Integral Representations ◽  
Complete Monotonicity ◽  
Modified Bessel Functions ◽  
Completely Monotonic ◽  
Completely Monotonic Degree

Integral Representations and Complete Monotonicity of Various Quotients of Bessel Functions

1977 ◽  
Vol 29 (6) ◽  
pp. 1198-1207 ◽  
Author(s):  
Mourad E. H. Ismail
Keyword(s):  
Laplace Transform ◽  
Bessel Functions ◽  
Asymptotic Methods ◽  
Laplace Transforms ◽  
Integral Representations ◽  
Complete Monotonicity ◽  
Vertical Line ◽  
Transcendental Functions ◽  
Complex Valued

Complete monotonicity of functions, Definition 3.1, is often proved by showing that their inverse Laplace transforms are nonnegative. There are relatively few simple functions whose inverse Laplace transforms can be expressed in terms of standard higher transcendental functions. Inverting a Laplace transform involves integrating a complex-valued function over a vertical line, and establishing the positivity of the resulting integral can be tricky. Sometimes asymptotic methods are helpful, see for example Fields and Ismail [6].

Special functions, infinite divisibility and transcendental equations

1979 ◽  
Vol 85 (3) ◽  
pp. 453-464 ◽  
Author(s):  
Mourad E. H. Ismail ◽  
C. Ping May
Keyword(s):  
Bessel Functions ◽  
Special Functions ◽  
Infinite Divisibility ◽  
Probability Density Functions ◽  
Integral Representations ◽  
Density Functions ◽  
Modified Bessel Functions ◽  
Explicit Formulas ◽  
Linear Combinations ◽  
Transcendental Equations

AbstractWe establish integral representations for quotients of Tricomi ψ functions and of several quotients of modified Bessel functions and of linear combinations of them. These integral representations are used to prove the complete monotonicity of the functions considered and to prove the infinite divisibility of a three parameter probability distribution. Limiting cases of this distribution are the hitting time distributions considered recently by Kent and Wendel. We also derive explicit formulas for the Kent–Wendel probability density functions.

scholarly journals Extended Mellin integral representations for the absolute value of the gamma function

Analysis ◽  
2018 ◽  
Vol 38 (1) ◽  
pp. 11-20
Author(s):  
Nicolas Privault
Keyword(s):  
Fourier Transform ◽  
Gamma Function ◽  
Bessel Functions ◽  
Planck Equation ◽  
Integral Representations ◽  
Modified Bessel Functions ◽  
Absolute Value ◽  
Fokker Planck Equation ◽  
The Absolute ◽  
Fokker Planck

AbstractWe derive Mellin integral representations in terms of Macdonald functions for the squared absolute value{s\mapsto|\Gamma(a+is)|^{2}}of the gamma function and its Fourier transform when{a<0}is non-integer, generalizing known results in the case{a>0}. This representation is based on a renormalization argument using modified Bessel functions of the second kind, and it applies to the representation of the solutions of a Fokker–Planck equation.

scholarly journals Integral Representations for Products of Two Bessel or Modified Bessel Functions

Mathematics ◽  
2019 ◽  
Vol 7 (10) ◽  
pp. 978
Author(s):  
Dragana Jankov Jankov Maširević ◽  
Tibor K. Pogány
Keyword(s):  
Bessel Functions ◽  
Integral Representations ◽  
Modified Bessel Functions ◽  
Similar Question

The first part of the article contains integral expressions for products of two Bessel functions of the first kind having either different integer orders or different arguments. A similar question for a product of modified Bessel functions of the first kind is solved next, when the input functions are of different integer orders and have different arguments.

Complete monotonicity of entropy production in chemical reaction

1981 ◽  
Vol 46 (2) ◽  
pp. 452-456
Author(s):  
Milan Šolc
Keyword(s):  
Entropy Production ◽  
Equilibrium Constant ◽  
Chemical Reaction ◽  
Relative Entropy ◽  
Order Reaction ◽  
Complete Monotonicity ◽  
First Order Reaction ◽  
First Order ◽  
Completely Monotonic ◽  
Derivatives Of

The successive time derivatives of relative entropy and entropy production for a system with a reversible first-order reaction alternate in sign. It is proved that the relative entropy for reactions with an equilibrium constant smaller than or equal to one is completely monotonic in the whole definition interval, and for reactions with an equilibrium constant larger than one this function is completely monotonic at the beginning of the reaction and near to equilibrium.

Asymptotic expansions and converging factors V. Lommel, Struve, modified Struve, Anger and Weber functions, and integrals of ordinary and modified Bessel functions

1959 ◽  
Vol 249 (1257) ◽  
pp. 284-292 ◽  
Keyword(s):  
Asymptotic Expansions ◽  
Bessel Functions ◽  
Modified Bessel Functions ◽  
Lommel Functions ◽  
Special Cases

A theory of Lommel functions is developed, based upon the methods described in the first four papers (I to IV) of this series for replacing the divergent parts of asymptotic expansions by easily calculable series involving one or other of the four ‘basic converging factors’ which were investigated and tabulated in I. This theory is then illustrated by application to the special cases of Struve, modified Struve, Anger and Weber functions, and integrals of ordinary and modified Bessel functions.

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