Abstract
The interconnection between the Liouville–Weyl fractional integral and the Lambert function is studied. The class of modified Abel equations of the first kind is solved. A new integral formula for the Gamma function and possibly new transform pairs for the Laplace and Mellin transform have been found. <br />
Highlights
For a study of the interconnection between fractional integrals [1] and the Lambert function [2], we start with the following variant of the fractional integral, known as the Liouville, Liouville–Weyl or Weyl fractional integral [1]: (I−ν F )(y) = 1 Γ(ν) ∞F (x)(x − y)ν−1 dx = y Γ(ν)F (u + y)uν−1 du, (1.1)where ν > 0 and y > −∞
The integral on the right of (1.1) is the Mellin transform of the shifted function F (x). This means that the Mellin transform of function F (·) is its Liouville–Weyl fractional integral of the order ν at y = 0 times the Gamma function of argument ν
Theorem 2.2 means that the Liouville–Weyl diagonal fractional integral of the unilateral Laplace transform can be interpreted as a two-sided Laplace transform with the region of convergence β0 < y < β1
Summary
For a study of the interconnection between fractional integrals [1] and the Lambert function [2], we start with the following variant of the fractional integral, known as the Liouville, Liouville–Weyl or Weyl fractional integral [1]:. A completely monotone function is defined as follows [6, Def. 1.3, p. Because function f (x) is nonnegative by hypothesis, the Liouville–Weyl fractional integral defined by (1.3) is the Laplace transform of a nonnegative function. (1.) the Liouville–Weyl fractional integral of function F (·) is a completely monotone function;. This implies that function f (t) in (1.2) is nonnegative This means that function F (·) is completely monotone according to the Bernstein theorem. Monotone functions play a substantial role in probability theory, measure theory, etc. They can be found in technological practice - we mention time-dependent shear, bulk and the Young moduli of linear viscoelastic materials, because they are Laplace transforms of the relevant nonnegative relaxation spectra [7]
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
