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Register a new userThe change of variable formula for Riemann-Stieltjes integrals
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Authors
Alberto Torchinsky
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We consider general formulations of the change of variable formula for the Riemann-Stieltjes integral, including the case when the substitution is not invertible.
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This note concerns the general formulation by Preiss and Uher of Kestelmans influential result pertaining the change of variable, or substitution, formula for the Riemann integral.
In this paper, motivated by physical considerations, we introduce the notion of modified Riemann sums of Riemann-Stieltjes integrable functions, show that they converge, and compute them explicitely under various assumptions.
We generalize Warnaars elliptic extension of a Macdonald multiparameter summation formula to Riemann surfaces of arbitrary genus.
Several approaches to the formulation of a fractional theory of calculus of variable order have appeared in the literature over the years. Unfortunately, most of these proposals lack a rigorous mathematical framework. We consider an alternative view on the problem, originally proposed by G. Scarpi in the early seventies, based on a naive modification of the representation in the Laplace domain of standard kernels functions involved in (constant-order) fractional calculus. We frame Scarpis ideas within recent theory of General Fractional Derivatives and Integrals, that mostly rely on the Sonine condition, and investigate the main properties of the emerging variable-order operators. Then, taking advantage of powerful and easy-to-use numerical methods for the inversion of Laplace transforms of functions defined in the Laplace domain, we discuss some practical applications of the variable-order Scarpi integral and derivative.
We consider two integrals over $xin [0,1]$ involving products of the function $zeta_1(a,x)equiv zeta(a,x)-x^{-a}$, where $zeta(a,x)$ is the Hurwitz zeta function, given by $$int_0^1zeta_1(a,x)zeta_1(b,x),dxquadmbox{and}quad int_0^1zeta_1(a,x)zeta_1(b,1-x),dx$$ when $Re (a,b)>1$. These integrals have been investigated recently in cite{SCP}; here we provide an alternative derivation by application of Feynman parametrization. We also discuss a moment integral and the evaluation of two doubly infinite sums containing the Riemann zeta function $zeta(x)$ and two free parameters $a$ and $b$. The limiting forms of these sums when $a+b$ takes on integer values are considered.
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