Related papers: A New Integral Equation and Some Integrals Associa…
We offer a solution to a functional equation using properties of the Mellin transform. A new criteria for the Riemann Hypothesis is offered as an application of our main result, through a functional relationship with the Riemann xi…
In this paper we comment the Post inversion formula for Laplace transform, and its possible application to the branch of Analytic Number theory (Arithmetical functions, RH and PNT), involving a condition in the form of iterated limit to…
An analog of the Riemann hypothesis is proved in this paper. Some new integral equations for the functions $\pi(x)$ and $R(x)$ follows. A new effect that is shown is that these function - with essentially different behavior - are the…
An elementary analytic proof of the famous Riemann hypothesis is given. The main "accent" of the proof is a both using of the 2-dimensional double real and complex Laplace integral representations of the Green function $\mid z \mid^{-2}$.
An important class of fractional differential and integral operators is given by the theory of fractional calculus with respect to functions, sometimes called $\Psi$-fractional calculus. The operational calculus approach has proved useful…
We introduce the linear operators of fractional integration and fractional differentiation in the framework of the Riemann-Liouville fractional calculus. Particular attention is devoted to the technique of Laplace transforms for treating…
We use the Laplace transform and the Gamma function to introduce a new integral transform and name it the Laplace-type transform possessing the property of mapping a function to a functional sequence, which cannot be achieved by the Laplace…
A new parametric integral is obtained as a consequence of the Riemann hypothesis. An asymptotic multiplicability is the main property of this integral.
New numerical algorithms based on rational functions are introduced that can solve certain Laplace and Helmholtz problems on two-dimensional domains with corners faster and more accurately than the standard methods of finite elements and…
We present a method derived from Laplace transform theory that enables the evaluation of fractional integrals. This method is adapted and extended in a variety of ways to demonstrate its utility in deriving alternative representations for…
We generalize the harmonic continuation of the Riemann xi-function to the $n$-dimension case, to obtain the solution to the Dirichlet problem on $\mathbb{R}_{+}^{n+1}.$ We also provide a new expansion for the harmonic continuation of the…
In this paper, we solve Laplace equation analytically by using differential transform method. For this purpose, we consider four models with two Dirichlet and two Neumann boundary conditions and obtain the corresponding exact solutions. The…
A class of Laplace transforms is examined to show that particular cases of this class are associated with production-destruction and reaction-diffusion problems in physics, study of differences of independently distributed random variables…
We first strictly expressed the basic notions and research methods of abstract operators, which systematically expounded the main results of abstract operator theory. By combining abstract operators with the Laplace transform, we can easily…
In this paper, we achieve a Reilly type integral formula associated with the $\phi$-Laplacian. As its applications, we obtain Heintze-Karcher and Minkowski type inequalities. Furthermore, almost Schur lemmas are also given. They recover the…
The Mellin transform and several Dirichlet series related with the Riemann zeta function are used to deduce some identities similar to the classical M\"untz formula [4]. These formulas are derived in the critical strip and in the half-plane…
In this paper, we begin by applying the Laplace transform to derive closed forms for several challenging integrals that seem nearly impossible to evaluate. By utilizing the solution to the Pythagorean equation $a^2 + b^2 = c^2$, these…
We consider some properties of integrals considered by Hardy and Koshliakov, and which have also been further extended recently by Dixit. We establish a new general integral formula from some observations about the digamma function. We also…
While the definition of a fractional integral may be codified by Riemann and Liouville, an agreed-upon fractional derivative has eluded discovery for many years. This is likely a result of integral definitions including numerous constants…
In this note it is worked out a new set of Laplace-Like equations for quaternions through Riemann-Cauchy hypercomplex relations otained earlier \cite{BorgesZeMarcio}. As in the theory of functions of a complex variable, it is expected that…