Related papers: An Alternate proof for a case of a Malmsten integr…
This paper discusses generalizations of logarithmic and hyperbolic integrals. A new proof for an integral presented by Vardi and several other integrals in relation to known mathematical constants are discovered. We introduce the signed…
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…
This paper uses the convolution theorem of the Laplace transform to derive new inverse Laplace transforms for the product of two parabolic cylinder functions in which the arguments may have opposite sign. These transforms are subsequently…
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…
The inversion of nabla Laplace transform, corresponding to a causal sequence, is considered. Two classical methods, i.e., residual calculation method and partial fraction method are developed to perform the inverse nabla Laplace transform.…
When introduced in a 2018 article in the American Mathematical Monthly, the omega integral was shown to be an extension of the Riemann integral. Although results for continuous functions such as the Fundamental Theorem of Calculus follow…
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…
We utilize a combination of integral transforms, including the Laplace transform, with some classical results in analytic number theory concerning the Riemann $\xi$-function, to obtain a new integral equation. We also provide a new proof of…
To test kinetic theories, simple and practical setups are proposed. It turns out that these setups cannot be treated by Boltzmann's equation. An alternative method, called the path-integral approach, is then employed and a number of…
A direct generalization of Laplace invariants to the case of hyperbolic partial differential systems is considered. The proof of the following statement is given: the determinant of a Laplace invariant vanishes if the corresponding system…
A new simple proof of Stirling's formula via the partial fraction expansion for the tangent function is presented.
We give an alternative proof of a sharp generalization of an integral inequality for the dyadic maximal operator due to which the evaluation of the Bellman function of this operator with respect to two variables, is possible. This last…
This paper is about a method for solving infinite series in closed form by using inverse and forward Laplace transforms. The resulting integral is to be solved instead. The method is extended by parametrizing the series. A further Laplace…
The Lamb-Bateman integral equation was introduced to study the solitary wave diffraction and its solution was written in terms of an integral transform. We prove that it is essentially the Abel integral equation and its solution can be…
The Laplace transform of $|\zeta(1/2+it)|$ is investigated, for which a precise expression is obtained, valid in a certain region in the complex plane. The method of proof is based on complex integration and spectral theory of the…
We present a proof of the functional equation of the Riemann zeta-function or more precisely the Dirichlet eta-function, which proof seems to be new but follows almost immediately from Malmst\`en's paper ``De integralibus quibusdam…
This note presents a new equivalence to the Riemann Hypothesis by means of the Salem integral equation.
This paper considers various integrals where the integrand includes the log gamma function (or its derivative, the digamma function) multiplied by a trigonometric or hyperbolic function. Some apparently new integrals and series are…
We discuss alternative iteration methods for differential equations. We provide a convergence proof for exactly solvable examples and show more convenient formulas for nontrivial problems.
In this article, we propose an integral expression of the Catalan numbers, based on Malmst\'en's definite-integral representation of $\ln\left[\Gamma(x)\right]$, $\Gamma$ being the usual Gamma function. The obtained expression is likely to…