[8] viXra:1712.0539 [pdf] submitted on 2017-12-20 06:47:39
Authors: Martin Nicholson
Comments: 8 Pages.
We study several integrals that contain the infinite product ${\displaystyle\prod_{n=0}^\infty}\left[1+\left(\frac{x}{b+n}\right)^3\right]$ in the denominator of their integrand. These considerations lead to closed form evaluation $\displaystyle\int_{-\infty}^\infty\frac{dx}{\left(e^x+e^{-x}+e^{ix\sqrt{3}}\right)^2}=\frac{1}{3}$ and to some other formulas.
Category: Functions and Analysis
[7] viXra:1712.0519 [pdf] submitted on 2017-12-19 19:49:52
Authors: Seong Won Cha
Comments: 11 Pages.
We show that some interesting properties of the bilateral Laplace transform of even and positive functions both on the line z=x+iy0 and on a circle. We also show the Riemann hypothesis is true using these properties.
Category: Functions and Analysis
[6] viXra:1712.0486 [pdf] replaced on 2019-12-28 14:52:22
Authors: Martin Nicholson
Comments: 12 Pages. 6 new theorems added, presentation and notation improved, more details added in the abstract, typo corrected
Several infinite products are studied that satisfy the transformation relation of the type $f(\alpha)=f(1/\alpha)$. For certain values of the parameters these infinite products reduce to modular forms. Finite counterparts of these infinite products are motivated by solution of Dirichlet boundary problem on a rectangular grid. These finite product formulas give an elementary proof of several modular transformations.
Category: Functions and Analysis
[5] viXra:1712.0478 [pdf] replaced on 2020-01-13 12:11:22
Authors: Martin Nicholson
Comments: 12 Pages. 2 references and a new section is adde, presentation and notation improved, typos corrected, abstract rewritten to better reflect the contents of the paper.
Several Fourier transformations of functions of one and two variables are evaluated andthen used to derive some integral and series identities. It is shown that certain double Mordellintegrals can be reduced to a sum of products of one-dimensional Mordell integrals. As aconsequence of this reduction, a quadratic polynomial identity is found connecting productsof certain one-dimensional Mordell integrals. An integral that depends on one real valuedparameter is calculated reminiscent of an integral previously calculated by Ramanujan andGlasser. Some connections to elliptic functions and lattice sums are discussed.
Category: Functions and Analysis
[4] viXra:1712.0463 [pdf] submitted on 2017-12-16 01:01:52
Authors: Carl Wigert, Quincy-Howard Xavier
Comments: 1 Page.
In this paper, we define very small numbers and very very small numbers and use them to construct derivatives as ratios of real numbers. We then use that result to rigorously prove that the chain rule treats derivatives as fractions being multiplied.
Category: Functions and Analysis
[3] viXra:1712.0355 [pdf] submitted on 2017-12-08 19:58:12
Authors: Seong Won Cha
Comments: 9 Pages.
This is a brief report before writing a full paper.
We proved the Riemann hypothesis using the properties of the bilateral Laplace transform.
Category: Functions and Analysis
[2] viXra:1712.0113 [pdf] submitted on 2017-12-04 21:50:14
Authors: D Williams
Comments: 8 Pages.
An overview of some types of multiplicative infinitesimal calculi is given. Analogs of standard results ("Simpson's" Product, "Maclurin's" Product, fundamental theorems, etc) are shown. An area that deserves more attention.
Category: Functions and Analysis
[1] viXra:1712.0019 [pdf] submitted on 2017-12-02 12:52:22
Authors: Antoine Balan
Comments: 2 pages, written in french
It is showed that a large class of functions defined by integrals verify the Riemann Hypothesis.
Category: Functions and Analysis