Functions and Analysis

1712 Submissions

[7] viXra:1712.0539 [pdf] submitted on 2017-12-20 06:47:39

Integrals Containing the Infinite Product $\prod_{n=0}^\infty\left[1+\left(\frac{x}{b+n}\right)^3\right]$

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

[6] viXra:1712.0519 [pdf] submitted on 2017-12-19 19:49:52

The Bilateral Laplace Transform of the Positive Even Functions and a Proof of Riemann Hypothesis

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

[5] viXra:1712.0478 [pdf] submitted on 2017-12-15 08:30:49

Two-Dimensional Fourier Transformations and Mordell Integrals

Authors: Martin Nicholson
Comments: 10 Pages.

Several Fourier transformations of functions of one and two variables are evaluated and then used to derive some integral and series identities. It is shown that certain two- dimensional Mordell integrals factorize into product of two integrals and that the square of the absolute value of the Mordell integral can be reduced to a single one-dimensional integral. 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

Proof that a Derivative is a Fraction, and the Chain Rule is the Product of Such Fractions

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

On_the_bilateral_laplace_transform_of_the_positive_even_functions_and_proof_of_the_riemann_hypothesis

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

Multiplicative Versions of Infinitesimal Calculus

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

The Troncated Integral

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