[6] **viXra:1806.0326 [pdf]**
*submitted on 2018-06-22 12:30:06*

**Authors:** Tejas Chandrakant Thakare

**Comments:** 3 Pages. Please feel free to comment on this study

Using method of integration as the limit of sum we can easily evaluate sum of an infinite series in which 1/n is common from every term such that n→∞ (n∈N). However in this method we do some rigorous calculations before integration. In this paper, in order to minimize the labor involved in this process I propose an alternative new method for finding the sum of an infinite series in which 1/n is common from every term such that n→∞.

**Category:** Functions and Analysis

[5] **viXra:1806.0239 [pdf]**
*submitted on 2018-06-17 23:43:19*

**Authors:** Michael Parfenov

**Comments:** 18 Pages.

This paper is the third paper of the cycle devoted to the theory of essentially adequate quaternionic differentiability. It is established that the quaternionic holomorphic (ℍ -holomorphic) functions, satisfying the essentially adequate generalization of Cauchy-Riemann’s equations, make up a very remarkable class: generally non-commutative quaternionic multiplication behaves as commutative in the case of multiplication of ℍ -holomorphic functions. Everyone can construct such ℍ-holomorphic functions by replacing a complex variable as a single whole by a quaternionic one in expressions for complex holomorphic functions, and thereafter verify their commutativity. This property, which is confirmed by a lot of ℍ-holomorphic functions, gives conclusive evidence that the developed theory is true. The rules for quaternionic differentiation of combinations of ℍ-holomorphic functions find themselves similar to those from complex analysis: the formulae for differentiation of sums, products, ratios, and compositions of H-holomorphic functions as well as quaternionic power series, are fully identical to their complex analogs. The example of using the deduced rules is considered and it is shown that they reduce essentially the volume of calculations. The base notions of complex Maclaurin series expansions are adapted to the quaternion case.

**Category:** Functions and Analysis

[4] **viXra:1806.0140 [pdf]**
*replaced on 2018-06-11 13:57:22*

**Authors:** Hassine Saidane

**Comments:** 3 Pages.

Abstract
In this paper, we present a short and easy proof of the Riemann Hypothesis using a simple constrained optimization approach based on the properties of the Riemann Zeta function and the properties of its nontrivial zeros.
Keywords: Riemann Zeta function, Riemann Hypothesis, Constrained Optimization.

**Category:** Functions and Analysis

[3] **viXra:1806.0082 [pdf]**
*replaced on 2018-06-09 05:42:18*

**Authors:** Jonathan W. Tooker

**Comments:** 4 Pages. two figures

This paper examines some familiar results from complex analysis in the framework of hypercomplex analysis. It is usually taught that the oscillatory behavior of sine waves means that they have no limit at infinity but here we derive definite limits. Where a central element in the foundations of complex analysis is that the complex conjugate of a $\mathbb{C}$-number is not analytic at the origin, we introduce the tools of hypercomplex analysis to show that the complex conjugate of a $^\star\mathbb{C}$-number is analytic at the origin.

**Category:** Functions and Analysis

[2] **viXra:1806.0067 [pdf]**
*submitted on 2018-06-07 04:22:20*

**Authors:** Claude Michael Cassano

**Comments:** 9 Pages.

Theorems establishing exact solution for any linear ordinary differential equation of arbitrary order (homogeneous and inhomogeneous) are presented and proven.

**Category:** Functions and Analysis

[1] **viXra:1806.0047 [pdf]**
*submitted on 2018-06-06 04:42:10*

**Authors:** Claude Michael Cassano

**Comments:** 18 Pages.

Further development of exactly solving second order linear ordinary differential equations, and related non-linear ordinary differential equations.

**Category:** Functions and Analysis