scholarly journals Finite Integral Involving the Modified Generalized Aleph-Function of Two Variables and Elliptic Integral of First Species I

2021 ◽  
Vol 8 (2) ◽  
pp. 1-10
Author(s):  
Frédéric Ayant ◽  
Prvindra Kumar
Keyword(s):  
Elliptic Integral ◽  
Elliptic Integrals ◽  
Finite Integral ◽  
Function Of Two Variables

In the present paper, we evaluate the general finite integral involving the elliptic integrals of first species and the generalized modified Aleph- function of two variables. At the end, we shall see several corollaries and remarks.  

scholarly journals Quantum walks and elliptic integrals

2010 ◽  
Vol 20 (6) ◽  
pp. 1091-1098 ◽  
Author(s):  
NORIO KONNO
Keyword(s):  
Random Walk ◽  
Generating Function ◽  
Elliptic Integral ◽  
Quantum Walk ◽  
Quantum Walks ◽  
Elliptic Integrals ◽  
Two Dimensional ◽  
One Dimensional ◽  
Similar Expression ◽  
Return Probability

Pólya showed in his 1921 paper that the generating function of the return probability for a two-dimensional random walk can be written in terms of an elliptic integral. In this paper we present a similar expression for a one-dimensional quantum walk.

scholarly journals On the Complete Elliptic Integrals and Babylonian Identity I: The 1/Π Formulaes Involving Gamma Functions and Summations

2013 ◽  
Vol 6 ◽  
pp. 40-46
Author(s):  
Edigles Guedes ◽  
Raja Rama Gandhi
Keyword(s):  
Elliptic Integral ◽  
Elliptic Integrals ◽  
Complete Elliptic Integral ◽  
Gamma Functions ◽  
Complete Elliptic Integrals

We evaluate the constant 1/Π using the Babylonian identity and complete elliptic integral of first kind. This resulted in two representations in terms of the Euler’s gamma functions and summations.

scholarly journals Answers to three conjectures on convexity of three functions involving complete elliptic integrals of the first kind

2020 ◽  
Vol 14 (1) ◽  
pp. 255-271 ◽  
Author(s):  
Miao-Kun Wang ◽  
Hong-Hu Chu ◽  
Yong-Min Li ◽  
Yu-Ming Chu
Keyword(s):  
Elliptic Integral ◽  
Elliptic Integrals ◽  
Complete Elliptic Integral ◽  
Complete Elliptic Integrals ◽  
Absolutely Monotonic

In the article, we prove that the function x ? (1-x)pK(?x) is logarithmically concave on (0,1) if and only if p ? 7/32, the function x ? K(?x)/log(1+4/?1-x) is convex on (0,1) and the function x ? d2/dx2 [K(?x)- log (1+4/?1-x) is absolutely monotonic on (0,1), where K(x) = ??/20 (1-x2 sin2t)-1/2 dt (0 < x < 1) is the complete elliptic integral of the first kind.

scholarly journals Researches on the geometrical properties of elliptic integrals

1854 ◽  
Vol 6 ◽  
pp. 143-145
Keyword(s):  
Elliptic Integral ◽  
Elliptic Functions ◽  
Second Order ◽  
Elliptic Integrals ◽  
Geometrical Properties ◽  
Entire Class ◽  
The Subject

In this paper the author proposes to investigate the true geometrical basis of that entire class of algebraical expressions, known to mathematicians as elliptic functions or integrals. He sets out by showing what had already been done in this department of the subject by preceding geometers. That the elliptic integral of the second order represented an arc of a plane ellipse, was evident from the beginning.

XVIII. Researches on the geometrical properties of elliptic integrals

1852 ◽  
Vol 142 ◽  
pp. 311-416 ◽  
Keyword(s):  
Elliptic Integral ◽  
Plane Curve ◽  
Elliptic Functions ◽  
Elliptic Integrals ◽  
Mathematical Science ◽  
Third Order ◽  
Geometrical Properties ◽  
First Order ◽  
The Third ◽  
The Right

I. In placing before the Royal Society the following researches on the geometrical types of elliptic integrals, which nearly complete my investigations on this interesting subject, I may be permitted briefly to advert to what bad already been effected in this department of geometrical research. Legendre, to whom this important branch of mathematical science owes so much, devised a plane curve, whose rectification might be effected by an elliptic integral of the first order. Since that time many other geometers have followed his example, in contriving similar curves, to represent, either by their quadrature or rectification, elliptic functions. Of those who have been most successful in devising curves which should possess the required properties, may be mentioned M. Gudermann, M. Verhulst of Brussels, and M. Serret of Paris. These geometers however have succeeded in deriving from those curves scarcely any of the properties of elliptic integrals, even the most elementary. This barrenness in results was doubtless owing to the very artificial character of the genesis of those curves, devised, as they were, solely to satisfy one condition only of the general pro­blem. In 1841 a step was taken in the right direction. MM. Catalan and Gudermann, in the journals of Liouville and Crelle, showed how the arcs of spherical conic sec­tions might be represented by elliptic integrals of the third order and circular form. They did not, however, extend their investigations to the case of elliptic integrals of the third order and logarithmic form; nor even to that of the first order. These cases still remained, without any analogous geometrical representative, a blemish to the theory.

scholarly journals On Alzer and Qiu's Conjecture for Complete Elliptic Integral and Inverse Hyperbolic Tangent Function

2011 ◽  
Vol 2011 ◽  
pp. 1-7 ◽  
Author(s):  
Yu-Ming Chu ◽  
Miao-Kun Wang ◽  
Ye-Fang Qiu
Keyword(s):  
Open Problem ◽  
Elliptic Integral ◽  
Elliptic Integrals ◽  
Complete Elliptic Integral ◽  
Hyperbolic Tangent ◽  
Hyperbolic Tangent Function ◽  
Double Inequality ◽  
Tangent Function ◽  
Complete Elliptic Integrals

We prove that the double inequality(π/2)(arthr/r)3/4+α*r<K(r)<(π/2)(arthr/r)3/4+β*rholds for allr∈(0,1)with the best possible constantsα*=0andβ*=1/4, which answer to an open problem proposed by Alzer and Qiu. Here,K(r)is the complete elliptic integrals of the first kind, and arth is the inverse hyperbolic tangent function.

scholarly journals Definite integrals of generalized certain class of incomplete elliptic integrals

2013 ◽  
Vol 44 (2) ◽  
pp. 197-208
Author(s):  
Ravi Shanker Dubey ◽  
V. B. L.Chaurasia
Keyword(s):  
Electromagnetic Waves ◽  
Elliptic Integral ◽  
Elliptic Integrals ◽  
Nuclear Technology ◽  
Radiation Physics ◽  
Systematic Account ◽  
Definite Integrals ◽  
Crack Problems ◽  
The Subject ◽  
Incomplete Elliptic Integrals

Elliptic- integral have their importance and potential in certain problems in radiation physics and nuclear technology, studies of crystallographic minimal surfaces, the theory of scattering of acoustic or electromagnetic waves by means of an elliptic disk, studies of elliptical crack problems in fracture mechanics. A number of earlier works on the subject contains remarkably large number of general families of elliptic- integrals and indeed also many definite integrals of such families with respect to their modulus (or complementary modulus) are known to arise naturally. Motivated essentially by these and many other potential avenues of their applications, our aim here is to give a systematic account of the theory of a certain family of generalized incomplete elliptic integrals in a unique and generalized manner. The results established in this paper are of manifold generality and basic in nature. By making use of the familiar Riemann-Liouville fractional differ integral operators, we establish many explicit hypergeometric representations and apply these representation in deriving several definite integrals pertaining to their, not only with respect to the modulus (or complementary modulus), but also with respect to the amplitude of generalized incomplete elliptic integrals involved therein.

scholarly journals On the Complete Elliptic Integrals and Babylonian Identity II: An Approximation for the Complete Elliptic Integral of the First Kind

2013 ◽  
Vol 6 ◽  
pp. 47-54
Author(s):  
Edigles Guedes ◽  
Raja Rama Gandhi
Keyword(s):  
Closed Form ◽  
Bessel Functions ◽  
Elliptic Integral ◽  
Elliptic Integrals ◽  
Complete Elliptic Integral ◽  
Complete Elliptic Integrals

Using the Theorem 4 of previous paper, I evaluate the complete elliptic integral of the first kind in approximate analytical closed form, by means of Bessel functions.

Sign in / Sign up

Export Citation Format

Share Document