Sum Kind of Asymptotic Trouble

The Mathematical Gazette, 2009

2024

Consider the graphs of f(x)=x^x, f(x)=x^1/x, f(x)=1/x^x, and f(x)=1/x^1/x (where y is equivalent to f(x) and its variants) superimposed onto one another. This is for all { x ∈ ℝ ∣ x > 0 }. The stationary points of each of these graphed functions form a rectilinear quadrilateral (referred to as Yaniv’s Rectangle throughout the paper), and our goal is to calculate the area and perimeter of said quadrilateral. To find the stationary points, the 4 functions are differentiated and their gradient functions are set equal to 1. The four stationary points are the four vertices of Yaniv's Rectangle. Upon calculating the area of Yaniv’s Rectangle, a 6-decimal-point value of 1.768801 is yielded, which is approximately equal to sqrt(pi), which has a 6-decimal-point value of 1.772454. The difference in the two values is approximately 0.003853 correct to 6 decimal places (which can be seen as negligible in certain trivial situations). Meanwhile, when Yaniv’s rectangle’s perimeter is calculated, a value of 6.205739 is obtained, which is approximately equal to 2*pi (ie- 6.283185). A difference of 0.0774461 can be seen. As a result, barring the slight differences, the following identities (known in this paper as Yaniv’s First and Second Approximation Identities) can be derived and can be used only for situations in which extremely precise values are not necessary.

Bulletin of the American Mathematical Society, 2013

This paper has two parts. The first part surveys Euler's work on the constant γ = 0.57721... bearing his name, together with some of his related work on the gamma function, values of the zeta function, and divergent series. The second part describes various mathematical developments involving Euler's constant, as well as another constant, the Euler-Gompertz constant. These developments include connections with arithmetic functions and the Riemann hypothesis, and with sieve methods, random permutations, and random matrix products. It also includes recent results on Diophantine approximation and transcendence related to Euler's constant.

In a recent study of representing Dirac's delta distribution using q-exponentials, M. Jauregui and C. Tsallis experimentally discovered formulae for π as hypergeometric series as well as certain integrals. Herein, we offer rigorous proofs of these identities using various methods and our primary intent is to lay down an illustration of the many technical underpinnings of such evaluations. This includes an explicit discussion of creative telescoping and Carlson's Theorem. We also generalize the Jauregui-Tsallis identities to integrals involving Chebyshev polynomials. In our pursuit, we provide an interesting tour through various topics from classical analysis to the theory of special functions.

2022

The Basel problem, solved by Leonhard Euler in 1734, asks to resolve ζ (2), the sum of the reciprocals of the squares of the natural numbers, i.e. the sum of the infinite series:

Illinois Journal of Mathematics

Mathematics of Computation, 1989

be a vector of real numbers, x is said to possess an integer relation if there exist integers a,-not all zero such that ain + a2X2 + ■ ■ ■ + anxn = 0. Beginning ten years ago, algorithms were discovered by one of us which, for any n, are guaranteed to either find a relation if one exists, or else establish bounds within which no relation can exist. One of those algorithms has been employed to study whether or not certain fundamental mathematical constants satisfy simple algebraic polynomials. Recently, one of us discovered a new relation-finding algorithm that is much more efficient, both in terms of run time and numerical precision. This algorithm has now been implemented on high-speed computers, using multiprecision arithmetic. With the help of these programs, several of the previous numerical results on mathematical constants have been extended, and other possible relationships between certain constants have been studied. This paper describes this new algorithm, summarizes the numerical results, and discusses other possible applications. In particular, it is established that none of the following constants satisfies a simple, low-degree polynomial: 7 (Euler's constant), log'y, log7r, pi (the imaginary part of the first zero of Riemann's zeta function), logpi, f (3) (Riemann's zeta function evaluated at 3), and logç(3). Several classes of possible additive and multiplicative relationships between these and related constants are ruled out. Results are also cited for Feigenbaum's constant, derived from the theory of chaos, and two constants of fundamental physics, derived from experiment.

Applied Mathematics and Computation, 2017

Recently various approximation formulas for some mathematical constants have been investigated and presented by many authors. In this paper, we first find that the relationship between the coefficients p j and q j is such that ψ x ∞ j=0 q j x − j ∼ ln x ∞ j=0 p j x − j , x → ∞ , where ψ is the logarithmic derivative of the gamma function (often referred to as psi function) and p 0 = q 0 = 1. Next, by using this result, we give a unified treatment of several asymptotic expansions concerning the Euler-Mascheroni constant, Landau and Lebesgue constants, Glaisher-Kinkelin constant, and Choi-Srivastava constants.

Applicable Algebra in Engineering, Communication and Computing, 2002

A fruitful interaction between a new randomized WZ procedure and other computer algebra programs is illustrated by the computer proof of a series evaluation that originates from a definite integration problem.