Wallis Formula Research Articles

Finding finite sums, products, and limits of some numerical sequences. Part 2. Application of methods of higher mathematics. Finding sums of series

Compared to the methods of elementary mathematics, the methods of higher mathematics significantly expand the range of possibilities when finding sums and products of elements of some numerical sequences and sums of numerical or functional series. This article examines the application of de Moivre’s, Euler’s formulas and Newton's binomial, examples are given. Considering a suitable function on the segment $[a,b]$ and forming the corresponding integral Riemann sum for it, we find its value by integration. Using the Taylor & Maclaurin series of some known functions in $R$ or $C$, integrating and differentiating the corresponding power and functional series, we obtain the sums we are looking for. An example of the application of the Wallis formula is given. Expanding some functions into a Fourier series, we find for certain values ​​of the argument the sum of many interesting numerical series. An example of using Lyapunov's equality to calculate the sum of trigonometric series is given. The use of operational calculus methods for finding sums of numerical, functional and trigonometric series is considered, an example of the use of Dirac's $\delta $-function and its properties is given.

Euler’s reflection formula, infinite product formulas, and the correspondence principle of quantum mechanics

We generalize the derivation of the Wallis formula for π from a variational computation of the spectrum of the hydrogen atom. We obtain infinite product formulas for certain combinations of gamma functions, which include irrational numbers such as 2 as well as some nested radicals. We also derive Euler’s reflection formula for reciprocals of positive even integers. We show that Bohr’s correspondence principle allows us to derive our product formulas and the reflection formula without the need for the limit definition of the gamma function.

A Generalized Wallis Formula

This article generalizes the famous Wallis’s formula for k ≥ 0 , to an integral over the unit sphere Sn-1. An application to the integral of polynomials over Sn-1 is discussed.

Wallis formula from the harmonic oscillator

We show that the asymptotic formula for $\pi$, the Wallis formula, that was related with quantum mechanics and the hydrogen atom in \cite{HF}, can also be related to the harmonic oscillator using a quantum duality between these two systems. As a corollary we show that this very interesting asymptotic formula is not related with the hydrogen atom or quantum mechanics itself but with a clever choice of a trial function and a potential in the Schroedinger equation when we use the variational approach to calculate the ground state energy associated with the given potential function.

On the quantum mechanical derivation of the Wallis formula for π

We comment on the Friedmann and Hagen's quantum mechanical derivation of the Wallis formula for π. In particular, we demonstrate that not only the Gaussian trial function, used by Friedmann and Hagen, but also the Lorentz trial function can be used to get the Wallis formula. The anatomy of the integrals leading to the appearance of the Wallis ratio is carefully revealed.

Pad\xe9 approximant related to the Wallis formula

Based on the Padé approximation method, in this paper we determine the coefficients a_{j} and b_{j} such that \t\t\tπ=((2n)!!(2n−1)!!)2{nk+a1nk−1+⋯+aknk+1+b1nk+⋯+bk+1+O(1n2k+3)},n→∞,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\pi= \\biggl(\\frac{(2n)!!}{(2n-1)!!} \\biggr)^{2} \\biggl\\{ \\frac {n^{k}+a_{1}n^{k-1}+\\cdots+a_{k}}{n^{k+1}+b_{1}n^{k}+\\cdots+b_{k+1}}+O \\biggl(\\frac{1}{n^{2k+3}} \\biggr) \\biggr\\} ,\\quad n\\to\\infty, $$\\end{document} where kgeq0 is any given integer. Based on the obtained result, we establish a more accurate formula for approximating π, which refines some known results.

On q-advanced spherical Bessel functions of the first kind and perturbations of the Haar wavelet

For q > 1 , the n th order q -advanced spherical Bessel functions of the first kind, j n ( q ; t ) , are introduced. Smooth perturbations, H q ( ω ) , of the Haar wavelet are derived. The inverse Fourier transforms F − 1 [ j n ( q ; t ) ] ( ω ) are expressed in terms of the Jacobi theta function and are shown to give genesis to the q -advanced Legendre polynomials P ˜ n ( q ; ω ) . The wavelet F − 1 [ sin ⁡ ( t ) j 0 ( q ; t ) ] ( ω ) is studied and shown to generate H q ( ω ) . For each n ≥ 1 , F − 1 [ j n ( q ; t ) ] ( ω ) is shown to be a Schwartz wavelet with vanishing j th moments for 0 ≤ j ≤ n − 1 and non-vanishing n th moment. Wavelet frame properties are developed. The family { 2 j / 2 H q ( 2 j ω − k ) | j , k ∈ Z } is seen to be a nearly orthonormal frame for L 2 ( R ) and a perturbation of the Haar basis. The corresponding multiplicatively advanced differential equations (MADEs) satisfied by these new functions are presented. As the parameter q → 1 + , convergence of the q -advanced functions to their classical counterparts is shown. A q -Wallis formula is given. Symmetry of the Jacobi theta function is shown to preclude Gibb's type phenomena. A Schwartz function with lower moments vanishing is shown to be a mother wavelet for a frame generating L 2 ( R ) .

Quantum mechanical derivation of the Wallis formula for π

A famous pre-Newtonian formula for π is obtained directly from the variational approach to the spectrum of the hydrogen atom in spaces of arbitrary dimensions greater than one, including the physical three dimensions.

The probabilistic nature of Wallis's formula

The beautiful representation of π by Wallis's product formula has attracted many authors who have given a variety of proofs (see [1, 2, 3, 4, 5, 6, 7]). The purpose of this note is to revisit some elementary derivations with a focus on probabilistic/statistical flavours of this formula. The famous Wallis's product formula for is given byThere are a number proofs available for this formula scattered in the literature. Here we will concentrate on the probabilistic nature of this formula.

Parametric improper integrals, Wallis formula and Catalan numbers

We present a sequence of improper integrals, for which a closed formula can be computed using Wallis formula and a non-straightforward recurrence formula. This yields a new integral presentation for Catalan numbers.

Stirred, Not Shaken, by Stirling's Formula

SummaryIn this note an elementary proof of Stirling's asymptotic formula for n! is given. The proof uses the Wallis formula for π and the trapezoidal rule for the calculation of a definite integral, with error estimate.

Integral presentations of Catalan numbers and Wallis formula

Looking for closed forms for an integral depending on one parameter, we obtain integral identities and new integral presentations of Catalan numbers. Wallis formula appears as an important tool.

A Probabilistic Proof of Wallis's Formula for π

(2008). A Probabilistic Proof of Wallis's Formula for π. The American Mathematical Monthly: Vol. 115, No. 8, pp. 740-745.

On Estimates of Factorial Quotients

We consider the factorial quotients (2n − 1)!!/(2n)!! in connection with the Wallis formula n−1(2n)!!2/(2n − 1)!!2 → π. We improve the Wallis inequalities (n + 1/2)−1(2n)!!2/(2n − 1)!!2 < π < n−1(2n)!!2/(2n − 1)!!2 for π and obtain new estimates of factorial quotients with error order not worse than 1/n2.

A generalization of the Wallis formula

A generalization of the Wallis formula

The wallis formula for PI, including two new expressions involving irrationals

Two new expressions related to Wallis's formula for π are obtained using methods of the calculus. When truncated, one of these expressions yields upper while the second results in lower bounds for π. Additionally these expressions involve ratios of irrationals which makes them more quickly convergent than Wallis's formula.

Two doubly infinite sets of series for π

It is shown how two doubly infinite sets of series involving π may be obtained using closure and a particular set of orthonormal functions. The first set of series converge to π from below, while the second set converge to π from above. The leading terms of the simplest of these series are alternately lower and upper bounds to, and successive truncations of Wallis's formula. Some convergence properties of these series are examined.

Absorption intensity measurements of the first overtone band of CO

The absolute intensity A2–0 of the 2–0 band of carbon monoxide has been measured with helium and argon as pressure broadening gases at pressures up to 600 amagat. A separate band intensity value has been derived from the measurements of several P-branch line intensity data and the Herman–Wallis formula. The best value obtained is A2–0 = (2.11 ± 0.08) cm−2 amagat−1. The square of the rotationless matrix element, [Formula: see text] has been found to be (4.39 ± 0.02)10−5 D2, as estimated from the measured line intensity values.

Mean-square atomic displacements in f.c.c. crystals

The mean-square displacements of volume atoms, surface atoms and atoms near a vacancy are calculated for f.c.c. metals and rare gas solids, using a simple series expansion of the Wallis formula. We employ a model with rotationally invariant bond bending forces, unlike the de Launay model used by Masri and Dobrzynski. For atoms near a vacancy and for rare gas solids this method has been applied for the first time. The calculations are done to the R 4 term in the expansion. The inclusion of the R 4 term together with temperature-dependent elastic constants give results which are in better agreement with experiment. For rare gas solids Lindemann ratios calculated by this method agree closely with the values obtained from elaborate computations as well as those deduced from entropy data. The present calculation also supports the view that the melting of solids is initiated at the crystal surfaces and in the neighbourhood of vacancies.