相关论文: From $e$ to $\pi$: Derivation of the Wallis formul…
This short note delivers, via elementary calculations, a product representation of pi.
A famous pre-Newtonian formula for $\pi$ 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.
We use well-known limit theorems in probability theory to derive a Wallis-type product formula for the gamma function. Our result immediately provides a probabilistic proof of Wallis's product formula for $\pi$, as well as the duplication…
In this article, we give another visual proof of $\pi^e < e^\pi$.
Using mostly elementary results and functions from probability, we prove Wallis's formula for pi: pi/2 = prod_n (2n * 2n) / ((2n-1) * (2n+1)). The proof involves normalization constants and the Gamma function, Standard normal, and the…
There are several reformulations of the Vi\`ete's formula for pi that have been reported in the modern literature. In this paper we show another analog to the Vi\`ete's formula for pi by Chebyshev polynomials of the first kind.
In this article, we derive, using Fourier series and multiple derivative of the function $\pi/\sin(\pi x)$, series representations for positive powers of $\pi$. We also show that the Euler-Wallis product can be easily obtained from the same…
In this note we show that any proof of Wallis's formula or of the probability integral formula proves both assertions.
We comment on the Friedmann and Hagen's quantum mechanical derivation of the Wallis formula for $\pi$. In particular, we demonstrate that not only the Gaussian trial function, used by Friedmann and Hagen, but also the Lorentz trial function…
Stirling's formula is a powerful asymptotic approximation of the factorial function. Many well-known proofs of this formula are grounded in integral calculus. In this paper, we present an alternative proof of Stirling's formula using only…
In this paper we give another proof of the Chudnovsky formula for calculating $\pi$ - a proof in detail with means of basic complex analysis. With the exception of the tenth chapter, the proof is self-contained, with proofs provided for all…
This note states and proves a representation theorem for regular quantity functions, based on the theory of quantity spaces, thereby giving a new perspective on dimensional analysis and the classical $\pi$ theorem.
This note presents an especially short and direct variant of Hermite's proof of the simple continued fraction expansion e = [2,1,2,1,1,4,1,1,6,...] and explains some of the motivation behind it.
A family of original formulae for computing number PI and its proof are presented. An algorithm is proposed to validate the results of this new algorithm.
We obtained a new formula for $\pi$.
We generalize the derivation of the Wallis formula for $\pi$ 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…
In this note, we derive an elementary version of the coarea formula by considering the mass of a solid body with density $g (x)$. Then we present an rigorous proof using the changing variable formula. To this end we construct the…
A simple way is shown to construct the length $\pi$ from the unit length with 4 digits accuracy.
In our previous publication we have shown a method for calculating series of even powers of $\pi$ based on the product representation of the $sinc$ function. We refer the readers to [1] for more details. In this work we apply the method to…
We provide a simplified approach to the the stable Hopf invariant. We provide short elementary proofs of the Cartan Formula, the Composition Formula, and the Transfer formula. In addition, when $\pi$ is a discrete group, we show how to…