相关论文: A New Proof of Stirling's Formula
By changing variables in a suitable way and using dominated convergence methods, this note gives a short proof of Stirling's formula and its refinement.
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…
The monotonicity properties of remainder of Stirling's formula for the gamma function are simply obtained by using the integral transforms with series.
In the paper, the authors provide four alternative proofs of an explicit formula for computing Bernoulli numbers in terms of Stirling numbers of the second kind.
Stirling's formula, the asymptotic expansion of $n!$ for $n$ large, or of $\Gamma(z)$ for $z\to \infty$, is derived directly from the recursion equation $\Gamma(z+1) =z \Gamma(s)$ and the normalization condition $\Gamma ({1/2})…
In the paper we present some new inversion formulas and two new formulas for Stirling numbers.
This article provides a simple proof of the quadratic formula, which also produces an efficient and natural method for solving general quadratic equations. The derivation is computationally light and conceptually natural, and has the…
A new method for continuing the usual Dirichlet series that defines the Riemann zeta function ${\zeta}(s)$ is presented. Numerical experiments demonstrating the computational efficacy of the resulting continuation are discussed.
We prove a continued fraction expansion for a certain $q$-tangent function that was conjectured by the present writer, then proved by Fulmek, now in a completely elementary way.
We prove a continued fraction expansion for a certain q--tangent function that was conjectured by Prodinger.
We present a survey on recent results about Stirling's formula. More exactly, we reffer to a method using a form of Cesaro-Stolz lemma firstly introduced in [C. Mortici Product approximations via asymptotic integration Amer. Math. Monthly…
We obtain another proof of Hermite's integral for the Hurwitz zeta function.
In the note, the author discovers an explicit formula for computing Bernoulli numbers in terms of Stirling numbers of the second kind.
A concise and elementary derivation of the complete asymptotic expansion for the factorial function $n!$ is presented. This treatment produces a new expression for the coefficients, and it brings to light the simple relationship between the…
We present a proof given by Euler in his paper {\it ``De serierum determinatione seu nova methodus inveniendi terminos generales serierum"} \cite{E189} (E189:``On the determination of series or a new method of finding the general terms of…
We prove some new results related to Tanaka's formula.
We give a new proof of Tietze Theorem on the convergence of infinite semi-regular continued fractions.
In this paper, we find a new recurrence formula fo the Euler zeta functions.
This note contains a short proof of the functional equation for the zeta function.
We derive and prove a new formulation of the Lerch zeta function as a fractional derivative of an elementary function. We demonstrate how this formulation interacts very naturally with basic known properties of Lerch zeta, and use the…