共查询到20条相似文献,搜索用时 453 毫秒
1.
研究了等差项乘积∏ni=1a_i的渐进估计.首先给出了一系列关于等差项乘积的不等式,继而应用Euler-Maclaurin求和公式及Γ函数的Stirling公式:Γ(x+1)~(2πx)~(1/2)(x/e)~x(x→+∞),推导出了∏ni=1a_i的较精确的渐进式,最后,得到了精确化的Wallis公式. 相似文献
2.
数学力学系卷1 1.解方程(3~(8xctgπx))~x27~(5xctgπx)=9~(ctgπx)。2.解不等式[log(8-2)~(1/2)(2x-1)][log_2(1+2x-x~2)]≥0。3.一条直线经过△ABC的顶点A和平行于边AC的中位线的中点,问这条直线分这个三角形成两部分的面积之比是多少? (如图1) 相似文献
3.
4.
5.
陆善镇 《数学年刊B辑(英文版)》1986,(3)
设K(x)=P(x/|x|)|x|~(-n)为一球调和核,P(x)为一m次齐次调和多项式。f(x)在R~n上的δ阶共轭Bochner-Riesz平均记为 (_(1/ε)~δf)(x)=∫_(|t|<1/ε)(t)(t)(1-|εt|~2)~δe~(iαt)dt.作者在本文中得到如下的弱型估计: |{x∈R~n:sup ε>0|(_(1/ε)~δf)(x)-_ε(x)|>λ}|≤C(‖f‖_(H~p)/λ)~p,此处δ=(n/p)-(n 2)/2,n/(n 1)≤p<1,f∈H~p(R~n),以及 _ε(x)=(2π)~(-n)∫_(|y|>ε)f(x-y)K(y)dy 。设f∈L(R~n),其δ阶的Bochner-Riesz平均为 (σ_(1/ε)~δf)(x)=∫_(|t|<1/ε)(t)(1-|εt|~2)~δe~(iαt)dt. 相似文献
6.
In this paper we obtain the best approximation constant of function f(x)(∈C_(2π))by theJackson's type operator J_(π3)(f;x),i.e.‖J_(n,3)(f,x)-f(x)‖_c≤(4-6/π)ω(f,1/n),‖J_(n,3)(f,x)-f(x)‖_c≤(8-17/π)ω_2(f,1/n) 相似文献
7.
§1 反三角函数的概念一、选择题 1.适合不等式arccos3x〉3的x的集合是( ) (A){x|0≤x相似文献
8.
<正> 运用重要极限lim(1+x)~(1/x)=e(或lim(1+1/x)~x=e)求极限是求未定式“1~∞”型极限的一个重要方法。例如: 相似文献
9.
Let f(n)be a multiplicative function satisfying |f(n)|≤1,q(≤N~2)be a positive integer and a be an integer with(a,q)= 1.In this paper,we shall prove that ∑n≤N(n,q)=1f(n)e(an/q)■(1/2)(τ(q)/q)N loglog(6N)+ q~(1/4+ε/2)N~(2/1)(log(6N))~(1/2)+N/(1/2)(loglog(6N)),where n is the multiplicative inverse of n such that nn ≡ 1(mod q),e(x)= exp(2πix),and τ(·)is the divisor function. 相似文献
10.
I.Schur问题的推广及证明 总被引:1,自引:0,他引:1
徐晓泉 《数学的实践与认识》1993,(4)
本文推广了Ⅰ.Schur 关于数列的三个结果,证明了函数 f(x)=(1+1/x)~(x+p_1)(x>0),g(x)=(1+1/x)~x(1+(p_2)/x)(x>0)与 h(x)=(1+p_3/x)~(x+1)(x>max{0,-P_3})单调下降充要条件,分别为 p_1≥1/2,p_2≥1/2与0相似文献
11.
We prove that, for all integers \(n\ge 1\), and with the best possible constants
相似文献
$$\begin{aligned} \Big (\sqrt{2\pi n}\Big )^{\frac{1}{n(n+1)}}\left( 1-\frac{1}{n+a}\right) <\frac{\root n \of {n!}}{\root n+1 \of {(n+1)!}}\le \Big (\sqrt{2\pi n}\Big )^{\frac{1}{n(n+1)}}\left( 1-\frac{1}{n+b}\right) \end{aligned}$$
$$\begin{aligned} \big (\sqrt{2\pi n}\big )^{1/n}\left( 1-\frac{1}{2n+\alpha }\right) <\left( 1+\frac{1}{n}\right) ^{n}\frac{\root n \of {n!}}{n}\le \big (\sqrt{2\pi n}\big )^{1/n}\left( 1-\frac{1}{2n+\beta }\right) , \end{aligned}$$
$$\begin{aligned}&a=\frac{1}{2},\quad b=\frac{1}{2^{3/4}\pi ^{1/4}-1}=0.807\ldots ,\quad \alpha =\frac{13}{6} \\&\text {and}\quad \beta =\frac{2\sqrt{2}-\sqrt{\pi }}{\sqrt{\pi }-\sqrt{2}}=2.947\ldots . \end{aligned}$$
12.
Cristinel Mortici 《Archiv der Mathematik》2009,93(1):37-45
We prove in this paper that for every x ≥ 0,
where and α = 1.072042464..., then
where and β = 0.988503589... Besides the simplicity, our new formulas are very accurate, if we take into account that they are much stronger
than Burnside’s formula, which is considered one of the best approximation formulas ever known having a simple form.
相似文献
13.
\small\zihao{-5}\begin{quote}{\heiti 摘要:} 设$M$为$n+1$维单位球面$S^{n+1}(1)$中的一个极小闭超曲面,如果 $ n \le S \le n+\frac{2}{3}$, 则有 $S=n$ 且 $M$ 与某一Clifford 环面 $S^m(\sqrt{m/n}) \times S^{n-m}(\sqrt{(n-m)/n})$等距. 相似文献
14.
In this paper, we establish two families of approximations for the gamma function: $$ \begin{array}{lll} {\varGamma}(x+1)&=\sqrt{2\pi x}{\left({\frac{x+a}{{\mathrm{e}}}}\right)}^x {\left({\frac{x+a}{x-a}}\right)}^{-\frac{x}{2}+\frac{1}{4}} {\left({\frac{x+b}{x-b}}\right)}^{\sum\limits_{k=0}^m\frac{{\beta}_k}{x^{2k}}+O{{\left(\frac{1}{x^{2m+2}}\right)}}},\\ {\varGamma}(x+1)&=\sqrt{2\pi x}\cdot(x+a)^{\frac{x}{2}+\frac{1}{4}}(x-a)^{\frac{x}{2}-\frac{1}{4}} {\left({\frac{x-1}{x+1}}\right)}^{\frac{x^2}{2}}\\ &\quad\times {\left({\frac{x-c}{x+c}}\right)}^{\sum\limits_{k=0}^m\frac{{\gamma}_k}{x^{2k}}+O{\left({\frac{1}{x^{2m+2}}}\right)}}, \end{array}$$ where the constants ${\beta }_k$ and ${\gamma }_k$ can be determined by recurrences, and $a$ , $b$ , $c$ are parameters. Numerical comparison shows that our results are more accurate than Stieltjes, Luschny and Nemes’ formulae, which, to our knowledge, are better than other approximations in the literature. 相似文献
15.
Bang-He Li 《数学研究》2016,49(4):319-324
Let $ζ(s)$ be the Riemann zeta function, $s=\sigma+it$. For $0 < \sigma < 1$, we expand $ζ(s)$ as the following series convergent in the space of slowly increasing distributions
with variable $t$ : $$ζ(\sigma+it)=\sum\limits^∞_{n=0}a_n(\sigma)ψ_n(t),$$ where $$ψ_n(t)=(2^nn!\sqrt{\pi})^{-1 ⁄ 2}e^{\frac{-t^2}{2}}H_n(t),$$ $H_n(t)$ is the Hermite polynomial, and $$a_n(σ)=2\pi(-1)^{n+1}ψ_n(i(1-σ))+(-i)^n\sqrt{2\pi}\sum\limits^∞_{m=1}\frac{1}{m^σ}ψ_n(1nm).$$ This paper is concerned with the convergence of the above series for $σ > 0.$ In the deduction,
it is crucial to regard the zeta function as Fourier transfomations of Schwartz'
distributions. 相似文献
16.
A new Stirling series as continued fraction 总被引:1,自引:0,他引:1
Cristinel Mortici 《Numerical Algorithms》2011,56(1):17-26
We introduce the following new Stirling series as a continued fraction, which is faster than the classical Stirling series.
相似文献
$ n!\sim \sqrt{2\pi n}\left( \frac{n}{e}\right) ^{n}\exp \frac{1}{12n+\frac{ \frac{2}{5}}{n+\frac{\frac{53}{210}}{n+\frac{\frac{195}{371}}{n+\frac{\frac{ 22,\!999}{22,\!737}}{n+\ddots}}}}}, $
17.
In this paper, the authors give the local L~2 estimate of the maximal operator S_(φ,γ)~* of the operator family {S_(t,φ,γ)} defined initially by ■which is the solution(when n = 1) of the following dispersive equations(~*) along a curve γ:■where φ : R~+→R satisfies some suitable conditions and φ((-?)~(1/2)) is a pseudo-differential operator with symbol φ(|ξ|). As a consequence of the above result, the authors give the pointwise convergence of the solution(when n = 1) of the equation(~*) along curve γ.Moreover, a global L~2 estimate of the maximal operator S_(φ,γ)~* is also given in this paper. 相似文献
18.
We study the Γ-convergence of the following functional (p > 2)
$F_{\varepsilon}(u):=\varepsilon^{p-2}\int\limits_{\Omega}
|Du|^p d(x,\partial \Omega)^{a}dx+\frac{1}{\varepsilon^{\frac{p-2}{p-1}}}
\int\limits_{\Omega}
W(u) d(x,\partial \Omega)^{-\frac{a}{p-1}}dx+\frac{1}{\sqrt{\varepsilon}}
\int\limits_{\partial\Omega}
V(Tu)d\mathcal{H}^2,$F_{\varepsilon}(u):=\varepsilon^{p-2}\int\limits_{\Omega}
|Du|^p d(x,\partial \Omega)^{a}dx+\frac{1}{\varepsilon^{\frac{p-2}{p-1}}}
\int\limits_{\Omega}
W(u) d(x,\partial \Omega)^{-\frac{a}{p-1}}dx+\frac{1}{\sqrt{\varepsilon}}
\int\limits_{\partial\Omega}
V(Tu)d\mathcal{H}^2, 相似文献
19.
Horst Alzer 《Advances in Computational Mathematics》2010,33(3):349-379
We present various inequalities for the error function. One of our theorems states: Let α?≥?1. For all x,y?>?0 we have $$ \delta_{\alpha} < \frac{ \mbox{erf} \left( x+ \mbox{erf}(y)^{\alpha}\right) +\mbox{erf}\left( y+ \mbox{erf}(x)^{\alpha}\right) } {\mbox{erf}\left( \mbox{erf}(x)+\mbox{erf}(y)\right) } < \Delta_{\alpha} $$ with the best possible bounds $$ \delta_{\alpha}= \left\{ \begin{array}{ll} 1+\sqrt{\pi}/2, & \ \ \textrm{{if} $\alpha=1$,}\\ \sqrt{\pi}/2, & \ \ \textrm{{if} $\alpha>1$,}\\ \end{array}\right. \quad{\mbox{and} \,\,\,\,\, \Delta_{\alpha}=1+\frac{1}{\mbox{erf}(1)}.} $$ 相似文献
20.
Adi Ben-Israel 《Annals of Operations Research》2013,208(1):147-153
The Cauchy distribution $$\mathfrak {C}(a,b)(x)=\frac{1}{\pi b(1+(\frac{x-a}{b})^2)},\quad -\infty < x <\infty,$$ with a,b real, b>0, has no moments (expected value, variance, etc.), because the defining integrals diverge. An obvious way to “concentrate” the Cauchy distribution, in order to get finite moments, is by truncation, restricting it to a finite domain. An alternative, suggested by an elementary problem in mechanics, is the distribution $${\mathfrak {C}}_g(a,b)(x)=\frac{\sqrt{1+2 b g}}{\pi b (1+(\frac{x-a}{b})^2)\sqrt{1-2 b g(\frac{x-a}{b})^2}},\quad a-\sqrt{\frac{b}{2g}}<x<a+\sqrt{\frac{b}{2g}},$$ with a,b as above and a third parameter g≥0. It has the Cauchy distribution C(a,b) as the special case with g=0, and for any g>0, ? g (a,b) has finite moments of all orders, while keeping the useful “fat tails” property of ?(a,b). 相似文献
|