Two asymptotic expansions for gamma function developed by Windschitl's formula

In this paper, we develop Windschitl's approximation formula for the gamma function to two asymptotic expansions by using a little known power series. In particular, for $n\in \mathbb{N}$ with $n\geq 4$, we have \begin{equation*} \Gamma \left( x+1\right) =\sqrt{2\pi x}\left( \tfrac{x}{e}\right) ^{x}\left( x\sinh \tfrac{1}{x}\right) ^{x/2}\exp \left( \sum_{k=3}^{n-1}\tfrac{\left( 2k\left( 2k-2\right) !-2^{2k-1}\right) B_{2k}}{2k\left( 2k\right) !x^{2k-1}} +R_{n}\left( x\right) \right) \end{equation*} with \begin{equation*} \left| R_{n}\left( x\right) \right| \leq \frac{\left| B_{2n}\right| }{2n\left( 2n-1\right) }\frac{1}{x^{2n-1}} \end{equation*} for all $x>0$, where $B_{2n}$ is the Bernoulli number. Moreover, we present some approximation formulas for gamma function related to Windschitl's approximation one, which have higher accuracy.Comment: 14 page

An accurate approximation formula for gamma function

In this paper, we present a very accurate approximation for gamma function: \begin{equation*} \Gamma \left( x+1\right) \thicksim \sqrt{2\pi x}\left( \dfrac{x}{e}\right) ^{x}\left( x\sinh \frac{1}{x}\right) ^{x/2}\exp \left( \frac{7}{324}\frac{1}{ x^{3}\left( 35x^{2}+33\right) }\right) =W_{2}\left( x\right) \end{equation*} as $x\rightarrow \infty$, and prove that the function $x\mapsto \ln \Gamma \left( x+1\right) -\ln W_{2}\left( x\right)$ is strictly decreasing and convex from $\left( 1,\infty \right)$ onto $\left( 0,\beta \right)$, where \begin{equation*} \beta =\frac{22\,025}{22\,032}-\ln \sqrt{2\pi \sinh 1}\approx 0.00002407. \end{equation*}Comment: 9 page

Study on Evaluation System of Sustainable Development Capability of Chinese Property and Casualty Insurance Enterprises

It is important but difficult for evaluating sustainable development capability of Chinese property and casualty insurance enterprises. From the perspective of core capability and based on previous study about factors and capability structure model, the paper builds up the first grade indexes of the evaluation, which include financial capability, learning and innovation capability, environment capability, management capability and corporate governance capability. Furthermore, the present study proposes the second grade indexes of first grade indexes

Basic Factors Influencing the Sustainable Development Capability of Chinese Property and Casualty Insurance Enterprises

Under the influence of financial crisis, the performance of Chinese property insurance enterprises keeps falling. The enterprises face the challenge of sustainable development. Based on capability theory, this paper analyzes the connotation of sustainable development ability; determines three main factors (capability resources, capability level and capability environment) and their sub-factors, lays out the structural model of the sustainable development capability of property insurance enterprises. In this model, capability environment is the basis and capability resources and capability level are the pillars. Through learning, enterprises can effectively combine capability resources and capability level，thus forms core competency

One-step implementation of multi-qubit conditional phase gating with nitrogen-vacancy centers coupled to a high-Q silica microsphere cavity

The diamond nitrogen-vacancy (NV) center is an excellent candidate for quantum information processing, whereas entangling separate NV centers is still of great experimental challenge. We propose an one-step conditional phase flip with three NV centers coupled to a whispering-gallery mode cavity by virtue of the Raman transition and smart qubit encoding. As decoherence is much suppressed, our scheme could work for more qubits. The experimental feasibility is justified.Comment: 3 pages, 2 figures, Accepted by Appl. Phys. Let