5,516 research outputs found
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 with , 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 , where
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 , and prove that the function
is strictly
decreasing and convex from onto , 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
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