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

Bills of Lading as Documents of Title — Chinese Law and Policy

In modern maritime commerce, the bill of lading is usually regarded as a document of title.This is also the case under Chinese law. However, unlike other jurisdictions such as the UK and the US, the exact effects of the bill of lading as a document of title have never been clarified in China due to the silence of Chinese law in this regard. In consequence of that, various explanations on the notion “bill of lading as document of title” have been developed.Unfortunately, none of them is fully compatible with the existing Chinese law and the reality of trade. To understand the notion “bill of lading as document of title” precisely, this article revisits English law and American law from which Chinese law borrows the notion and uncovers that the definition of the notion in national law may be influenced by the following factors: the legislative intent of the national law on the usage of the bill of lading, the freedom of contract in transfer of property rights in goods, and the domestic law on doctrine of good faith purchase. This article then analyses these factors in the context of Chinese law and policy,seeking to find a proper way to clarify the notion “bill of lading as document of title” and localise such a notion to suit Chinese legal and economic background

The insurer’s primary obligation to pay valid claims in a timely manner

This article considers the English and Chinese laws relating to the insurer�s primary obligation and the consequences for non-performance of the obligation. English law characterises an insurer�s primary obligation as a duty to "hold the insured harmless" against the insured event and there is no damage for late payment of insurance claims. Chinese law requires the insurer to pay valid claims in a timely manner and also sets out time-limits for an insurer to assess and pay claims. The strengths in setting out such time-limits and the difficulties without such time-limits are discussed. This article also considers how Chinese courts interpret and apply these time-limits. Finally, suggestions are made on further improvement of Chinese law and of the Law Commissions� recommendation that insurers should have an obligation to pay valid claims "within a reasonable time"