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Results in Mathematics

, 74:44 | Cite as

Inequalities and Asymptotic Expansions Related to the Volume of the Unit Ball in \(\pmb {\mathbb {R}}^{{{\varvec{n}}}}\)

  • Chao-Ping ChenEmail author
  • Richard B. Paris
Article
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Abstract

Let \(\Omega _{n}=\pi ^{n/2}/\Gamma (\frac{n}{2}+1) \, (n \in \mathbb {N})\) denote the volume of the unit ball in \(\mathbb {R}^{n}\). In this paper, we present asymptotic expansions and inequalities related to \(\Omega _{n}\) and the quantities:
$$\begin{aligned} \frac{\Omega _{n-1}}{\Omega _{n}}, \quad \frac{\Omega _{n}}{\Omega _{n-1}+\Omega _{n+1}} \quad \text {and}\quad \frac{\Omega _n^{1/n}}{\Omega _{n+1}^{1/(n+1)}}. \end{aligned}$$

Keywords

Volume of the unit n-dimensional ball gamma function asymptotic expansions inequalities 

Mathematics Subject Classification

Primary 33B15 Secondary 41A60 26D15 
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Notes

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.School of Mathematics and InformaticsHenan Polytechnic UniversityJiaozuoChina
  2. 2.Division of Computing and MathematicsAbertay UniversityDundeeUK

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