The Holtsmark distribution has applications in plasma physics, for the electric-microfield distribution involved in spectral line shapes for instance, as well as in astrophysics for the distribution of gravitating bodies. It is one of the few examples of a stable distribution for which a closed-form expression of the probability density function is known. However, the latter is not expressible in terms of elementary functions. In the present work, we mention that the Holtsmark probability density function can be expressed in terms of hypergeometric function $_2F_2$ and of Airy function of the second kind $\mathrm{Bi}$ and its derivative. The new formula is simpler than the one proposed by Lee involving $_2F_3$ and $_3F_4$ hypergeometric functions.
翻译:Holtsmark分布在等离子物理、光谱线形状(例如光谱线形状)所涉及的电微场分布以及引力体分布的天体物理学中都有应用,这是已知概率密度函数闭式表达的少数稳定分布实例之一,但后者不能以基本功能表示。在目前的工作中,我们提到,Holtsmarks概率密度函数可以用超测函数$_2F_2美元和第二类气态函数$\mathrm{Bi}及其衍生值表示。新公式比Lee提议的2F_3美元和$3F_4美元超测函数简单。