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Expression of the Holtsmark function in terms of hypergeometric $_2F_2$ and Airy $\mathrm{Bi}$ functions

Jean-Christophe Pain

Abstract

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.

Expression of the Holtsmark function in terms of hypergeometric $_2F_2$ and Airy $\mathrm{Bi}$ functions

Abstract

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 and of Airy function of the second kind and its derivative. The new formula is simpler than the one proposed by Lee involving and hypergeometric functions.

Paper Structure

This paper contains 3 sections, 36 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Analytical expressions of $S(\beta)$ and $H(\beta)$
  3. Conclusion

Figures (2)

  • Figure 1: Expansion (\ref{['smallarg']}) for different orders: 4 (red curve), 16 (green curve) and 64 (blue curve) compared to the exact probability density function $S(\beta)$ (black curve).
  • Figure 2: Expansion (\ref{['bigarg']}) for different orders: 4 (red curve), 16 (green curve) and 64 (blue curve) compared to the exact probability density function $S(\beta)$ (black curve).