Expression of the Holtsmark function in terms of hypergeometric $$_{2}F_2$$ and Airy 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 2F2 and of Airy function of the second kind Bi and its derivative. The new formula is simpler than the one proposed by Lee involving 2F3 and 3F4 hypergeometric functions.

Extracting Physical Information from the Voigt Profile Using the Lambert W Function

“…Finally, it is important to keep in mind that properly accounting for the Stark effect requires the convolution of the Voigt profile with an electric-microfield distribution. It would be interesting to repeat the present study with a simple modeling of the latter, for instance, by resorting to the Holtsmark function, for which appropriate analytical formulations may be derived [88,89].…”