In probability theory, the Holtsmark distribution is a probability distribution named after Johan Peter Holtsmark, In 1919.
It's especially used in astrophysics for modeling gravitational bodies.

This distribution is a special case of a stable distribution with shape parameter α = 3/2 and skewness parameter β = 0.

Such a distribution with β = 0 is called symmetric alpha-stable distribution.

• α = 2: Normal distribution
• α = 1: Cauchy distribution

The Holtsmark distribution, like these distribution, has a closed-from expression, but it can't be expressed in terms of elementary function.

The Holtsmark distribution, generalized to a stable distribution by introducing position and scale parameters, is as follows:

Since scaling and translations allow for standardization, standard parameters are discussed here.

Using hypergeometric function, it's obtained as follows:

The following series expression can also be used:

When x is large, the following equation can be used as an asymptotic expression:

Evaluating only one term of this equation yields the tail power of the distribution:

PDF Precision 150
CDF Precision 150
Quantile Precision 142

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ResearchGate
TechRxiv

J.Holtsmark, "Uber die Verbreiterung von Spektrallinien" (1919)
J.C.Pain "Expression of the Holtsmark function in terms of hypergeometric 2F2 and Airy Bi functions" (2010)

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