where 
is a dimensionless number and is proportional to the energy loss, and c is any real positive number. Other expressions and formulae are indicated in [Kölbig83]
(with program implementations) and [Leo94]; e.g.
Programs also exist for the distribution (integral function) of 
, for its derivative, and for the first two moments.
[Moyal55] has given a closed analytic form
with
This curve reproduces the gross asymmetric features of the Landau distribution and avoids the pitfalls of numerical integration;
it is, however, too low in the tail and unrelated to as defined above.
On the other hand, Moyal's curve is useful in some situations:
as the layer over which energy loss is integrated becomes thicker, the tail of the Landau distribution (according to the central limit theorem of statistics) has a tendency to diminish:
the Landau distribution becomes the more general Vavilov distribution . Vavilov introduced the additional parameter 
, with 
the average energy loss over the layer, and 
the maximum energy loss in a single collision.
The Landau distribution is the limiting case for k = 0 and a good approximation for k<0.01; the Gaussian is the limiting case for 
and a reasonable approximation for k>10.
For more details, see [Leo94], [Schorr74], the latter with programs.