  The incomplete version of the Macdonald function has various appellations in
literature and earns a well-deserved reputation of being a computational
challenge. This paper ties together the previously disjoint literature and
presents the basic properties of the incomplete Macdonald function, such as
recurrence and differential relations, series and asymptotic expansions. This
paper also shows that the incomplete Macdonald function, as a simple
closed-form expression, is a particular solution to a parabolic partial
differential equation, which arises naturally in a wide variety of transient
and diffusion-related phenomena.
