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For binary stars, the mass function (or binary mass function) is a formula of observable values that yields constraints on the masses of the two stars, usable if their radial velocities can be determined through Doppler shifts.
P ——— V³ 2πG
This value is equal to the following constraint on the masses:
m³ —————— sin³i (M+m)²
The inclination is generally presumed to be ~90° if it also transits, and otherwise may be roughly approximated by the inclination that produces the mean value of sin3i (best if also excluding possible inclinations that would yield an insignificant V and those that would presumably result in an observed transit: ~24° without such exclusions, and closer to 42° with them). For studies of entire groups of stars, useful statistics may be derived assuming the inclinations are random.
The phrase, mass function is also used in probability, for an incompatible concept: a probability mass function (PMF), which is a function yielding the probability that some discrete random variable takes on a given specific value.
In astrophysics, the term mass function, when qualified, is also widely used (perhaps its most common use) for a function that yields the distribution of masses of some type of astronomical object. In some cases, typically indicated by the adjective initial, it refers specifically to their mass at the time of creation, the relationship between such a function and one regarding observation-data depends upon the lifetimes of the objects. These mass functions can be considered un-normalized probability density functions (i.e., not PMFs). Examples include the initial mass function (IMF, stars), the conditional stellar mass function (CSMF), the dense core mass function (DCMF), the halo mass function (HMF), and the black hole mass function (BHMF).