 Initial mass function

The initial mass function specifies the distribution of stellar masses at birth. Writing $\xi (M)=dN/dM$ means that $\xi (M)dM$ counts the expected number born in a small mass interval. Counts per logarithmic interval differ: $dN/d\ln M=M\xi (M)$. Confusing these conventions changes the reported slope even for the same population.

For a chosen finite birth-mass range, $\int \xi (M)dM$ gives the number of stars and $\int M\xi (M)dM$ their total mass. The mass and luminosity weighting of a stellar initial mass function explains why a small number of luminous massive stars can dominate the light without dominating the count. A present-day distribution also reflects deaths, continued star formation, unresolved companions and observational selection, so it is not automatically the birth distribution.

The Initial Mass Function (IMF) is a crucial concept in astrophysics that describes the distribution of masses for a population of stars when they form. It provides a statistical representation of how many stars are born within a certain mass range in a stellar population, essentially outlining the relationship between the number of stars and their masses at the time of formation.