We derive a semi-empirical galactic initial mass function ( IMF ) from observational constraints .
We assume that the star formation rate in a galaxy can be expressed as the product of the IMF , \psi ( m ) , which is a smooth function of mass m ( in units of M _ { \odot } ) , and a time- and space-dependent total rate of star formation per unit area of galactic disk , \dot { \varsigma } _ { *T } .
The mass dependence of the proposed IMF is determined by five parameters : the low-mass slope \gamma , the high-mass slope - \Gamma ( taken to be the Salpeter value , -1.35 ) , the characteristic mass m _ { ch } ( which is close to the mass m _ { peak } at which the IMF turns over ) , and the lower and upper limits on the mass , m _ { \ell } ( taken to be 0.004 ) and m _ { u } ( taken to be 120 ) .
The star formation rate in terms of number of stars per unit area of galactic disk per unit logarithmic mass interval , is proportional to the IMF :

\dot { \varsigma } _ { * } ( m ) \equiv \frac { d ^ { 2 } \dot { \cal N } _ { * } ( m ) } { dA d \ln m } \equiv% \dot { \varsigma } _ { *T } \psi ( m ) = C \dot { \varsigma } _ { *T } m ^ { - \Gamma } \left \ { 1 - \exp% \left [ { - ( m / m _ { ch } ) ^ { \gamma + \Gamma } } \right ] \right \ } , where \cal N _ { * } is the number of stars , m _ { \ell } < m < m _ { u } is the range of stellar masses .
The values of \gamma and m _ { ch } are derived from two integral constraints : i ) the ratio of the number density of stars in the range m = 0.1 - 0.6 to that in the range m = 0.6 - 0.8 as inferred from the mass distribution of field stars in the local neighborhood , and ii ) the ratio of the number of stars in the range m = 0.08 - 1 to the number of brown dwarfs in the range m = 0.03 - 0.08 in young clusters .
The IMF satisfying the above constraints is characterized by the parameters \gamma = 0.51 and m _ { ch } = 0.35 ( which corresponds to m _ { peak } = 0.27 ) .
This IMF agrees quite well with the Chabrier ( 2005 ) IMF for the entire mass range over which we have compared with data , but predicts significantly more stars with masses < 0.03 M _ { \odot } ; we also compare with other IMFs in current use .
We give a number of important parameters implied by the IMF , such as the fractional number of brown dwarfs and high-mass stars formed at a given time , the average mass of a newly-formed star , and the mass of stars formed per high-mass star .