Luminosity of a collider with asymmetric beams

A formula for the collider luminosity is derived for a head-on collision of two beams with different parameters. The formula is valid for colliding and partially separated ("merging") beams. Three particular cases are presented: the collision of two identical axially symmetric bunches, the collision of a bunch with a disbanded beam, and the collision of two disbanded bunches. The colliding beams have coinciding longitudinal axes. The formula is valid for colliding both counter propagating and co-propagating beams.