A function (symbol Γ) which extends the notion of factorial n (written n!) from positive whole numbers to real and complex variables, given by Γ(z) = ∫∞ 0 t z − 1 e − t dt.
The function has the properties that Γ(z) = (z − 1)!, and Γ(z + 1) = zΓ(z), and is undefined for values of z that are negative whole numbers or zero. The "incomplete gamma function" is obtained by varying one or other of the limits of integration in the defining equation.
Mid 19th century; earliest use found in Reports of the British Association for the Advancement of Science.