)So a harmonic weak Maass form is annihilated by the differential operator

If "F" is any harmonic weak Maass form then the function "g" given by

is holomorphic and transforms like a modular form of weight "k", though it may not be holomorphic at cusps.If we can find any other function "g"* with the same image "g", then "F" − "g"* will be holomorphic.