 Such a function is given by inverting the differential operator by integration; for example we can define

where 

is essentially the incomplete gamma function.The integral converges whenever "g" has a zero at the cusp "i"∞, and the incomplete gamma function can be extended by analytic continuation, so this formula can be used to define the holomorphic part "g"* of "F" even in the case when "g" is meromorphic at "i"∞, though this requires some care if "k" is 1 or not integral or if "n" = 0.The inverse of the differential operator is far from unique as we can add any homomorphic function to "g"* without affecting its image, and as a result the function "g"* need not be invariant under the group Γ.