Cohomology of regular differential forms for affine curves

Let C be a complex affine reduced curve, and denote by H¹(C) its first truncated cohomology group, i.e. the quotient of all regular differential 1-forms by exact 1-forms. First we introduce a nonnegative invariant μ^′(C,x) that measures the complexity of the singularity of C at the point x, and we establish the following formula: