If is a zero of multiplicity , then and as , hence and . Thus, the residue of at is the multiplicity of (as a zero), i.e. the "number of zeroes at ", while the residue of is , i.e. the "sum of zeroes at " (counting multiplicity).

Thus, by the theorem of residues, the first integral is the number of zeroes inside the curve (counting multiplicity) and the second integral is the sum of these zeroes (counting multiplicity). With you would get the sum of their squares, and so on.