and f is a holomorphic function defined (at least) on D. The residue Res(f, c) of f at c is the coefficient a−1 of in the Laurent series expansion of f around c. Various methods exist for calculating this value, and the choice of which method to use depends on the function in question, and on the nature of the singularity. According to the residue theorem, we have: where γ traces out a circle around c in a counterclockwise manner. We may choose the path γ to be a circle of radius ε around c, where ε is as small as we desire. This may be used for calculation in cases where the integral can be calculated directly, but it is usually the case that residues are used to simplify calculation of integrals, and not the other way around. Removable singularities If the function f can be continued to a holomorphic function on the whole disk , then Res(f, c) = 0. The converse is not generally true. Simple poles At a simple pole c, the residue of f is given by: If that limit does not exist, there is an essential singularity there. If it is 0 then it is either analytic there or there is a removable singularity. If it is equal to infinity then the order is higher than 1. It may be that the function f can be expressed as a quotient of two functions, , where g and h are holomorphic functions in a neighbourhood of c, with h(c) = 0 and h'''(c) ≠ 0. In such a case, L'Hôpital's rule can be used to simplify the above formula to: Limit formula for higher-order poles More generally, if c is a pole of order n, then the residue of f around z = c'' can be found by the formula: This formula can be very useful in determining the residues for low-order poles. For higher-order poles, the calculations can become unmanageable, and series expansion is usually easier. For essential singularities, no such simple formula exists, and residues must usually be taken directly from series expansions. Residue at infinity In general, the residue at infinity is defined as: If the following condition is met: then the residue at infinity can be computed using the following formula: If instead then the residue at infinity is For holomorphic functions the sum of the residues at the isolated singularities