Let \(K\) be a field of characteristic 0, \(\overline K\) an algebraic closure of \(K\) and \(K(x)\) the rational function field in one variable \(x\) over \(K\). The paper describes an algorithm that computes for an element \(f\) of \(K(x)\) for each \(n\in\mathbb N\) and \(m\in\{1,\dots,n\}\) a univariate polynomial \(B_{nm}\) over \(K\) whose roots in \(\overline K\) are exactly the coefficients of \((x-\alpha)^{-m}\) in the Laurent expansions of \(f\) at its poles \(\alpha\in\overline K\) of order \(n\). The algorithm uses only rational operations in the field of coefficients of \(f\) and makes it possible to compute the principal parts of all the poles simultaneously. It yields a generalization of residue formulas used in symbolic integration [see e.g. \textit{M. Rothstein}, Proc. 1977 MACSYMA Users' Conference, 263--274] and also an improved version of the necessary conditions for the various cases of \textit{J. J. Kovacic}'s algorithm [J. Symb. Comput. 2, 3--43 (1986; Zbl 0603.68035)].