A general nth-order transfer function (TF) is derived, whose time-domain response approximates that of an integrator in an optimal manner, optimality being defined by maximal flatness of impulse response at t = 0⁺. It is shown that transformerless, passive, unbalanced realisability is guaranteed for n < 3, but for n > 3, the TF is unstable. For n = 3, the TF violates a passive realisability constraint; near optimum results can, however, be obtained, in this case only, by a small perturbation of the pole locations. Optimum transfer functions are also derived for the additional constraint of inductorless realisability. In this case, it is shown that TFs with n ≥ 2 are not realisable. For all n, however, near optimum results can be achieved by small perturbations of the pole locations; this is illustrated in the paper for n = 2. Network realisations are given for the following cases: n = 2, optimal, RLC; n = 3, suboptimal, RLC; and n = 2, suboptimal, RC.