  Generating an initial condition for a Langevin equation with memory is a non
trivial issue. We introduce a generalisation of the Laplace transform as a
useful tool for solving this problem, in which a limit procedure may send the
extension of memory effects to arbitrary times in the past. This method allows
us to compute average position, work, their variances and the entropy
production rate of a particle dragged in a complex fluid by an harmonic
potential, which could represent the effect of moving optical tweezers. For
initial conditions in equilibrium we generalise the results by van Zon and
Cohen, finding the variance of the work for generic protocols of the trap. In
addition, we study a particle dragged for a long time captured in an optical
trap with constant velocity in a steady state. Our formulas open the door to
thermodynamic uncertainty relations in systems with memory.
