This paper considers the solution of the equations for ruin probabilities
in infinite continuous time. Using the Fourier Transform and certain results
from the theory of complex functions, these solutions are obtained as com-
plex integrals in a form which may be evaluated numerically by means of
the inverse Fourier Transform. In addition the relationship between the re-
sults obtained for the continuous time cases, and those in the literature, are
compared. Closed form ruin probabilities for the heavy tailed distributions:
mixed exponential; Gamma (including Erlang); Lognormal; Weillbull; and
Pareto, are derived as a result (or computed to any degree of accuracy, and
without the use of simulations).