In the last few decades, accurate and efficient algorithms for the inversion of Laplace transforms have been developed, which are useful to study prohabilistic models for which explicit solutions are not available. The object of this paper is to specify technical conditions under which the Euler inversion algorithm can be extended to functions defined on the entire real line and compute bounds for corresponding discretization errors. An extension of the Euler algorithm to the entire real line makes it appealing for financial applications since it is often to derive Laplace transforms in the logarithm of a derivatives strike which can be inverted using Euler algorithm. Some applications of this extended Euler algorithm to the pricing of plain vanilla and path independent options are mentioned to highlight the accuracy of the method.