  A discrete Laplace transform and its inversion formula are obtained by using
a quadrature of the continuous Fourier transform which is given in terms of
Hermite polynomials and its zeros. This approach yields a convergent discrete
formula for the two-sided Laplace transform if the function to be transformed
falls off rapidly to zero and satisfy certain conditions of integrability,
achieving convergence also for singular functions. The inversion formula
becomes a quadrature formula for the Bromwich integral. This procedure also
yields a quadrature formula for the Mellin transform and its corresponding
inversion formula that can be generalized straightforwardly for functions of
several variables.
