Complex variable and regularization methods of inversion of the Laplace transform

In this paper three methods are derived for approximating f, given its Laplace transform g on $(0,\infty )$, i.e., $\smallint_0^\infty {f(t)\exp ( - st)\,dt = g(s)}$. Assuming that $g \in {L^2}(0,\infty )$, the first method is based on a Sinc-like rational approximation of g, the second on a Sinc solution of the integral equation $\smallint_0^\infty {f(t)\exp ( - st)\,dt = g(s)}$ via standard regularization, and the third method is based on first converting $\smallint_0^\infty {f(t)\exp ( - st){\mkern 1mu} dt = g(s)}$ to a convolution integral over $\mathbb{R}$, and then finding a Sinc approximation to f via the application of a special regularization procedure to solve the Fourier transform problem. We also obtain bounds on the error of approximation, which depend on both the method of approximation and the regularization parameter.