From the calculation of inversion of the Laplace transform without contour integration in the complex plane, the objective is to express the Mittag-Leffler function, suitable for problems arising from the fractional calculus as an improper integral. As a result, a variety of convergent improper integrals of functions in terms of trigonometric functions can be expressed through the Mittag-Leffler functions. Still using this method of inversion of the Laplace transform without contour integration, intended to obtain the inverse Laplace transform for the non-tabulated functions and in addition to developing an alternative method for the find the analytical solution of the ordinary or partial differential equation with the boundary conditions and/or initial conditions when we use the Laplace transform in resolution for the differential equation. This methodology is used, for example, to obtain information on the structure and molecular dynamics of physical, chemical, and biological systems that occurs in a luminescence decay law.
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