The exponential function is solution of a linear differential equation with constant coefficients, and the Mittag-Leffler function is solution of a fractional linear differential equation with constant coefficients. Using infinite series and Laplace transform, we introduce the Mittag-Leffler function as a generalization of the exponential function. Particular cases are recovered. © 2013 Taylor & Francis.454595604Ablowitz, M.J., Fokas, A.S., (1999) Complex variables: Introduction and applications (Cambridge texts in applied mathematics), , Cambridge (UK),: Cambridge University PressSneddon, I.N., (1995) Fourier transforms, , New York (NY),: Dover Publications IncCoddington, E.A., (1989) An introduction to ordinary differential equations, , New York (NY),: Dover Publications IncKevorkian, J., (1990) Partial differential equations, analytical solution techniques, , Belmont (CA),: Wadsworth & Brooks/Cole Advanced Book & SoftwareCamargo, R.F., Chiacchio, A.O., Oliveira, E.C., One-sided and two-sided Green's function (2013) Bound Value Probl, 2013, p. 45Miller, K.S., Ross, B., (1993) An introduction to the fractional calculus and fractional differential equations, , New York (NY),: WileyMittag-Leffler, G.M., A generalization of the Laplace-Abel integral (1903) C R Acad Sci Paris, 137, pp. 537-539Prabhakar, T.R., A singular integral equation with generalized Mittag-Leffler function in the kernel (1971) Yokohama Math J, 19, pp. 7-15Mainardi, F., (2010) Fractional calculus and waves in linear viscoelasticity, , London,: Imperial College PressCapelas de Oliveira, E., (2012) Special functions and applications, , 2nd, São Paulo,: Livraria Editora da Física, PortugueseMathai, A.M., Haubold, H.J., (2008) Special functions for applied scientists, , New York (NY),: Springer ScienceOliveira, E.C., Rodrigues Jr., W.A., (2006) Analytical functions and applications, , São Paulo,: Livraria Editora da Física, PortugueseMittag-Leffler, G.M., On the new function Eα(x) (1903) C R Acad Sci Paris, 137, pp. 554-558Humbert, P., Agarwal, R.P., On the Mittag-Leffler function and some of its generalizations (1953) Bull Sci Math Ser, 77, pp. 180-185Shukla, A.K., Prajapati, J.C., On a generalization of Mittag-Leffler function and its properties (2007) J Math Anal Appl, 336, pp. 797-811Gehlot, K.S., The generalized k-Mittag-Leffler function (2012) Int J Contemp Math Sci, 7, pp. 2213-2219Silva Costa, F., (2011) Fox's H function and applications, , Campinas: Unicamp, Portuguese