Rational Approximation for Oscillatory Mittag-Leffler Function

Mittag-Leffler function E_ and its generalizations are indispensable in fractional calculus. When 12, its behavior is so diverse since it could be nonmonotone with oscillatory profile and varying odd number of real roots. Due to these properties, it is a challenging task to approximate this function in this range of. In this paper, we develop a rational approximation for E_ (-z), z0, that captures the oscillatory behavior and the roots over extended intervals. The approximation is based on decomposing E_ into a combination of a two-parameter rootless Mittag-Leffler function and a polynomial. This type of approximations provides efficient implementations when numerically solving fractional oscillation equations.