The Wright Function: Its Properties, Applications, and Numerical Evaluation.

In this paper, some elements of the theory of the Wright function φ are discussed. The Wright function-along with the Mittag-Leffler function-plays a prominent role in the theory of the partial differential equations of the fractional order that are actively used nowadays for modeling of many phenomena including e.g. the anomalous diffusion processes or in the theory of the complex systems. This function appears there simultaneously as a Green function in the initial-value problems for the model linear equations with the constant coefficients and as a special solution invariant under the groups of the scaling transformations of the fractional differential equations. In this paper, both of these applications are shortly introduced. Whereas the analytical theory of the Wright function is already more or less well developed, its numerical evaluation is still an area of the active research. In this paper, the numerical evaluation of the Wright function is discussed with a focus on the case of the real axis that is very important for applications. In particular, several approaches are presented including the method of series summation, integral representations, and asymptotical expansions. In different parts of the complex plane different numerical techniques are employed. In each case, estimates for accuracy of the computations are provided. [ABSTRACT FROM AUTHOR]