The Mittag-Leffler function

Evaluation of the Mittag-Leffler function with 1, 2 or 3 parameters

Evaluation of the Mittag-Leffler (ML) function with 1, 2 or 3 parameters by means of the OPC algorithm [1]. The routine evaluates an approximation Et of the ML function E such that |E-Et|/(1+|E|) approx 1.0e-15

E = ML(z,alpha) evaluates the ML function with one parameter alpha for the corresponding elements of z; alpha must be a real and positive scalar. The one parameter ML function is defined as
E = sum_{k=0}^{infty} z^k/Gamma(alpha*k+1)
with Gamma the Euler's gamma function.

E = ML(z,alpha,beta) evaluates the ML function with two parameters alpha and beta for the corresponding elements of z; alpha must be a real and positive scalar and beta a real scalar. The two parameters ML function is defined as

E = sum_{k=0}^{infty} z^k/Gamma(alpha*k+beta)

E = ML(z,alpha,beta,gama) evaluates the ML function with three parameters alpha, beta and gama for the corresponding elements of z; alpha must be a real scalar such that 0<alpha<1, beta any real scalar and gama a real and positive scalar; the arguments z must satisfy |Arg(z)| > alpha*pi. The three parameters ML function is defined as

E = sum_{k=0}^{infty} Gamma(gama+k)*z^k/Gamma(gama)/k!/Gamma(alpha*k+beta)

NOTE: This routine implements the optimal parabolic contour (OPC) algorithm described in [1] and based on the inversion of the Laplace transform on a parabolic contour suitably choosen in one of the regions of analyticity of the Laplace transform.

REFERENCES:
[1] R. Garrappa, Numerical evaluation of two and three parameter Mittag-Leffler functions, SIAM Journal of Numerical Analysis, 2015, 53(3), 1350-1369

Please, report any problem or comment to : roberto dot garrappa at uniba dot it