## The Mittag-Leffler function

Version 1.3.0.0 (11.7 KB) by
Evaluation of the Mittag-Leffler function with 1, 2 or 3 parameters

Updated 7 Dec 2015

Evaluation of the Mittag-Leffler (ML) function with 1, 2 or 3 parameters by means of the OPC algorithm . 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  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:
 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

### Cite As

Roberto Garrappa (2023). The Mittag-Leffler function (https://www.mathworks.com/matlabcentral/fileexchange/48154-the-mittag-leffler-function), MATLAB Central File Exchange. Retrieved .

##### MATLAB Release Compatibility
Created with R2009b
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Version Published Release Notes
1.3.0.0

References updated
Description improved

1.2.0.0

References updated
Some bugs fixed and accuracy improved in some cases

1.1.0.0

Just the description of the code has been updated

1.0.0.0