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Numerical results for the generalized Mittag-Leffler function. (English) Zbl 1123.33018

The generalized Mittag-Leffler function is defined by the sum

E a,b (z)= n=0 z n Γ(an+b),a>0,bC,zC;

when b=1 this reduces to the ordinary Mittag-Leffler fuction E a,1 (z)E a (z). The authors point out that the solution of the eigenvalue equation D 0+ α,β f(x)=λf(x), where D denotes the Riemann-Liouville fractional derivative, is solved by

f(x)=x (1-β)(α-1) E α,α+β(1-α) (λx α )·

The aim of this paper is to study numerically the function E a,b (z) in the complex z plane. All calculations are presented for the particular case a=0·8, b=0·9. The authors employ the contour integral representation

E a,b (z)=1 2πi C s a-b e s s a -zds,

where C is a path that lies outside the disc |s||z| 1/a . The portion of the z plane studied is -8Re(z)5, -10Im(z)10 which is divided into a grid consisting of 801×481 points. Three-dimensional plots of the real and imaginary parts of E 0·8,0·9 (z) are presented. A contour plot of the real and imaginary parts is also given, which includes the first pair of complex conjugate zeros of E 0·8,0·9 (z) situated at approximately -1·09±4·20i.

MSC:
33E12Mittag-Leffler functions and generalizations