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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 $E_{a,b}(z)=\sum_{n=0}^\infty \frac{z^n}{\Gamma(a n+b)},\qquad a>0,\;b\in C, \;z\in C;$
when $$b=1$$ this reduces to the ordinary Mittag-Leffler fuction $$E_{a,1}(z)\equiv E_a(z)$$. The authors point out that the solution of the eigenvalue equation $$D_{0+}^{\alpha,\beta}\,f(x)=\lambda f(x)$$, where $$D$$ denotes the Riemann-Liouville fractional derivative, is solved by $f(x)=x^{(1-\beta)(\alpha-1)}E_{\alpha,\alpha+\beta(1-\alpha)}(\lambda x^\alpha).$ 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
$E_{a,b}(z)=\frac{1}{2\pi i}\int_C\frac{s^{a-b}e^s}{s^a-z}\,ds,$
where $$C$$ is a path that lies outside the disc $$|s|\leq |z|^{1/a}$$. The portion of the $$z$$ plane studied is $$-8\leq\text{Re}(z)\leq 5$$, $$-10\leq\text{Im}(z)\leq 10$$ which is divided into a grid consisting of $$801\times 481$$ 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 $$-1.09\pm 4.20i$$.

##### MSC:
 3.3e+13 Mittag-Leffler functions and generalizations
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