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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 $-1.09\pm 4.20i$.

33E12Mittag-Leffler functions and generalizations
Full Text: EuDML