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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 \(-1.09\pm 4.20i\).

33E12 Mittag-Leffler functions and generalizations
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