Mittag-Leffler function

In mathematics, the Mittag-Leffler functions are a family of special functions. They are complex-valued functions of a complex argument z, and moreover depend on one or two complex parameters. The one-parameter Mittag-Leffler function, introduced by Gösta Mittag-Leffler in 1903, can be defined by the Maclaurin series E α ( z ) = ∑ k = 0 ∞ z k Γ ( α k + 1 ) , {\displaystyle E_{\alpha }(z)=\sum _{k=0}^{\infty }{\frac {z^{k}}{\Gamma (\alpha k+1)}},} where Γ ( x ) {\displaystyle \Gamma (x)} is the gamma function, and α {\displaystyle \alpha } is a complex parameter with Re ⁡ ( α ) > 0 {\displaystyle \operatorname {Re} \left(\alpha \right)>0} . The two-parameter Mittag-Leffler function, introduced by Wiman in 1905, is occasionally called the generalized Mittag-Leffler function. It has an additional complex parameter β {\displaystyle \beta } , and may be defined by the series E α , β ( z ) = ∑ k = 0 ∞ z k Γ ( α k + β ) , {\displaystyle E_{\alpha ,\beta }(z)=\sum _{k=0}^{\infty }{\frac {z^{k}}{\Gamma (\alpha k+\beta )}},} When β = 1 {\displaystyle \beta =1} , the one-parameter function E α = E α , 1 {\displaystyle E_{\alpha }=E_{\alpha ,1}} is recovered. In the case α {\displaystyle \alpha } and β {\displaystyle \beta } are real and positive, the series converges for all values of the argument z {\displaystyle z} , so the Mittag-Leffler function is an entire function. This class of functions are important in the theory of the fractional calculus. See below for three-parameter generalizations.