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.
Some basic properties For α > 0 {\displaystyle \alpha >0} , the Mittag-Leffler function E α , β ( z ) {\displaystyle E_{\alpha ,\beta }(z)} is an entire function of order 1 / α {\displaystyle 1/\alpha } , and type 1 {\displaystyle 1} for any value of β {\displaystyle \beta } . In some sense, the Mittag-Leffler function is the simplest entire function of its order. The indicator function of E α ( z ) {\displaystyle E_{\alpha }(z)} is
h E α ( θ ) = { cos ( θ α ) , for | θ | ≤ 1 2 α π ; 0 , otherwise . {\displaystyle h_{E_{\alpha }}(\theta )={\begin{cases}\cos \left({\frac {\theta }{\alpha }}\right),&{\text{for }}|\theta |\leq {\frac {1}{2}}\alpha \pi ;\\0,&{\text{otherwise}}.\end{cases}}}
This result actually holds for β ≠ 1 {\displaystyle \beta \neq 1} as well with some restrictions on β {\displaystyle \beta } when α = 1 {\displaystyle \alpha =1} . The Mittag-Leffler function satisfies the recurrence property (Theorem 5.1 of )
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