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mittable
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There are 8 letters in MITTABLE ( A1B3E1I1L1M3T1 )
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In mathematics, the Mittag-Leffler function Eα,β is a special function, a complex function which depends on two complex parameters α and β. It may be defined by the following series when the real part of α is strictly positive:* * * * * 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 )}}.} * where * * * * Γ * * * {\displaystyle \Gamma } * is the Gamma function . * In the case α and β are real and positive, the series converges for all values of the argument z, so the Mittag-Leffler function is an entire function. This function is named after Gösta Mittag-Leffler. This class of functions are important in the theory of the fractional calculus. * For α > 0, the Mittag-Leffler function Eα,1 is an entire function of order 1/α, and is in some sense the simplest entire function of its order. * The Mittag-Leffler function satisfies the recurrence property * * * * * * E * * α * , * β * * * ( * z * ) * = * * * 1 * z * * * * E * * α * , * β * − * α * * * ( * z * ) * − * * * 1 * * z * Γ * ( * β * − * α * ) * , * * * * * * {\displaystyle E_{\alpha ,\beta }(z)={\frac {1}{z}}E_{\alpha ,\beta -\alpha }(z)-{\frac {1}{z\Gamma (\beta -\alpha ),}}} * from which the Poincaré asymptotic expansion * * * * * * E * * α * , * β * * * ( * z * ) * ∼ * − * * ∑ * * k * = * 1 * * * * * 1 * * * z * * k * * * Γ * ( * β * − * k * α * ) * * * * * * {\displaystyle E_{\alpha ,\beta }(z)\sim -\sum _{k=1}{\frac {1}{z^{k}\Gamma (\beta -k\alpha )}}} * follows, wh... |