Mittag-Leffler Functions
========================


If the Gamma function is the generalization of the factorial, the Mittag-Leffler function is the generalization of the exponential function. It is the "crown jewel" of fractional calculus.

Definition and Origin
---------------------

The Mittag-Leffler function :math:`E_{\alpha, \beta}(z)` is defined by the following power series
for :math:`\alpha > 0`:


.. math::

   E_{\alpha, \beta}(z) = \sum_{k=0}^{\infty} \frac{z^k}{\Gamma(\alpha k + \beta)}


* When :math:`\alpha = 1, \beta = 1`, it becomes the standard exponential: :math:`E_{1,1}(z) = e^z`.

* When :math:`\alpha = 2, \beta = 1`, it describes hyperbolic cosines: :math:`E_{2,1}(z^2) = \cosh(z)`.



Physical Significance: Fractional Calculus
------------------------------------------

While the standard exponential function describes "normal" relaxation
(like a cooling cup of coffee), the Mittag-Leffler function describes
"anomalous" relaxation.

* Viscoelasticity: Used to model materials that are halfway between a liquid and a solid (like polymers or human tissue).
* Fractional Diffusion: Describes how particles move in crowded environments (like proteins moving inside a biological cell).

3. Implementation in SepalSolver
--------------------------------

Unlike other scientific computing tools like matlab, in sepalsolver, the Mittag-Leffler function is
not part of the sepcial function library. And it is exposed in the SepalSolver.Math class. 


.. code-block:: csharp

   //Plotting E_{ a, 1}  (z) for varying alpha
   ColVec z = Linspace(0, 4);
   double[] alp = [0.5, 0.8, 1.0, 1.2];
   Matrix Y = alp.Select(a => Arrayfun(x => MettagLeffler(a, 1, x), z)).ToList();
   Plot(z, Y);
   Legend(alp.Select(a=>"E_{" + a +",1}(x)"));
   Title("Mittag-Leffler Function E_{\alpha, 1}(z)");

Key Properties
--------------

* Interpolation: It interpolates between a pure exponential and a power-law function.
* Laplace Transform: The Laplace transform of :math:E_{\alpha}( -at^\alpha ) is :math:`\frac{s^{\alpha-1}}{s^\alpha + a}`, which is vital for solving fractional differential equations.