MittagLefflerE¶

Mittag-Leffler function.

Two-argument form MittagLefflerE[a, z] is Sum[z^k / Gamma[a*k + 1], {k, 0, Infinity}]; the three-argument form MittagLefflerE[a, b, z] is Sum[z^k / Gamma[a*k + b], {k, 0, Infinity}].

For exact integer a in {0, 1, 2} the two-argument form has closed forms:

$ wo 'MittagLefflerE[0, z]'
(1 - z)^(-1)

$ wo 'MittagLefflerE[1, x]'
E^x

$ wo 'MittagLefflerE[2, z]'
Cosh[Sqrt[z]]

$ wo 'MittagLefflerE[2, 1]'
Cosh[1]

The three-argument form with b == 1 reduces to the two-argument form:

$ wo 'MittagLefflerE[2, 1, z]'
Cosh[Sqrt[z]]

Arbitrary-precision evaluation works through the closed form:

$ wo 'N[MittagLefflerE[2, 1], 20]'
1.5430806348152437784779056207570616826`20.

At z == 0 only the k == 0 term survives, so the two-argument form is 1 and the three-argument form is 1/Gamma[b]:

$ wo 'MittagLefflerE[a, 0]'
1

$ wo 'MittagLefflerE[a, b, 0]'
Gamma[b]^(-1)

The function threads over lists (it is Listable):

$ wo 'MittagLefflerE[2, {1.0, 2.0}]'
{1.5430806348152437, 2.178183556608571}

Symbolic arguments without a closed form stay unevaluated:

$ wo 'MittagLefflerE[a, z]'
MittagLefflerE[a, z]