Mittag-Leffler function
Encyclopedia
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:
In this case, 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.
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.
:
Error function
:
Sum of a geometric progression
:
Hyperbolic cosine:
where the contour C starts and ends at −∞ and circles around the singularities and branch points of the integrand.
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Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the Mittag-Leffler function Eα,β is a special function, a complex
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
which depends on two complex parameters α and β. It may be defined by the following series
Series (mathematics)
A series is the sum of the terms of a sequence. Finite sequences and series have defined first and last terms, whereas infinite sequences and series continue indefinitely....
when the real part of α is strictly positive:
In this case, the series converges for all values of the argument z, so the Mittag-Leffler function is an entire function
Entire function
In complex analysis, an entire function, also called an integral function, is a complex-valued function that is holomorphic over the whole complex plane...
. This function is named after Gösta Mittag-Leffler.
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.
Special cases
Exponential functionExponential function
In mathematics, the exponential function is the function ex, where e is the number such that the function ex is its own derivative. The exponential function is used to model a relationship in which a constant change in the independent variable gives the same proportional change In mathematics,...
:
Error function
Error function
In mathematics, the error function is a special function of sigmoid shape which occurs in probability, statistics and partial differential equations...
:
Sum of a geometric progression
Geometric progression
In mathematics, a geometric progression, also known as a geometric sequence, is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed non-zero number called the common ratio. For example, the sequence 2, 6, 18, 54, ... is a geometric progression...
:
Hyperbolic cosine:
Mittag-Leffler's integral representation
where the contour C starts and ends at −∞ and circles around the singularities and branch points of the integrand.
External links
- Mittag-Leffler function on MathWorld
- Mittag-Leffler function: MATLAB code
- Mittag-Leffler and stable random numbers: Continuous-time random walks and stochastic solution of space-time fractional diffusion equations
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