exponential integral
The antiderivative of the function
is not expressible in closed form. Thus such integrals (http://planetmath.org/ImproperIntegral) as
define certain non-elementary (http://planetmath.org/ElementaryFunction) transcendental functions. They are called exponential integrals and denoted usually and , respectively. Accordingly,
Then one has the connection
For positive values of the series expansion
where is the http://planetmath.org/node/1883Euler–Mascheroni constant, is valid.
Note: Some authors use the convention .
0.1 Laplace transform of
By the definition of Laplace transform,
The substitution (http://planetmath.org/ChangeOfVariableInDefiniteIntegral) gives
from which the substitution yields
i.e.
(1) |
Using the rule (http://planetmath.org/LaplaceTransformOfDerivative) , one easily derives from (1) the
(2) |
Title | exponential integral |
---|---|
Canonical name | ExponentialIntegral |
Date of creation | 2013-03-22 18:44:17 |
Last modified on | 2013-03-22 18:44:17 |
Owner | pahio (2872) |
Last modified by | pahio (2872) |
Numerical id | 8 |
Author | pahio (2872) |
Entry type | Definition |
Classification | msc 30A99 |
Classification | msc 26A36 |
Synonym | Ei |
Related topic | LogarithmicIntegral |
Related topic | TableOfLaplaceTransforms |
Related topic | IndexOfSpecialFunctions |