exponential integral

The antiderivative of the functionMathworldPlanetmath


is not expressible in closed form.  Thus such integralsDlmfPlanetmath (http://planetmath.org/ImproperIntegral) as


define certain non-elementary (http://planetmath.org/ElementaryFunction) transcendental functionsMathworldPlanetmath.  They are called exponential integralsDlmfDlmfDlmfMathworldPlanetmath and denoted usually E1 and Ei, respectively.  Accordingly,


Then one has the connection


For positive values of x the series expansion


where γ is the http://planetmath.org/node/1883Euler–Mascheroni constant, is valid.

Note: Some authors use the convention  Eix:=xe-tt𝑑t.

0.1 Laplace transform of 1t+a

By the definition of Laplace transformDlmfMathworldPlanetmath,


The substitution (http://planetmath.org/ChangeOfVariableInDefiniteIntegral)  t+a=u  gives


from which the substitution  su=t  yields



{1t+a}=easE1(as). (1)

Using the rule (http://planetmath.org/LaplaceTransformOfDerivative)  {f(t)}=sF(s)-f(0),  one easily derives from (1) the

{1(t+a)2}=1a-seasE1(as). (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