# exponential integral

The antiderivative of the function

 $x\mapsto\frac{e^{-x}}{x}$

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

 $\int_{x}^{\infty}\!\frac{e^{-t}}{t}\,dt\quad\mbox{and}\quad\int_{\infty}^{-x}% \!\frac{e^{-t}}{t}\,dt,$

define certain non-elementary (http://planetmath.org/ElementaryFunction) transcendental functions.  They are called exponential integrals and denoted usually ${\rm E}_{1}$ and ${\rm Ei}$, respectively.  Accordingly,

 ${\rm E}_{1}(x)\;:=\;\int_{x}^{\infty}\!\frac{e^{-t}}{t}\,dt$
 ${\rm Ei}\,x\;:=\;\int_{\infty}^{-x}\!\frac{e^{-t}}{t}\,dt\;=\;-\int_{-x}^{% \infty}\!\frac{e^{-t}}{t}\,dt\;:=\;\int_{-\infty}^{x}\!\frac{e^{-u}}{u}\,du.$

Then one has the connection

 ${\rm E}_{1}(x)\;=\;-{\rm Ei}\,(-x).$

For positive values of $x$ the series expansion

 ${\rm Ei}\,x\;=\;\gamma+\ln{x}+\sum_{j=1}^{\infty}\frac{x^{j}}{j!j},$

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

Note: Some authors use the convention  ${\rm Ei}\,x\,:=\,\int_{x}^{\infty}\!\frac{e^{-t}}{t}\,dt$.

## 0.1 Laplace transform of $\frac{1}{t+a}$

By the definition of Laplace transform,

 $\mathcal{L}\{\frac{1}{t\!+\!a}\}\;=\;\int_{0}^{\infty}\frac{e^{-st}}{t\!+\!a}% \,dt.$

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

 $\mathcal{L}\{\frac{1}{t\!+\!a}\}\;=\;\int_{a}^{\infty}\frac{e^{as-su}}{u}\,du% \;=\;e^{as}\int_{a}^{\infty}\frac{e^{-su}}{u}\,du,$

from which the substitution  $su=t$  yields

 $\mathcal{L}\{\frac{1}{t\!+\!a}\}\;=\;e^{as}\int_{as}^{\infty}\frac{e^{-t}}{t}% \,dt,$

i.e.

 $\displaystyle\mathcal{L}\{\frac{1}{t\!+\!a}\}\;=\;e^{as}{\rm E}_{1}(as).$ (1)

Using the rule (http://planetmath.org/LaplaceTransformOfDerivative)  $\mathcal{L}\{f^{\prime}(t)\}=sF(s)\!-\!f(0)$,  one easily derives from (1) the

 $\displaystyle\mathcal{L}\{\frac{1}{(t\!+\!a)^{2}}\}\;=\;\frac{1}{a}\!-\!se^{as% }{\rm E}_{1}(as).$ (2)
Title exponential integral ExponentialIntegral 2013-03-22 18:44:17 2013-03-22 18:44:17 pahio (2872) pahio (2872) 8 pahio (2872) Definition msc 30A99 msc 26A36 Ei LogarithmicIntegral TableOfLaplaceTransforms IndexOfSpecialFunctions