relative of exponential integral

Let $a$ and $b$ be positive numbers.  We want to calculate the value of the improper integral

${\displaystyle {\int }_0^{\mathrm{\infty }}}{\displaystyle \frac{e^{-ax}-e^{-bx}}{x}}𝑑x$ (1)

related to the exponential integral.

The value may be found e.g. by utilising the derivative of the integral

$$I(y):={\int }_0^{\mathrm{\infty }}{e}^{-xy}\cdot \frac{e^{-ax}-e^{-bx}}{x}𝑑x$$

which can be formed by differentiating under the integral sign (http://planetmath.org/DifferentiationUnderIntegralSign):

$I^{\prime }(y)$ $\mathrm{\hspace{0.33em}}={\displaystyle {\int }_0^{\mathrm{\infty }}}{e}^{-xy}(-x){\displaystyle \frac{e^{-ax}-e^{-bx}}{x}}𝑑x$

$\mathrm{\hspace{0.33em}}={\displaystyle {\int }_0^{\mathrm{\infty }}}\left(e^{-(y+b)x}-e^{-(y+a)x}\right)𝑑x$

$\mathrm{\hspace{0.33em}}=\underset{x=0}{\overset{\mathrm{\infty }}{/}}\left({\displaystyle \frac{e^{-(y+b)x}}{-(y+b)}}-{\displaystyle \frac{e^{-(y+a)x}}{-(y+a)}}\right)$

$\mathrm{\hspace{0.33em}}={\displaystyle \frac{1}{y+b}}-{\displaystyle \frac{1}{y+a}}$

Thus,

$$I(y)=\ln (y+b)-\ln (y+a)=\ln \frac{y+b}{y+a},$$

and the integral (1) has the value  $I(0)=\ln {\displaystyle \frac{b}{a}}$.

There is another method via Laplace transforms.  By the table of Laplace transforms, we have

$$ℒ\{e^{-at}-e^{-bt}\}=\frac{1}{s+a}-\frac{1}{s+b}$$

and therefore

$$ℒ\{\frac{e^{-at}-e^{-bt}}{t}\}={\int }_s^{\mathrm{\infty }}\left(\frac{1}{u+a}-\frac{1}{u+b}\right)𝑑u=\underset{u=s}{\overset{\mathrm{\infty }}{/}}\ln \frac{u+a}{u+b}=\ln \frac{s+b}{s+a},$$

i.e.

$${\int }_0^{\mathrm{\infty }}{e}^{-st}\cdot \frac{e^{-at}-e^{-bt}}{t}𝑑t=\ln \frac{s+b}{s+a}.$$

Letting  $s\to 0+$,  this yields the equation

$${\int }_0^{\mathrm{\infty }}\frac{e^{-ax}-e^{-bx}}{x}𝑑x=\ln \frac{b}{a}.$$

Title	relative of exponential integral
Canonical name	RelativeOfExponentialIntegral
Date of creation	2013-03-22 18:44:20
Last modified on	2013-03-22 18:44:20
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	12
Author	pahio (2872)
Entry type	Example
Classification	msc 44A10
Classification	msc 26A36
Related topic	SubstitutionNotation
Related topic	RelativeOfCosineIntegral
Related topic	IntegrationOfLaplaceTransformWithRespectToParameter
Related topic	IntegrationUnderIntegralSign