Exponential integral

In mathematics, the exponential integral Ei(x) is defined as

$\mbox{Ei}(x)=-\int_{-x}^{\infty} \frac{e^{-t}}{t} dt\,.$

Since 1/t diverges at t=0, the above integral has to be understood in terms of the Cauchy principal value.

The exponential integral arises also in the following sum:

$\sum_{k=1}^{\infty} \frac{x^k}{k k!} = \mbox{Ei}(x)+\gamma+\ln x\,,$

where γ is the Euler gamma constant.

The exponential integral is closely related to the logarithmic integral function li(x),

li(x) = Ei (ln (x)) for all positive real x ≠ 1.

Also closely related is a function which integrates over a different range:

${\rm E}_1(x) = \int_x^\infty \frac{e^{-t}}{t} dt\,.$

This function may be regarded as extending the exponential integral to the negative reals by $Ei( - x) = - E 1 (x)$ . We can express both of them in terms of an entire function,

${\rm Ein}(x) = \int_0^x (1-e^{-t})\frac{dt}{t} = \sum_{k=1}^\infty \frac{(-1)^{k+1}x^k}{k k!}$ .

Using this function, we then may define, using the logarithm, $E 1 (x) = - γ - ln x + Ein(x)$ and $Ei(x) = γ + ln x + Ein( - x)$ .

The exponential integral may also be generalized to $E_n(x) = \int_1^\infty \frac{e^{-xt}}{t} dt$ .

References

Milton Abramowitz and Irene A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972. (See Chapter 5)

Exponential integral

References

See also: Exponential integral, Cauchy principal value, Entire function, Euler-Mascheroni gamma constant, Handbook of Mathematical Functions, Logarithmic integral function, Mathematics