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 has the series representation:

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

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

${\rm Ei}(-x) = - {\rm E}_1(x).\,$

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,

${\rm E}_1(x) \,=\, -\gamma-\ln x + {\rm Ein}(x)$

and

${\rm Ei}(x) \,=\, \gamma+\ln x - {\rm Ein}(-x).$

The exponential integral may also be generalized to

$E_n(x) = \int_1^\infty \frac{e^{-xt}}{t^n}\, dt$

which is sometimes called Misra function $\varphi_m(x)$ , defined as

$\varphi_m(x)=E_{-m}(x)\,$

[edit] Applications

Time-dependent heat transfer
Nonequilibrium groundwater flow in the Theis solution (called a well function)

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