Exponential integral

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

<math> \mbox{Ei}(x)=-\int_{-x}^{\infty} \frac{e^{-t}}{t}\, dt\,.</math>

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:

<math>\mbox{Ei}(x) = \gamma+\ln x+

 \sum_{k=1}^{\infty} \frac{x^k}{k\; k!} \,,</math>

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:

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

This function may be regarded as extending the exponential integral to the negative reals by

We can express both of them in terms of an entire function,

<math>{\rm Ein}(x) = \int_0^x (1-e^{-t})\frac{dt}{t}

= \sum_{k=1}^\infty \frac{(-1)^{k+1}x^k}{k\; k!}</math>.

Using this function, we then may define, using the logarithm,

<math>{\rm E}_1(x) \,=\, -\gamma-\ln x + {\rm Ein}(x)</math>

and

<math>{\rm Ei}(x) \,=\, \gamma+\ln x - {\rm Ein}(-x).</math>

The exponential integral may also be generalized to

<math>E_n(x) = \int_1^\infty \frac{e^{-xt}}{t^n}\, dt.</math>

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)