Exponential integral
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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 arises also in the following sum:
- <math>\sum_{k=1}^{\infty} \frac{x^k}{k k!} = \mbox{Ei}(x)+\gamma+\ln x\,,<math>
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:
- <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 <math>{\rm Ei}(-x) = - {\rm E}_1(x)<math>. 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} 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) (http://www.math.sfu.ca/~cbm/aands/page_227.htm)