Exponential Integral

Let ${\rm E}_1(x)$ be the En-Function with $n=1$,

$${\rm E}_1(x) \equiv \int_1^\infty {e^{-tx}\,dt\over t} = \int_x^\infty {e^{-u}\,du\over u}.$$ (1)

Then define the exponential integral $\mathop{\rm ei}\nolimits (x)$ by

$${\rm E}_1(x) = -\mathop{\rm ei}\nolimits (-x),$$ (2)

where the retention of the $-\mathop{\rm ei}\nolimits (-x)$ Notation is a historical artifact. Then $\mathop{\rm ei}\nolimits (x)$ is given by the integral

$$\mathop{\rm ei}\nolimits (x)=-\int_{-x}^\infty {e^{-t}\,dt\over t}.$$ (3)

This function is given by the Mathematica ${}^{\scriptstyle\circledRsymbol}$ (Wolfram Research, Champaign, IL) function ExpIntegralEi[x]. The exponential integral can also be written

$$\mathop{\rm ei}\nolimits (ix) = \mathop{\rm ci}\nolimits (x)+i\mathop{\rm si}\nolimits (x),$$ (4)

where $\mathop{\rm ci}\nolimits (x)$ and $\mathop{\rm si}\nolimits (x)$ are Cosine and Sine Integral.

The real Root of the exponential integral occurs at 0.37250741078..., which is not known to be expressible in terms of other standard constants. The quantity $-e\mathop{\rm ei}\nolimits (-1)=0.596347362\ldots$ is known as the Gompertz Constant.

See also Cosine Integral, En-Function, Gompertz Constant, Sine Integral

References

Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 566-568, 1985.

Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 434-435, 1953.

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. ``Exponential Integrals.'' §6.3 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 215-219, 1992.

Spanier, J. and Oldham, K. B. ``The Exponential Integral Ei($x$) and Related Functions.'' Ch. 37 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 351-360, 1987.