## Exponential Integral

Let be the En-Function with ,

 (1)

Then define the exponential integral by
 (2)

where the retention of the Notation is a historical artifact. Then is given by the integral
 (3)

This function is given by the Mathematica (Wolfram Research, Champaign, IL) function ExpIntegralEi[x]. The exponential integral can also be written
 (4)

where and 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 is known as the Gompertz Constant.

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() and Related Functions.'' Ch. 37 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 351-360, 1987.