Exponential Integral

Min   Max       Re       Im

Let  be the En-function with ,

			(1)
			(2)

Then define the exponential integral  by

(3)

where the retention of the  notation is a historical artifact. Then  is given by the integral

(4)

This function is implemented in the Wolfram Language as ExpIntegralEi[x].

The exponential integral  is closely related to the incomplete gamma function  by

(5)

Therefore, for real ,

(6)

The exponential integral of a purely imaginary number can be written

(7)

for  and where  and  are cosine and sine integral.

Special values include

(8)

(OEIS A091725).

The real root of the exponential integral occurs at 0.37250741078... (OEIS A091723), which is , where  is Soldner's constant (Finch 2003).

The quantity  (OEIS A073003) is known as the Gompertz constant.

The limit of the following expression can be given analytically

			(9)
			(10)

(OEIS A091724), where  is the Euler-Mascheroni constant.

The Puiseux series of  along the positive real axis is given by

(11)

where the denominators of the coefficients are given by  (OEIS A001563; van Heemert 1957, Mundfrom 1994).

REFERENCES:

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

Finch, S. R. "Euler-Gompertz Constant." §6.2 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 423-428, 2003.

Harris, F. E. "Spherical Bessel Expansions of Sine, Cosine, and Exponential Integrals." Appl. Numer. Math. 34, 95-98, 2000.

Havil, J. Gamma: Exploring Euler's Constant. Princeton, NJ: Princeton University Press, pp. 105-106, 2003.

Jeffreys, H. and Jeffreys, B. S. "The Exponential and Related Integrals." §15.09 in Methods of Mathematical Physics, 3rd ed.Cambridge, England: Cambridge University Press, pp. 470-472, 1988.

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

Mundfrom, D. J. "A Problem in Permutations: The Game of 'Mousetrap.' " European J. Combin. 15, 555-560, 1994.

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.

Sloane, N. J. A. Sequences A001563/M3545, A073003, A091723, A091724, and A091725 in "The On-Line Encyclopedia of Integer Sequences."

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.

van Heemert, A. "Cyclic Permutations with Sequences and Related Problems." J. reine angew. Math. 198, 56-72, 1957.