Exponential Integral — from Wolfram MathWorld

Integral of 1/(1+e^x)
Integral of 1/(1+e^x)


Let E_1(x)
be the En-function with n=1,

Then define the exponential integral Ei(x) by



where the retention of the -Ei(-x) notation is a historical
artifact. Then Ei(x)
is given by the integral



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

The exponential integral Ei(z) is closely related to the incomplete
gamma function Gamma(0,z) by



Therefore, for real x,

 Gamma(0,x)={-Ei(-x)-ipi for x<0; -Ei(-x) for x>0.


The exponential integral of a purely imaginary
number can be written



for x>0
and where ci(x)
and si(x)
are cosine and sine

Special values include



(OEIS A091725).

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

The quantity -eEi(-1)=0.596347362...
(OEIS A073003) is known as the Gompertz

The limit of the following expression can be given analytically

(OEIS A091724), where gamma is the Euler-Mascheroni

The Puiseux series of Ei(z) along the positive real axis is given by



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

See also

Cosine Integral




Gompertz Constant


Gamma Function


Sine Integral



Related Wolfram sites


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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(x) 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.

Referenced on Wolfram|Alpha

Exponential Integral

Cite this as:

Weisstein, Eric W. “Exponential Integral.”
From MathWorld–A Wolfram Web Resource. https://mathworld.wolfram.com/ExponentialIntegral.html

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