Exponential Integral — from Wolfram MathWorld

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

ExponentialIntegral

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

Then define the exponential integral Ei(x) by

 E_1(x)=-Ei(-x),

(3)

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

 Ei(x)=-int_(-x)^infty(e^(-t)dt)/t.

(4)

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

 Gamma(0,z)=-Ei(-z)+1/2[ln(-z)-ln(-1/z)]-lnz.

(5)

Therefore, for real x,

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

(6)

The exponential integral of a purely imaginary
number can be written

 Ei(ix)=ci(x)+i[1/2pi+si(x)]

(7)

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

Special values include

 Ei(1)=1.89511781...

(8)

(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
constant.

The limit of the following expression can be given analytically

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

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

 Ei(z)=gamma+lnz+z+1/4z^2+1/(18)z^3+1/(96)z^4+1/(600)z^5+...,

(11)

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

See also

Cosine Integral

,

En-Function

,

Gompertz Constant

,

Incomplete
Gamma Function

,

Sine Integral

,

Soldner’s
Constant

Related Wolfram sites

http://functions.wolfram.com/GammaBetaErf/ExpIntegralEi/

Explore with Wolfram|Alpha

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(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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