TOPICS
Search

Exponential Integral

DOWNLOAD Mathematica NotebookDownload Wolfram Notebook
ExponentialIntegral
ExpIntReImAbs
Min Max
Re
Im Powered by webMathematica

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

E_1(x)=int_1^infty(e^(-tx)dt)/t
(1)
=int_x^infty(e^(-u)du)/u.
(2)

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

lim_(x->0^+)(e^(2Ei(-x)))/(x^2)=e^(2gamma)
(9)
=3.17221895...,
(10)

(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

Subject classifications