Let

be the En-function with ,

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

(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).

## 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() 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