# Exponential Integral ```Exponential Integral -- from Wolfram MathWorld
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12,637 entries
Sat Dec 23 2006
Min
Max
Re -5
5
Im -5
5
Let
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Replot
be the En-function with
,
(1)
Then define the exponential integral
by
(2)
where the retention of the
given by the integral
notation is a historical artifact. Then
is
(3)
This function is implemented in Mathematica as ExpIntegralEi[x].
http://mathworld.wolfram.com/ExponentialIntegral.html
25/12/2006
Exponential Integral -- from Wolfram MathWorld
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The exponential integral
by
is closely related to the incomplete gamma function
(4)
Therefore, for real
,
(5)
The exponential integral of a purely imaginary number can be written
(6)
for
and where
and
are cosine and sine integral.
Special values include
(7)
(Sloane's A091725).
The real root of the exponential integral occurs at 0.37250741078... (Sloane's
A091723), which is
, where is Soldner's constant (Finch 2003).
The quantity
Gompertz constant.
(Sloane's A073003) is known as the
The limit of the following expression can be given analytically
(8)
(9)
(Sloane's A091724), where
The Puiseux series of
is the Euler-Mascheroni constant.
along the positive real axis is given by
(10)
where the denominators of the coefficients are given by
van Heemert 1957, Mundfrom 1994).
(Sloane's A001563;
Function, Sine Integral, Soldner's Constant. [Pages Linking Here]
RELATED WOLFRAM SITES:
http://functions.wolfram.com/GammaBetaErf/ExpIntegralEi/
REFERENCES:
Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 566-568,
1985.
Finch, S. R. &quot;Euler-Gompertz Constant.&quot; &sect;6.2 in Mathematical Constants. Cambridge, England:
Cambridge University Press, pp. 423-428, 2003.
http://mathworld.wolfram.com/ExponentialIntegral.html
25/12/2006
Exponential Integral -- from Wolfram MathWorld
Page 3 of 3
Harris, F. E. &quot;Spherical Bessel Expansions of Sine, Cosine, and Exponential Integrals.&quot; Appl. Numer.
Math. 34, 95-98, 2000.
Havil, J. Gamma: Exploring Euler's Constant. Princeton, NJ: Princeton University Press, pp. 105106, 2003.
Jeffreys, H. and Jeffreys, B. S. &quot;The Exponential and Related Integrals.&quot; &sect;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. &quot;A Problem in Permutations: The Game of 'Mousetrap.' &quot; European J. Combin. 15,
555-560, 1994.
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. &quot;Exponential Integrals.&quot; &sect;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 &quot;The
On-Line Encyclopedia of Integer Sequences.&quot;
Spanier, J. and Oldham, K. B. &quot;The Exponential Integral Ei( ) and Related Functions.&quot; Ch. 37 in An
Atlas of Functions. Washington, DC: Hemisphere, pp. 351-360, 1987.
van Heemert, A. &quot;Cyclic Permutations with Sequences and Related Problems.&quot; J. reine angew. Math.
198, 56-72, 1957.