The function is defined by the integral
|
(1)
|
and is given by the Wolfram Language function ExpIntegralE[n,
x]. Defining
so that ,
|
(2)
|
For integer ,
|
(3)
|
Plots in the complex plane are shown above for .
The special case
gives
where
is the exponential integral and is an incomplete
gamma function. It is also equal to
|
(8)
|
where
is the Euler-Mascheroni constant.
where
and are the cosine
integral and sine integral.
The function satisfies the recurrence relations
In general,
can be built up from the recurrence
|
(13)
|
The series expansions is given by
|
(14)
|
and the asymptotic expansion by
|
(15)
|
See also
Cosine Integral,
Et-Function,
Exponential Integral,
Gompertz
Constant,
Sine Integral
Related Wolfram sites
http://functions.wolfram.com/GammaBetaErf/ExpIntegralE/
Explore with Wolfram|Alpha
References
Abramowitz, M. and Stegun, I. A. (Eds.). "Exponential Integral and Related Functions." Ch. 5 in Handbook
of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing.
New York: Dover, pp. 227-233, 1972.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.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.
Cite this as:
Weisstein, Eric W. "E_n-Function." From
MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/En-Function.html
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