The
function is defined by the integral
![\begin{displaymath}
\mathop{\rm E}\nolimits_n(x) \equiv \int_1^\infty {e^{-xt}\,dt\over t^n}
\end{displaymath}](e_48.gif) |
(1) |
and is given by the Mathematica
(Wolfram Research, Champaign, IL) function ExpIntegralE[n,x].
Defining
so that
,
![\begin{displaymath}
\mathop{\rm E}\nolimits_n(x) = \int^1_0 e^{-x/\eta} \eta^{n-2}\,d\eta
\end{displaymath}](e_52.gif) |
(2) |
![\begin{displaymath}
\mathop{\rm E}\nolimits_n(0) = {1\over n-1}.
\end{displaymath}](e_53.gif) |
(3) |
The function satisfies the Recurrence Relations
![\begin{displaymath}
\mathop{\rm E}\nolimits_n'(x) = -\mathop{\rm E}\nolimits_{n-1}(x)
\end{displaymath}](e_54.gif) |
(4) |
![\begin{displaymath}
n\mathop{\rm E}\nolimits_{n+1}(x) = e^{-x}-x\mathop{\rm E}\nolimits_n(x).
\end{displaymath}](e_55.gif) |
(5) |
Equation (4) can be derived from
and (5) using integrating by parts, letting
![\begin{displaymath}
u={1\over t^n} \qquad dv=e^{- t x}\,d t
\end{displaymath}](e_62.gif) |
(8) |
![\begin{displaymath}
du=-{n\over t^{n+1}}\,d t \qquad v=-{e^{- t x}\over x}
\end{displaymath}](e_63.gif) |
(9) |
gives
Solving (10) for
then gives (5). An asymptotic expansion gives
![\begin{displaymath}
(n-1)!\,\mathop{\rm E}\nolimits_n(x) = (-x)^{n-1}\mathop{\rm E}\nolimits_1(x)+e^{-x} \sum_{s=0}^n-2 (n-s-2)!(-x)^s,
\end{displaymath}](e_69.gif) |
(11) |
so
![\begin{displaymath}
\mathop{\rm E}\nolimits_n(x) = {e^{-x}\over x} \left[{1 - {n\over x} + {n(n+1)\over x^2} + \ldots}\right].
\end{displaymath}](e_70.gif) |
(12) |
The special case
gives
![\begin{displaymath}
\mathop{\rm E}\nolimits_1(x) \equiv -\mathop{\rm ei}\nolimit...
...nfty {e^{-tx}\,dt\over t} = \int_x^\infty {e^{-u}\,du\over u},
\end{displaymath}](e_72.gif) |
(13) |
where
is the Exponential Integral, which is also equal to
![\begin{displaymath}
\mathop{\rm E}\nolimits_1(x) = -\gamma - \ln x - \sum_{n=1}^\infty {(-1)^n x^n\over n!n},
\end{displaymath}](e_74.gif) |
(14) |
where
is the Euler-Mascheroni Constant.
where
and
are the Cosine Integral and Sine Integral.
See also Cosine Integral, Et-Function, Exponential Integral, Gompertz Constant,
Sine Integral
References
Abramowitz, M. and Stegun, C. 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.
© 1996-9 Eric W. Weisstein
1999-05-25