The
function is defined by the integral
|
(1) |
and is given by the Mathematica
(Wolfram Research, Champaign, IL) function ExpIntegralE[n,x].
Defining
so that
,
|
(2) |
|
(3) |
The function satisfies the Recurrence Relations
|
(4) |
|
(5) |
Equation (4) can be derived from
and (5) using integrating by parts, letting
|
(8) |
|
(9) |
gives
Solving (10) for
then gives (5). An asymptotic expansion gives
|
(11) |
so
|
(12) |
The special case gives
|
(13) |
where
is the Exponential Integral, which is also equal to
|
(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