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
![E_n(x)=1/((n-1)!)[(-x)^(n-1)E_1(x)+e^(-x)sum_(s=0)^(n-2)(n-s-2)!(-x)^s].](https://txtdot.deep-swarm.xyz/proxy/img?url=https%3A%2F%2Fmathworld.wolfram.com%2Fimages%2Fequations%2FEn-Function%2FNumberedEquation5.svg) |
(13) |
The series expansions is given by
![E_n(x)=x^(n-1)Gamma(1-n)+[-1/(1-n)+x/(2-n)-(x^2)/(2(3-n))+(x^3)/(6(4-n))-...]](https://txtdot.deep-swarm.xyz/proxy/img?url=https%3A%2F%2Fmathworld.wolfram.com%2Fimages%2Fequations%2FEn-Function%2FNumberedEquation6.svg) |
(14) |
and the asymptotic expansion by
![E_n(x)=(e^(-x))/x[1-n/x+(n(n+1))/(x^2)+...].](https://txtdot.deep-swarm.xyz/proxy/img?url=https%3A%2F%2Fmathworld.wolfram.com%2Fimages%2Fequations%2FEn-Function%2FNumberedEquation7.svg){kind=small} |
(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."
ยง6.3 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.
Referenced on Wolfram|Alpha
TagBox[E, MathPlain]_n-Function
Cite this as:
Weisstein, Eric W. "TagBox[E, MathPlain]_n-Function." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/En-Function.html