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E_n-Function


ExpIntegralE

The E_n(x) function is defined by the integral

 E_n(x)=int_1^infty(e^(-xt)dt)/(t^n)
(1)

and is given by the Wolfram Language function ExpIntegralE[n, x]. Defining t=eta^(-1) so that dt=-eta^(-2)deta,

 E_n(x)=int_0^1e^(-x/eta)eta^(n-2)deta
(2)

For integer n>1,

 E_n(0)=1/(n-1).
(3)
EnFunctionReIm
EnFunctionContours

Plots in the complex plane are shown above for E_0(z).

The special case n=1 gives

E_1(x)=-Ei(-x)
(4)
=Gamma(0,x)
(5)
=int_1^infty(e^(-tx)dt)/t
(6)
=int_x^infty(e^(-u)du)/u,
(7)

where Ei(x) is the exponential integral and Gamma(a,z) is an incomplete gamma function. It is also equal to

 E_1(x)=-gamma-lnx-sum_(n=1)^infty((-1)^nx^n)/(n!n),
(8)

where gamma is the Euler-Mascheroni constant.

E_1(0)=infty
(9)
E_1(ix)=-ci(x)+isi(x),
(10)

where ci(x) and si(x) are the cosine integral and sine integral.

The function satisfies the recurrence relations

E_n^'(x)=-E_(n-1)(x)
(11)
nE_(n+1)(x)=e^(-x)-xE_n(x).
(12)

In general, E_(n+1)(x) 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].
(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))-...]
(14)

and the asymptotic expansion by

 E_n(x)=(e^(-x))/x[1-n/x+(n(n+1))/(x^2)+...].
(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(x) 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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