E_n-Function
المؤلف:
Abramowitz, M. and Stegun, I. A.
المصدر:
"Exponential Integral and Related Functions." Ch. 5 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover
الجزء والصفحة:
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31-7-2019
2316
E_n-Function
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].](http://mathworld.wolfram.com/images/equations/En-Function/NumberedEquation5.gif) |
(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))-...]](http://mathworld.wolfram.com/images/equations/En-Function/NumberedEquation6.gif) |
(14)
|
and the asymptotic expansion by
![E_n(x)=(e^(-x))/x[1-n/x+(n(n+1))/(x^2)+...].](http://mathworld.wolfram.com/images/equations/En-Function/NumberedEquation7.gif) |
(15)
|
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.
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