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Date: 17-9-2018
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Date: 15-6-2019
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Date: 30-3-2019
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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
(4) |
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(5) |
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(6) |
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(7) |
where is the exponential integral and is an incomplete gamma function. It is also equal to
(8) |
where is the Euler-Mascheroni constant.
(9) |
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(10) |
where and are the cosine integral and sine integral.
The function satisfies the recurrence relations
(11) |
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(12) |
In general, can be built up from the recurrence
(13) |
The series expansions is given by
(14) |
and the asymptotic expansion by
(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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شراء وقود الطائرات المستدام.. "الدفع" من جيب المسافر
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