## En-Function

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
 (6) (7)

and (5) using integrating by parts, letting
 (8)

 (9)

gives
 (10)

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.
 (15) (16)

where and are the Cosine Integral and 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.