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A341908
Decimal expansion of Integral_{x=0..1} x/(exp(x)-1) dx.
0
7, 7, 7, 5, 0, 4, 6, 3, 4, 1, 1, 2, 2, 4, 8, 2, 7, 6, 4, 1, 7, 5, 8, 6, 5, 4, 5, 4, 2, 5, 7, 1, 0, 5, 0, 7, 1, 9, 2, 4, 7, 7, 2, 9, 6, 2, 2, 9, 0, 0, 0, 8, 6, 9, 1, 7, 9, 4, 9, 4, 5, 4, 1, 0, 6, 9, 9, 6, 6, 8, 4, 8, 8, 6, 2, 4, 9, 8, 0, 3, 7, 6, 8, 7, 7, 1, 1
OFFSET
0,1
REFERENCES
Alvaro Meseguer, Fundamentals of Numerical Mathematics for Physicists and Engineers, Wiley, 2020, Chapter 4, exercise 12, p. 128.
John Michael Rassias, Geometry, Analysis, and Mechanics, World Scientific, 1994, p. 14.
LINKS
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972, p. 998.
Mathematics MI, Integral x/(e^x - 1) from 0 to 1, YouTube video, May 8 2021.
Voodooguru, Collaboration between Integration and Summation, Mathematical Meanderings, May 9 2021.
Eric Weisstein's World of Mathematics, Debye Functions.
Eric Weisstein's World of Mathematics, Dilogarithm.
Eric Weisstein's World of Mathematics, Polylogarithm.
Wikipedia, Debye function.
Wikipedia, Polylogarithm.
Wikipedia, Spence's function.
FORMULA
Equals D_1(1) = Sum_{k>=0} A120082(k)/A120083(k), where D_n(x) are the Debye functions.
Equals Li_2(1-1/e) = -1/2 - Li_2(1-e) = Pi^2/6 - 1 + log(e-1) - Li_2(1/e), where Li_2(x) is the dilogarithm function.
Equals Sum_{k>=0} B(k)/(k+1)! = -1/2 + Sum_{k>=0} (-1)^k*B(k)/(k+1)! = -1/4 + Sum_{k>=0} B(2*k)/(2*k+1)!, where B(k) is the k-th Bernoulli number.
Equals Sum_{k>=1} (1 - (k+1)*exp(-k))/k^2.
EXAMPLE
0.77750463411224827641758654542571050719247729622900...
MAPLE
evalf(-dilog(exp(1))-1/2, 140); # Alois P. Heinz, Jun 04 2021
MATHEMATICA
RealDigits[PolyLog[2, 1-1/E], 10, 100][[1]]
PROG
(PARI) intnum(x=0, 1, x/(exp(x)-1)) \\ Michel Marcus, Jun 04 2021
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Amiram Eldar, Jun 04 2021
STATUS
approved