A341908 Decimal expansion of Integral_{x=0..1} x/(exp(x)-1) dx.
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
Examples
0.77750463411224827641758654542571050719247729622900...
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.
Programs
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Maple
evalf(-dilog(exp(1))-1/2, 140); # Alois P. Heinz, Jun 04 2021
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Mathematica
RealDigits[PolyLog[2, 1-1/E], 10, 100][[1]]
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PARI
intnum(x=0, 1, x/(exp(x)-1)) \\ Michel Marcus, Jun 04 2021
Formula
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.