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A355234
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Decimal expansion of Li_2(-1/2), the dilogarithm of (-1/2) (negated).
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4
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4, 4, 8, 4, 1, 4, 2, 0, 6, 9, 2, 3, 6, 4, 6, 2, 0, 2, 4, 4, 3, 0, 6, 4, 4, 0, 5, 9, 1, 5, 7, 7, 4, 3, 2, 0, 8, 3, 4, 2, 6, 9, 9, 4, 1, 3, 4, 9, 1, 9, 9, 1, 2, 8, 5, 0, 1, 7, 4, 6, 3, 7, 1, 3, 1, 6, 8, 2, 4, 3, 7, 2, 2, 5, 5, 7, 2, 0, 3, 1, 2, 3, 8, 9, 8, 6, 5, 1, 6, 5, 1, 8, 6, 6, 5, 3, 3, 1, 0, 6, 6, 9, 0, 2, 8
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OFFSET
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0,1
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LINKS
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Eric Weisstein's World of Mathematics, Dilogarithm, eq. (26).
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FORMULA
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From Shamos (2011):
Equals -Li_2(1/3) - log(3/2)^2/2.
Equals Li_2(2/3) + log(3)^2/2 - log(2)^2/2 - Pi^2/6.
Equals Li_2(1/4)/2 + log(2)^2/2 - Pi^2/12.
Equals -Sum_{k>=1} (-1)^(k+1)/(2^k*k^2) = -Sum_{k>=1} (-1)^(k+1)/A007758(k).
Equals -Sum_{k>=1} H(k)/(k*3^k), where H(k) = A001008(k)/A002805(k) is the k-th harmonic number.
Equals -Integral_{x=0..1} log(x)^2/(x+2)^2 dx.
Equals -Integral_{x>=1} log(x)^2/(2*x+1)^2 dx.
Equals Integral_{x=0..1} log(x)/(x+2) dx.
Equals -Integral_{x>=0} log(1 + exp(-x)/2) dx.
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EXAMPLE
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-0.44841420692364620244306440591577432083426994134919...
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MATHEMATICA
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RealDigits[PolyLog[2, -1/2], 10, 100][[1]]
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PROG
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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