A152115 -id:A152115 - cp's OEIS frontend

A355234 Decimal expansion of Li_2(-1/2), the dilogarithm of (-1/2) (negated).

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

Offset: 0

Amiram Eldar, Jun 25 2022

			-0.44841420692364620244306440591577432083426994134919...

Michael Ian Shamos, Shamos's Catalog of the Real Numbers, 2011, p. 456.
Eric Weisstein's World of Mathematics, Dilogarithm, eq. (26).
Wikipedia, Spence's function.

Cf. A001008, A002805, A007758.

Other values of Li_2: A072691, A076788, A152115, A242599, A242600.

RealDigits[PolyLog[2, -1/2], 10, 100][[1]]

-dilog(-1/2) \\ Michel Marcus, Jun 25 2022

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.

A242599 Decimal expansion of dilog(phi-1) = polylog(2, 2-phi) with phi = (1 + sqrt(5))/2.

Original entry on oeis.org

4, 2, 6, 4, 0, 8, 8, 0, 6, 1, 6, 2, 0, 9, 6, 1, 8, 2, 0, 9, 2, 0, 3, 6, 9, 9, 5, 4, 2, 6, 8, 7, 7, 3, 1, 5, 6, 7, 1, 1, 7, 3, 6, 1, 0, 4, 3, 3, 4, 2, 0, 5, 0, 4, 2, 7, 8, 9, 2, 2, 0, 6, 3, 0, 5, 8, 2, 0, 7, 6, 4, 2, 5, 9, 4, 3, 1, 8, 5, 3, 6, 5, 4, 8, 3, 9, 7, 0, 1, 3, 1, 6, 1, 5, 1, 5, 0, 8, 7, 0, 6, 5, 8, 5, 8, 5, 5

Offset: 0

Wolfdieter Lang, Jun 16 2014

dilog(phi-1) = polylog(2, 2-phi) = sum((2-phi)^k/k^2 , k =1 ..infinity) = sum((1 - 2*sin(Pi/10))^(2*k)/k^2, k=1..infinity) = Pi^2/15 - (log(phi-1))^2 = Pi^2/15 - (2/5)*log(phi-1)*(log(2-phi) + log(phi-1)/2).

See the Jolley reference pp. 66-69, (360)(e), and the Abramowitz-Stegun link, p. 1004, eqs. 27.7.3 - 27.7.6 with x = phi-1, solving for dilog(x) = f(x).

			0.42640880616209618209...

L. B. W. Jolley, Summation of Series, Dover, 1961.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

Cf. A152115, A076788 (dilog(1/2)), A242600.

phi := (1+sqrt(5))/2 ; dilog(phi-1) ; evalf(%) ; # R. J. Mathar, Jun 10 2024

RealDigits[PolyLog[2, 2 - GoldenRatio], 10, 120][[1]] (* Amiram Eldar, May 27 2023 *)

polylog(2, 2 - (1+sqrt(5))/2) \\ Gheorghe Coserea, Sep 30 2018

sumpos(k=1, (1 - 2*sin(Pi/10))^k/k^2) \\ Gheorghe Coserea, Sep 30 2018

dilog(phi-1) = polylog(2, 2-phi) = Sum_{k>=1} (2-phi)^k/k^2 = Sum_{k>=1} (1 - 2*sin(Pi/10))^k/k^2.

A242600 Decimal expansion of -dilog(phi) = -polylog(2, 1-phi) with phi = (1 + sqrt(5))/2.

Original entry on oeis.org

5, 4, 2, 1, 9, 1, 2, 1, 6, 4, 5, 0, 6, 9, 3, 3, 7, 8, 3, 4, 0, 5, 0, 1, 5, 3, 1, 0, 4, 2, 6, 4, 3, 6, 9, 5, 6, 7, 9, 3, 7, 6, 7, 8, 5, 4, 5, 8, 0, 6, 9, 9, 3, 9, 6, 8, 6, 5, 7, 2, 6, 7, 7, 4, 0, 3, 1, 0, 5, 3, 1, 5, 3, 7, 7, 7, 9, 9, 4, 4, 3, 0, 4, 0, 9, 2, 4, 2, 8, 6, 7, 0, 4, 7, 0, 9, 2, 8, 4, 5, 9, 3, 7, 3, 0, 1, 3

Offset: 1

Wolfdieter Lang, Jun 16 2014

This solution for -Sum_{k>=1} (-2*sin(Pi/10)^k/k^2) should also have been mentioned in the Jolley reference pp. 66-69 under (360).

			0.542191216450693...

L. B. W. Jolley, Summation of Series, Dover, 1961.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

Cf. A001622, A076788 (polylog(2,1/2)), A152115, A242599.

RealDigits[PolyLog[2, 1 - GoldenRatio], 10, 120][[1]] (* Amiram Eldar, May 30 2023 *)

Equals -Sum_{k>=1} (1-phi)^k/k^2 = Pi^2/15 - (log(phi-1)^2)/2, with the golden section phi = (1 + sqrt(5))/2. See the Abramowitz-Stegun link, p. 1004, eqs. 27.7.3 - 27.7.6 with x = phi-1, solving for -dilog(x+1) = -f(1+x), using log(2-phi) = 2*log(phi-1).

cp's OEIS Frontend

A355234 Decimal expansion of Li_2(-1/2), the dilogarithm of (-1/2) (negated).

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A242599 Decimal expansion of dilog(phi-1) = polylog(2, 2-phi) with phi = (1 + sqrt(5))/2.

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A242600 Decimal expansion of -dilog(phi) = -polylog(2, 1-phi) with phi = (1 + sqrt(5))/2.

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