OFFSET
1,1
COMMENTS
Also equals the area under the peak-shaped even function f(x)=x/sinh(x).
Proof: For the upper half of the integral, write f(x) = 2x*exp(-x)/(1-exp(-2x)) = sum_{k=1..infinity} 2x*exp(-(2k-1)x) and integrate term by term from zero to infinity. - Stanislav Sykora, Nov 01 2013
Volume of the 4-dimensional unit sphere; the volume of the n-dimensional unit sphere is Pi^(n/2)/gamma(n/2+1) (see n-ball link and A164103). - Rick L. Shepherd, Jun 22 2017
Pi^2/2 is the squared side-length of a square with diagonal Pi. - Wesley Ivan Hurt, Jan 28 2022
REFERENCES
J. Rivaud, Analyse, Séries, Equations différentielles, Mathématiques Supérieures et Spéciales, Premier Cycle Universitaire, Vuibert, 1981, Exercice 2, p. 135.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Middlesex, England: Penguin Books, 1986, p. 53.
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..10000
T. Amdeberhan, L. Medina and V. H. Moll, The integrals in Gradshteyn and Ryzhik. Part 5: Some trigonometric integrals, equation 2.39, arXiv:0705.2379 [math.CA], 2007.
Renzo Sprugnoli, Sums of reciprocals of the central binomial coefficients, El. J. Combin. Numb. Th. 6 (2006) # A27
Eric Weisstein's World of Mathematics, Hypersphere.
Eric Weisstein's World of Mathematics, Trigamma Function.
Wikipedia, Hypersphere.
Wikipedia, Volume of an n-ball.
FORMULA
Equals psi_1(1/2), where psi_1(x) is the second logarithmic derivative of GAMMA(x).
Equals the volume of revolution of the sine or cosine curve for one half period, Integral_{0,Pi} Sin(x)^2 dx. - Robert G. Wilson v, Dec 15 2005
Equals Sum_{k >=1} 4^k/(k^2*binomial(2*k,k)) [Amdeberhan, Sprugnoli]. - R. J. Mathar, Sep 28 2007
Equals 4*Sum_{k >=1} 1/(2k-1)^2 [Wells].
From Peter Bala, Nov 05 2019: (Start)
Pi^2/2 = Integral_{x = 0..inf} cosh(x)*x^2/sinh(x)^2 dx.
Pi^2/2 = 5*sum_{k >= 0} binomial(2*k,k)(-1/16)^k*1/(2*k+1)^2.
Pi^2/2 = 10*Integral_{x = 0..1/2} 1/x*log(x + sqrt(1 + x^2)) dx. (End)
Pi^2/20 = 0.1 * Pi^2/2 = Sum_{k>=1} 1/A026424(k)^2. - Amiram Eldar, Aug 17 2020
Conjecture: Pi^2/2 = Sum_{n = -oo..oo} ( cos(Pi*sqrt(n^2+1)) - cos(Pi*n) ) (using the Eisenstein summation convention). - Peter Bala, Oct 08 2021
Pi^2/2 = Integral_{x = -oo..oo} x/sinh(x) dx (see Rivaud reference). - Bernard Schott, Jan 28 2022
EXAMPLE
4.9348022005446793094172454999380755676568497036203953132066746881100\ 224112096026215008867018592761159120129568870115720388....
MATHEMATICA
RealDigits[Pi^2/2, 10, 111][[1]] (* Robert G. Wilson v, Dec 15 2005 *)
PROG
(PARI) Pi^2/2 \\ Michel Marcus, Sep 04 2015
CROSSREFS
KEYWORD
cons,nonn
AUTHOR
Jun Mizuki (suzuki32(AT)sanken.osaka-u.ac.jp), Feb 10 2005
STATUS
approved