A258408 Decimal expansion of Integral_{x=0..1} Product_{k>=1} (1-x^(2*k)) dx.

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Offset: 0

Author

Vaclav Kotesovec, May 29 2015

Keywords

• nonn
• cons

Comments

In general, Integral_{x=0..1} Product_{k>=1} (1-x^(m*k)) dx = Sum_{n} (-1)^n / (m*n*(3*n-1)/2 + 1) is equal to

if 0

(sqrt((24-m)*m) * (2*cosh(Pi/3*sqrt(24/m-1))-1))

if m = 24: Pi^2/(6*sqrt(3)) = A258414

if m > 24: 8*sqrt(3)*Pi*sin(Pi/6*sqrt(1-24/m)) /

(sqrt((m-24)*m) * (2*cos(Pi/3*sqrt(1-24/m))-1)).

Integral_{x=0..1} Product_{k=1..n} (1+x^(m*k)) dx, where m >= 1, is asymptotic to 2*(m+1)^(n+1)/(m*n^2).

Integral_{x=-1..1} Product_{k>=1} (1-x^(2*k)) dx = 8*Pi*sqrt(3/11) * sinh(sqrt(11)*Pi/6) / (2*cosh(sqrt(11)*Pi/3)-1) = 1.154664240367959678... . - Vaclav Kotesovec, Jun 02 2015

Examples

			0.5773321201839797055525469620159041550801193138356349245589088...
		

Links

• Vaclav Kotesovec, The integration of q-series

Crossrefs

Cf. A258232, A258406, A258414.

Programs

Maple

evalf(4*Pi*sqrt(3/11) * sinh(sqrt(11)*Pi/6) / (2*cosh(sqrt(11)*Pi/3) - 1), 120);

Mathematica

RealDigits[4*Pi*Sqrt[3/11]*Sinh[Sqrt[11]*Pi/6] / (2*Cosh[Sqrt[11]*Pi/3] - 1),10,120][[1]]

Formula

Equals 4*Pi*sqrt(3/11) * sinh(sqrt(11)*Pi/6) / (2*cosh(sqrt(11)*Pi/3) - 1).

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