OFFSET
0,2
COMMENTS
In general, Integral_{x=0..1} Product_{k>=1} (1-x^(m*k))^3 dx = Sum_{n>=0} (-1)^n * (2*n+1) / (m*n*(n+1)/2 + 1) is equal to
if 0<m<8: (2*Pi / (m * cosh(Pi/2*sqrt(8/m-1)))
if m = 8: Pi/4
if m > 8: (2*Pi / (m * cos(Pi/2*sqrt(1-8/m)))
Special values: m=4: Pi/(2*cosh(Pi/2)), m=9: 4*Pi/(9*sqrt(3)).
---
Integral_{x=-1..1} Product_{k>=1} (1-x^k)^3 dx = 2*Pi*(1 + sqrt(2) * cosh(sqrt(7)*Pi/4)) / cosh(sqrt(7)*Pi/2) = 1.32639350417409769439126... . - Vaclav Kotesovec, Jun 02 2015
LINKS
Vaclav Kotesovec, The integration of q-series
FORMULA
Equals 2*Pi/cosh(sqrt(7)*Pi/2).
Equals Sum_{n>=0} (-1)^n * (2*n+1) / (n*(n+1)/2 + 1).
EXAMPLE
0.1968806153145889753533513584769666829667343178391757586093357...
MAPLE
evalf(2*Pi/cosh(sqrt(7)*Pi/2), 120);
evalf(Sum((-1)^n * (2*n+1) / (n*(n+1)/2 + 1), n=0..infinity), 120);
MATHEMATICA
RealDigits[2*Pi*Sech[(Sqrt[7]*Pi)/2], 10, 105][[1]]
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Vaclav Kotesovec, May 29 2015
STATUS
approved