Infinite Cosine Product Integral

Author

Eric W. Weisstein
January 14, 2005

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Calculus/InfiniteCosineProductIntegral.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/InfiniteCosineProductIntegral.html.

©2005 Wolfram Research, Inc. except for portions noted otherwise

Variants

Borwein and Bailey (Experimental Math 1, p. 26)

(∫_0^∞)(∏_(n=1)^∞)cos((x/n))dx

Product[Cos[x/n],{n,∞}]

NIntegrate[Evaluate[Product[Cos[x/n],{n,1000}]],{x,0,∞}]

N[Pi/4,20]

NIntegrate[Evaluate[Product[Cos[x/n],{n,1000}]],{x,0,∞},WorkingPrecision->10]//Timing

NIntegrate[Evaluate[Product[Cos[x/n],{n,1000}]],{x,0,∞},WorkingPrecision->15]//Timing

NIntegrate[Evaluate[Product[Cos[x/n],{n,1000}]],{x,0,∞},WorkingPrecision->20]//Timing

(∫_0^∞)cos(x) (∏_(n=1)^∞)cos((x/n))dx

NIntegrate[Evaluate[Cos[x]Product[Cos[x/n],{n,1000}]],{x,0,∞}]

Pi/4-%32

(∫_0^∞)cos(2 x) (∏_(n=1)^∞)cos((x/n))dx

From Michael Trott, January 22, 2004

0.392699081698724154807830422909937860524645434187231595926812285162093247139938546179016512747455366777507395573123898403582027154856887609365093419344

NIntegrate[Evaluate[Cos[2x]Product[Cos[x/n],{n,1000}]],{x,0,∞}]

c=Table[Cos[x/n],{n,5}]

l=Reverse[Times@@@NestList[Rest,c,4]];

l

<<Graphics`Colors`

Plot[Evaluate[Cos[2x]l],{x,0,30},PlotStyle->{Red,Orange,Yellow,Green,Blue}]

Plot[Evaluate[Table[Cos[2x]Product[Cos[x/k],{k,n}],{n,10,50,10}]],{x,0,40},PlotStyle->{Red,Orange,Yellow,Green,Blue},PlotRange->{-.2,.1}]

Plot[Evaluate[Cos[2x]Product[Cos[x/k],{k,100}]],{x,0,40},PlotStyle->Red,PlotRange->{-.2,.1}]

Plot[Evaluate[Cos[2x]Product[Cos[x/k],{k,500}]],{x,0,40},PlotStyle->Red,PlotRange->{-.2,.1}]

NIntegrate[Evaluate[Cos[2x]Product[Cos[x/n],{n,1000}]],{x,0,∞}]

NIntegrate[Evaluate[Product[Cos[x/n],{n,5000}]],{x,0,∞}]

Pi/4.

TrigReduce[Cos[2x]Cos[x]Cos[x/2]Cos[x/3]]

Simplify[(1/8) (Cos[(x/6)]+Cos[((5 x)/6)]+Cos[((7 x)/6)]+Cos[((11 x)/6)]+Cos[((13 x)/6)]+Cos[((17 x)/6)]+Cos[((19 x)/6)]+Cos[((23 x)/6)])]

Simplify[TrigReduce[Cos[2x]Cos[x]Cos[x/2]Cos[x/3]]]

Simplify[TrigReduce[Cos[2x]Cos[x]Cos[x/2]Cos[x/3]Cos[x/4]]]

FullSimplify[TrigReduce[Cos[2x]Cos[x]Cos[x/2]Cos[x/3]Cos[x/4]Cos[x/5]]]

Select[Range[25],GCD[#,6]==1&]

TrigReduce[Cos[2x]Cos[x]Cos[x/2]Cos[x/3]Cos[x/4]]

Select[Range[49],GCD[#,12]==1&]

TrigReduce[Cos[2x]Cos[x]Cos[x/2]Cos[x/3]Cos[x/4]Cos[x/5]]

Select[Range[260],GCD[#,60]==1&]

<<Graphics`Colors`

correct up to Pi/2

cosProduct[x_?NumericQ,o_]:=Exp[Sum[((-1)^k*2^(2*k-1)*(4^k-1)*BernoulliB[2*k]*x^(2*k)*Zeta[2*k])/(k*(2*k)!),{k,o}]]

Sum[((-1)^k*2^(2*k-1)*(4^k-1)*BernoulliB[2*k]*x^(2*k)*Zeta[2*k])/(k*(2*k)!),{k,∞}]

Plot[cosProduct[x,20],{x,0,2},PlotStyle->Red,PlotRange->All]

Plot[cosProduct[x,10]-Product[Cos[x/n],{n,1000}],{x,0,2},PlotRange->All]

Plot[cosProduct[x,10]-Product[Cos[x/n],{n,2000}],{x,0,5},PlotRange->All]

NIntegrate[Cos[2x]cosProduct[x,100],{x,0,5}]

NIntegrate[Cos[2x]cosProduct[x,10],{x,0,∞}]

v=0.392699081698724154807830422909937860524645434187231595926812285162093247139938546179016512747455366777;

RealDigits[v][[1]]//Length

N[Pi/8-v,20]

c=7.4073465663169505578887638380364584`20.*^-43;

Sqrt[c]

Log[7.4073465663169505578887638380364584`20.*^-43]

Log[Log[7.4073465663169505578887638380364584`20.*^-43]]

c=First[RealDigits[v,10]]

pi8={3,9,2,6,9,9,0,8,1,6,9,8,7,2,4,1,5,4,8,0,7,8,3,0,4,2,2,9,0,9,9,3,7,8,6,0,5,2,4,6,4,6,1,7,4,9,2,1,8,8,8,2,2,7,6,2,1,8,6,8,0,7,4,0,3,8,4,7,7,0,5,0,7,8,5,7,7,6,1,2,4,8,2,8,5,0,4,3,5,3,1,6,7,7,6,4,6,3,3}

Length/@{c,pi8}

Take[c,99]-pi8

First[Position[%15,_?(#≠0&)]//Flatten]

(∫_0^∞)cos(4x) (∏_(n=1)^∞)cos((x/n))dx

NIntegrate[Evaluate[Cos[4x]Product[Cos[x/n],{n,2000}]],{x,0,100},MaxRecursion->20]//Timing