Lebesgue Constants

Author

Eric W. Weisstein
October 18, 2008

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Constants/LebesgueConstants.nb.

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

©2008 Wolfram Research, Inc. except for portions noted otherwise

(1/π)(∫_0^π)((Abs[sin((1/2) (2 n+1) θ)])/(sin((θ/2))))dθ

LebesgueL[n_]:=Module[{θ},Integrate[Abs[Sin[(2n+1)θ/2]]/Sin[θ/2],{θ,0,π}]]/π

LebesgueL[n]//Timing

n==0

LebesgueL[0]

n==1

LebesgueL[1]//Simplify

n==2

LebesgueL[2]//FullSimplify

n==3

V4.2

With[{n=3},Integrate[Abs[Sin[(2n+1)θ/2]]/Sin[θ/2],{θ,0,π}]/π]//Timing

FullSimplify[%]

N[%]

With[{n=3},NIntegrate[Abs[Sin[(2n+1)θ/2]]/Sin[θ/2],{θ,0,π}]/π]//Timing

V5

With[{n=3},Integrate[Abs[Sin[(2n+1)θ/2]]/Sin[θ/2],{θ,0,π}]/π]//Timing

V7

LebesgueL[3]//FullSimplify//Expand

TraditionalForm[%]

N[%]

n==4

V4.2

With[{n=4},Integrate[Abs[Sin[(2n+1)θ/2]]/Sin[θ/2],{θ,0,π}]/π]//Timing

FullSimplify[Last[%]]

RootReduce[Simplify[RootReduce[TrigToExp[%29]]]]

N[%]

With[{n=4},NIntegrate[Abs[Sin[(2n+1)θ/2]]/Sin[θ/2],{θ,0,π}]/π]//Timing

V5.0

With[{n=4},Integrate[Abs[Sin[(2n+1)θ/2]]/Sin[θ/2],{θ,0,π}]/π]//Timing

Simplify[L[5]]

V7

LebesgueL[4]//FullSimplify//Expand

TraditionalForm[%]

N[%]

n==5

V4.2

FullSimplify[L[5]]

V7

LebesgueL[5]//FullSimplify//Expand

TraditionalForm[%]

n==6

V4.2

FullSimplify[L[6]]

Table[N[L[n],20],{n,6}]

V7

LebesgueL[6]//FullSimplify//Expand

TraditionalForm[%]

(Subscript[L, n])==4 (∫_0^∞)((tanh((2 n+1) x))/((4 (x^2)+(π^2)) tanh(x)))dx

LebesgueL[n_]:=Module[{x},4Integrate[((Tanh[(2n+1)x])/(Tanh[x]))(1/((π^2)+4(x^2))),{x,0,∞}]]

LebesgueL[0]

LebesgueL[1]//Timing

LebesgueL[2]//Timing

Table[4NIntegrate[((Tanh[(2n+1)x])/(Tanh[x]))(1/((π^2)+4(x^2))),{x,0,∞}],{n,0,5}]

Analytic Sums

(Subscript[L, n])==(2/π) (∑_k^n)((tan(((π k)/(2 n+1))))/k)+(1/(2 n+1))

V6

(1/(2n+1))+(2/π)Sum[(1/k)Tan[((π k)/(2n+1))],{k,n}]

V7

(1/(2n+1))+(2/π)Sum[(1/k)Tan[((π k)/(2n+1))],{k,n}]

Table[(1/(2n+1))+(2/π)Sum[(1/k)Tan[((π k)/(2n+1))],{k,n}],{n,3}]//FullSimplify

(Subscript[L, n])==((16 )/(π^2))(∑_k^∞)(∑_j^(k (2 n+1)))(1/((2 j-1) (4 (k^2)-1)))

V7

Table[(16/(π^2))Sum[(1/(4(k^2)-1))(1/(2j-1)),{k,∞},{j,(2n+1)k}],{n,0,1}]//Timing

Table[(16/(π^2))NSum[(1/(4(k^2)-1))(1/(2j-1)),{k,∞},{j,(2n+1)k},WorkingPrecision->50],{n,0,4}]

(16/(π^2))Sum[(1/(4(k^2)-1))(1/(2j-1)),{k,∞},{j,(2n+1)k}]

n==1

(16/(π^2))NSum[(1/(4(k^2)-1))(1/(2j-1)),{k,∞},{j,k}]

(16/(π^2))Sum[(1/(4(k^2)-1))(1/(2j-1)),{k,1000},{j,k}]//N

TimeConstrained[(16/(π^2))Sum[(1/(4(k^2)-1))(1/(2j-1)),{k,∞},{j,k}],120]//Timing

Sum[(1/(2j-1)),{j,k}]//FullSimplify

(16/(π^2))Sum[(((1/2) HarmonicNumber[-(1/2)+k]+Log[2])/(4(k^2)-1)),{k,∞}]//Timing

(16/(π^2))NSum[(((1/2) HarmonicNumber[-(1/2)+k]+Log[2])/(4(k^2)-1)),{k,∞}]

(16/(π^2))NSum[(((1/2) HarmonicNumber[-(1/2)+k]+Log[2])/(4(k^2)-1)),{k,10^5}]//N

Constant c

NSum[((Log[k])/(4(k^2)-1)),{k,∞}]

(4/Pi^2)*(EulerGamma + Log[4] + 2*NSum[Log[k]/(-1 + 4*k^2), {k, Infinity}])

V6

Sum[((Log[k])/(4(k^2)-1)),{k,∞}]

Sum[((DirichletLambda[2j+2]-1)/(2j+1)),{j,0,∞}]

(4/(π^2))(2Log[2]+EulerGamma)+(8/(π^2))N[Plus@@Table[(((1-(2^(-2-2 j))) Zeta[2+2 j]-1)/(2j+1)),{j,0,1000}],110]

c=(4/(π^2))(2Log[2]+EulerGamma)+(8/(π^2))N[Plus@@Table[(((1-(2^(-2-2 j))) Zeta[2+2 j]-1)/(2j+1)),{j,0,1000}],120]

V7

(8/(π^2))Sum[((Log[k])/(4(k^2)-1)),{k,∞}]-(4/(π^2))((Gamma'[(1/2)])/(Gamma[(1/2)]))

FullSimplify[%]

Sum[((DirichletLambda[2j+2]-1)/(2j+1)),{j,0,∞}]