Log Cosine Function

Author

Eric W. Weisstein
April 19, 2011

This notebook downloaded from http://mathworld.wolfram.com/notebooks/SpecialFunctions/LogCosineFunction.nb.

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

©2011 Wolfram Research, Inc. except for portions noted otherwise

Function

Integral

Table[Plot[Through[{Re,Im}[Log[Cos[x]]^n]],{x,0,π},Evaluated->True,PlotStyle->{Directive[Thick,Red],Directive[Thick,Yellow]}],{n,3}]

Plot[Table[Log[Cos[x]]^n,{n,5}],{x,0,π/2},Evaluated->True,PlotStyle->{Red,Orange,Yellow,Green,Blue},AxesLabel->{x,Log[Cos[x]]^n}]

V8

Integrate[Log[Cos[x]]^n,{x,0,π},Assumptions->n>0]

Special Cases

<<Utilities`Simplify`

V8

Table[Print[Integrate[Log[Cos[x]]^n,{x,0,π/2}]//LogExpand//Expand//Timing],{n,3}];

Table[NIntegrate[Log[Cos[x]]^n,{x,0,π/2}],{n,5}]

Table[NIntegrate[Log[Sin[x]]^n,{x,0,π}],{n,10}]/Table[NIntegrate[Log[Cos[x]]^n,{x,0,π/2}],{n,10}]

Integrals

(∫_0^(π/2))(x/((x^2)+(log^2)(2 cos(x))))dx

Tewodros Amdeberhan, Apr. 18, 2011

Plot[{(x/((x^2)+((Log[2Cos[x]])^2))),(x/((x^2)+((Log[2Sin[x]])^2)))},{x,0,π/2}]

Integrate[(x/((x^2)+((Log[2Cos[x]])^2))),{x,0,π/2}]

NIntegrate[(x/((x^2)+((Log[2Cos[x]])^2))),{x,0,π/2}]

Integrate[((1/(Log[1+x]))-(1/x)-(1/2))(1/x),{x,0,1}]

FullSimplify[((1/(Log[1+x]))-(1/x)-(1/2))(1/x),0<x<1]

Integrate[-((2+x-((2 x)/(Log[1+x])))/(2 (x^2))),{x,0,1}]

Integrate[-((2+x-((2 x)/(Log[1+x])))/(2 (x^2))),x]

ExpToTrig[-((2+x-((2 x)/(Log[1+x])))/(2 (x^2)))]

Integrate[-(1/(x^2))-(1/(2 x))+(1/(x Log[1+x])),{x,0,1}]

(1/4)(3+Log[2π]-EulerGamma)-(1/2)NIntegrate[((1/(Log[1+x]))-(1/x)-(1/2))(1/x),{x,0,1}]

Integrate[Pochhammer[t,n],{t,0,1}]

Integrate[FunctionExpand@Pochhammer[t,n]//FullSimplify,{t,0,1}]

a[n_]:=Integrate[Pochhammer[t,n],{t,0,1}]

Integrate[Pochhammer[t,n],t]

s=Sum[(((a[n])/(n!))-((a[n+1])/((n+1)!)))((((-1))^(n-1))/n),{n,50}]//N

(1/4)(3+Log[2π]-EulerGamma)+(1/2)s

(∫_0^(π/2))((x^4)/((x^2)+(log^2)(2 sin(x))))dx

Oloa, Apr. 18, 2011

Plot[{((x^4)/((x^2)+((Log[2Cos[x]])^2))),((x^4)/((x^2)+((Log[2Sin[x]])^2)))},{x,0,π/2}]

(32/π)Integrate[((x^4)/((x^2)+((Log[2Sin[x]])^2))),{x,0,π/2}]//Timing

(32/π)Integrate[((x^4)/((x^2)+((Log[2Cos[x]])^2))),{x,0,π/2}]//Timing

{NIntegrate[((x^4)/((x^2)+((Log[2Cos[x]])^2))),{x,0,π/2}],NIntegrate[((x^4)/((x^2)+((Log[2Sin[x]])^2))),{x,0,π/2}]}

N[%]

MZV[s,1,1]

Refine[MZV[s,1,1],s>0]

D[sum[sum[sum[((((a[1])^(-s)) )/(a[2] a[3])),{a[3],1,-1+a[2]}],{a[2],1,-1+a[1]}],{a[1],1,∞}]/.∞->50/.sum->Sum,s]/.s->0.

Refine[sum[((((a[1])^(-Abs[s])) ((Sign[s])^(a[1])))/(a[2] a[3])),{a[3],1,-1+a[2]}],s>0]/.sum->Sum

Sum[((((a[1])^(-s)) HarmonicNumber[-1+a[2]])/(a[2])),{a[2],1,-1+a[1]}]

Sum[(1/2) ((a[1])^(-s)) (((HarmonicNumber[-1+a[1]])^2)-HarmonicNumber[-1+a[1],2]),{a[1],k}]

Sum[(1/2) ((a[1])^(-s)) (((HarmonicNumber[-1+a[1]])^2)-HarmonicNumber[-1+a[1],2]),{a[1],∞}]

FullSimplify[%]

Sum[(1/2) ((a[1])^(-s)) (-HarmonicNumber[-1+a[1],2]+(((EulerGamma+PolyGamma[0,a[1]]))^2)),{a[1],1,∞}]

...hangs...

Sum[(1/2) ((a[1])^(-s)) (-HarmonicNumber[-1+a[1],2]),{a[1],1,∞}]

Sum[(1/2) ((a[1])^(-s)) (((EulerGamma+PolyGamma[0,a[1]]))^2),{a[1],1,∞}]

Sum[((HarmonicNumber[k-1])/k),{k,n-1}]

FullSimplify[%]

-Sum[((Log[n])/(n^s))(1/2) (-HarmonicNumber[-1+n,2]+(((EulerGamma+PolyGamma[0,n]))^2)),{n,20}]/.s->0.

12Zeta[2]Log[2π]-18Zeta[2]EulerGamma+4Zeta[3]+2(EulerGamma^3)+12approx+9Zeta[2]-(3/2)(EulerGamma^2)//N

FullSimplify[Sum[((HarmonicNumber[k-1])/k),{k,n-1}]]

FullSimplify[(EulerGamma-HarmonicNumber[n]+Log[n]+(1/(2n)))((1/2) (-HarmonicNumber[-1+n,2]+(((EulerGamma+PolyGamma[0,n]))^2)))]

f[n_]:=(1/2) (EulerGamma+(1/(2 n))-HarmonicNumber[n]+Log[n]) (-HarmonicNumber[-1+n,2]+(((EulerGamma+PolyGamma[0,n]))^2))

f/{100.,10^5.}

Plot[(1/2) (EulerGamma+(1/(2 n))-HarmonicNumber[n]+Log[n]) (-HarmonicNumber[-1+n,2]+(((EulerGamma+PolyGamma[0,n]))^2)),{n,1,20},PlotRange->All]

h=Sum[(1/2) (EulerGamma+(1/(2 n))-HarmonicNumber[n]+Log[n]) (-HarmonicNumber[-1+n,2]+(((EulerGamma+PolyGamma[0,n]))^2)),{n,1000}]//N

Sum[(1/2) (EulerGamma+(1/(2 n))-HarmonicNumber[n]+Log[n]) (-HarmonicNumber[-1+n,2]+(((EulerGamma+PolyGamma[0,n]))^2)),{n,∞}]//Timing

Sum[(1/2.) (EulerGamma+(1/(2 n))-HarmonicNumber[n]+Log[n]) (-HarmonicNumber[-1+n,2]+(((EulerGamma+PolyGamma[0,n]))^2)),{n,10^3}]//Timing

(h=NSum[(1/2) (EulerGamma+(1/(2 n))-HarmonicNumber[n]+Log[n]) (-HarmonicNumber[-1+n,2]+(((EulerGamma+PolyGamma[0,n]))^2)),{n,10^5}])//Timing

NSum[(1/2) (EulerGamma+(1/(2 n))-HarmonicNumber[n]+Log[n]) (-HarmonicNumber[-1+n,2]+(((EulerGamma+PolyGamma[0,n]))^2)),{n,∞}]//Timing

-1-(1/4)Zeta[2]+(1/4)(EulerGamma^2)-(1/12)(EulerGamma^3)+(1/4)EulerGamma Zeta[2]-(1/6)Zeta[3]-h

z=1.3969896209263858690159994844722589525351856547313582554652;

l=Table[((1`100)/2) (EulerGamma+(1/(2 n))-HarmonicNumber[n]+Log[n]) (-HarmonicNumber[-1+n,2]+(((EulerGamma+PolyGamma[0,n]))^2)),{n,1000}];

Plot[f[n]-((6 (EulerGamma^2)-(π^2)-12 EulerGamma Log[(1/n)]+6 ((Log[(1/n)])^2))/(144 (n^2))),{n,200,1000}]

s=Sum[((6 (EulerGamma^2)-(π^2)-12 EulerGamma Log[(1/n)]+6 ((Log[(1/n)])^2))/(144 (n^2))),{n,∞}]

N[s]

Sum[f[n],{n,k}]

s

h=With[{max=5000},
Sum[f[n],{n,max}]+NSum[((6 (EulerGamma^2)-(π^2)-12 EulerGamma Log[(1/n)]+6 ((Log[(1/n)])^2))/(144 (n^2))),{n,max+1,∞},WorkingPrecision->100,PrecisionGoal->20]
]

-1-(1/4)Zeta[2]+(1/4)(EulerGamma^2)-(1/12)(EulerGamma^3)+(1/4)EulerGamma Zeta[2]-(1/6)Zeta[3]-h

z

Other MZV

ζ'(0)

Refine[MZV[s],s>0]/.sum->Sum

D[%,s]

%/.s->0

Refine[MZV[s],s>0]

D[((a[1])^(-s)),s]

Sum[-((a[1])^(-s)) Log[a[1]],{a[1],1,∞}]/.s->0

ζ'(0,1)

Refine[MZV[s,1],s>0]

Sum[(((a[1])^(-s))/(a[2])),{a[1],1,∞},{a[2],1,-1+a[1]}]//Timing

Sum[(((a[1])^(-s))/(a[2])),{a[2],1,-1+a[1]}]

Sum[((a[1])^(-s)) HarmonicNumber[-1+a[1]],{a[1],∞}]//Timing

D[((a[1])^(-s)) ,s]

Sum[-((a[1])^(-s)) Log[a[1]] HarmonicNumber[-1+a[1]],{a[1],∞}]//Timing