Hadjicostas's Formula

Author

Eric W. Weisstein
January 25, 2005

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

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

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

Hadjicostas's Formula

Graphics

<<Utilities`Plot`

ComplexContourPlot[Gamma[s+2](Zeta[s+2]-(1/(s+1))),{s,-5I-2,5I+5},PlotPoints->250]//Timing

ComplexPlot3D[Gamma[s+2](Zeta[s+2]-(1/(s+1))),{s,-5I-2,5I+5},PlotPoints->300]//Timing

Table

Integrate[((1-x)/(1-x y))(-Log[x y])^s,{x,0,1},{y,0,1}]//Timing

Table[Integrate[((1-x)/(1-x y))(-Log[x y])^s,{x,0,1},{y,0,1}]//Timing,{s,-1,3}]//ColumnForm

Table[Integrate[((1-x)/(1-x y))(-Log[x y])^s,{y,0,1},{x,0,1}]//Timing,{s,-1,3}]//ColumnForm

Table[{NIntegrate[((1-x)/(1-x y))(-Log[x y])^s,{x,0,1},{y,0,1}],Gamma[s+2](Zeta[s+2]-(1/(s+1)))},
{s,-1,5}]//N

Gamma[s+2](Zeta[s+2]-(1/(s+1)))/.s->-1

Limit[Gamma[s+2](Zeta[s+2]-(1/(s+1))),s->-1]

Sondow's Formula

Graphics

<<Utilities`Plot`

ComplexContourPlot[Gamma[s+2](DirichletEta[s+2]+((1-2DirichletEta[s+1])/(s+1))),{s,-5I-3,5I+5},PlotPoints->250,ComplexPlotLabel->Gamma[s+2](η[s+2]+((1-2η[s+1])/(s+1)))]//Timing

ComplexPlot3D[Gamma[s+2](DirichletEta[s+2]+((1-2DirichletEta[s+1])/(s+1))),{s,-5I-3,5I+5},ComplexPlotLabel->Gamma[s+2](η[s+2]+((1-2η[s+1])/(s+1))),PlotPoints->300]//Timing

Table

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

Table[Integrate[((1-x)/(1+x y))(-Log[x y])^s,{x,0,1},{y,0,1}]//Timing,{s,-1,3}]//ColumnForm

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

N[Log[(((Sqrt[π])(Glaisher^6))/((2^(7/6))E))],20]

N[(((Sqrt[π])(Glaisher^6))/((2^(7/6))E)),20]

With[{s=-2},NIntegrate[((1-x)/(1+x y))(-Log[x y])^s,{y,0,1},{x,0,1}]]

Table[Integrate[((1-x)/(1+x y))(-Log[x y])^s,{y,0,1},{x,0,1}]//Timing,{s,-2,3}]//ColumnForm

Table[{NIntegrate[((1-x)/(1+x y))(-Log[x y])^s,{x,0,1},{y,0,1}],Gamma[s+2](DirichletEta[s+2]+((1-2DirichletEta[s+1])/(s+1)))},
{s,-2,5}]//N