Log Gamma Function

Author

Eric W. Weisstein
November 9, 2008

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

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

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

Plots

Real Axis

Plot[LogGamma[z],{z,0,5},PlotStyle->Red,AxesLabel->{x,ln[Gamma[x]]}]

Complex Plane

Log of Gamma

ComplexPlot3D[Log[Gamma[z]],{z,-5,5},ComplexPlotLabel->ln[Gamma[z]]]//Timing

ComplexContourPlot[Log[Gamma[z]],{z,-5,5},ComplexPlotLabel->ln[Gamma[z]]]//Timing

LogGamma

ComplexPlot3D[LogGamma[z],{z,-5,5},ComplexPlotLabel->ln[Gamma[z]]]//Timing

ComplexContourPlot[LogGamma[z],{z,-5,5},ComplexPlotLabel->ln[Gamma[z]]]//Timing

Difference

ComplexPlot3D[LogGamma[z]-Log[Gamma[z]],{z,-5,5},
PlotPoints->150]//Timing

ComplexContourPlot[LogGamma[z]-Log[Gamma[z]],{z,-5,5}]//Timing

LogGamma[z]-Log[Gamma[z]]/.z->10.I

Sums

(∑_k^∞)((z/k)-ln((z/k)+1))-ℽ z-ln(z)==(LogGamma[z])

Boros and Moll 2004, p. 204

V5

-Log[x]-EulerGamma x-Sum[Log[1+(x/n)]-(x/n),{n,∞}]

With[{x=.3},{Log[Gamma[x]],-Log[x]-EulerGamma x-NSum[Log[1+(x/n)]-(x/n),{n,∞}]}]

V7

-Log[x]-EulerGamma x-Sum[Log[1+(x/n)]-(x/n),{n,∞}]

Integrals

(∫_1^z)(PolyGamma[t])dt==(LogGamma[z])

Integrate[PolyGamma[t],{t,1,z}]

2 (∫_0^∞)(((tan^(-1))((z/a)))/(exp(2 π z)-1))dz-a+(a-(1/2)) ln(a)+(1/2) ln(2 π)==(LogGamma[a])

Binet's Log Gamma Formula

V5

(a-1/2)Log[a]-a+Log[2π]/2+2Integrate[ArcTan[z/a]/(Exp[2π z]-1),{z,0,∞}]

V7

(a-1/2)Log[a]-a+Log[2π]/2+2Integrate[ArcTan[z/a]/(Exp[2π z]-1),{z,0,∞}]

(∫_0^∞)((exp(-t) ((z-1)-((1-exp(t (-(z-1))))/(1-exp(-t)))))/t)dt

Malmstén's Formula

V5

Integrate[((z-1)-(1-Exp[-(z-1)t])/(1-Exp[-t]))Exp[-t]/t,{t,0,∞}]

V7

Integrate[((z-1)-(1-Exp[-(z-1)t])/(1-Exp[-t]))Exp[-t]/t,{t,0,∞}]

(∫_0^1)ln((Gamma[x]))dx==(1/2) ln(2 π)

V5.2

{Integrate[LogGamma[x],{x,0,1}],Integrate[Log[Gamma[x]],{x,0,1}]}

{N[%],(1/2)Log[2π]//N}

-Zeta'[0]

V6

{Integrate[LogGamma[x],{x,0,1}],Integrate[Log[Gamma[x]],{x,0,1}]}

V7

{Integrate[LogGamma[x],{x,0,1}],Integrate[Log[Gamma[x]],{x,0,1}]}

(∫_0^1)(log^2)(Γ(x))dx

V5.2

{Integrate[Log[Gamma[x]]^2,{x,0,1}],Integrate[LogGamma[x]^2,{x,0,1}]}

{NIntegrate[Log[Gamma[x]]^2,{x,0,1},WorkingPrecision->20],NIntegrate[LogGamma[x]^2,{x,0,1},WorkingPrecision->20]}//Timing

N[(1/12)(EulerGamma^2)+((π^2)/48)+(1/6)EulerGamma Log[2π]+(1/3)Log[2π]^2-(EulerGamma+Log[2π])((Zeta'[2])/(π^2))+((Zeta''[2])/(2(π^2))),15]

V6

{Integrate[Log[Gamma[x]]^2,{x,0,1}],Integrate[LogGamma[x]^2,{x,0,1}]}

V7

{Integrate[Log[Gamma[x]]^2,{x,0,1}],Integrate[LogGamma[x]^2,{x,0,1}]}//Timing

(∫_0^(1/2))(LogGamma[x+1])dx==((3 ln(A))/2)-(1/2)-(1/24) (7 ln(2))+((ln(π))/4)

Integrate[LogGamma[x+1],{x,0,1/2}]//FullSimplify//Expand