G-Function

Author

Eric W. Weisstein
August 18, 2008

This notebook downloaded from http://mathworld.wolfram.com/notebooks/SpecialFunctions/G-Function.nb.

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

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

Definition

G[z_?NumericQ]:=-PolyGamma[0,(z/2)]+PolyGamma[0,((1+z)/2)]

G[z_?NumericQ]:=2Hypergeometric2F1[1,z,1+z,-1]/z

Plots

<<Utilities`Plot`
<<Utilities`Typesetting`

Real Axis

PlotDiscontinuous[GFunction[z],{z,Sequence@@Range[-5,0],5},
VerticalAsymptotes->Range[-5,0],
PlotRange->{Automatic,{-20,20}},AxesLabel->TraditionalForm/@{x,G[x]}]

Complex Plane

ComplexPlot3D[GFunction[z],{z,-5,5},PlotPoints->100]//Timing

ComplexContourPlot[GFunction[z],{z,-5,5}]//Timing

Integral Representations

2 (∫_0^1)((t^(z-1))/(t+1))dt

FullSimplify[2Integrate[t^(z-1)/(1+t),{t,0,1}]]

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

FullSimplify[2Integrate[Exp[-z t]/(1+Exp[-t]),{t,0,∞}]]

Sums

2 (∑_(n=0)^∞)((((-1))^n)/(n+z))

V6

FullSimplify[2Sum[(-1)^n/(z+n),{n,0,∞}]]

V7

FullSimplify[2Sum[(-1)^n/(z+n),{n,0,∞}]]

FullSimplify[%]

Functional Relations

FullSimplify[G[1+z]-(2/z-G[z])]

FullSimplify[G[1-z]-(2Pi Csc[Pi z]-G[z])]

FullSimplify[-PolyGamma[0,(z/2)]+PolyGamma[0,((1+z)/2)]-2Hypergeometric2F1[1,z,1+z,-1]/z]

Special Values

G[z_]:=PolyGamma[0,(1+z)/2]-PolyGamma[0,z/2]

Table[G[z],{z,5}]/.Log[4]:>2 Log[2]//ColumnForm