Tetrahedral Vacuum Feynman Diagram

Author

Eric W. Weisstein
January 1, 2009

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

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

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

V==(∑_k^(j-1))(((((-1))^j) cos(((2 π k)/3)))/((j^3) k))==(Piecewise[{{-(((((-1))^j) ((PolyGamma2[0, ((j+1)/3)])+(PolyGamma2[0, ((j+2)/3)])-2 (PolyGamma2[0, (j/3)])+27 log))/(6 (j^3))), (Mod[j, 3])==0}, {-(((((-1))^j) ((PolyGamma2[0, ((j+1)/3)])-2 (PolyGamma2[0, ((j+2)/3)])+(PolyGamma2[0, (j/3)])+27 log))/(6 (j^3))), (Mod[j, 3])==1}, {-(((((-1))^j) (-2 (PolyGamma2[0, ((j+1)/3)])+(PolyGamma2[0, ((j+2)/3)])+(PolyGamma2[0, (j/3)])+27 log))/(6 (j^3))), (Mod[j, 3])==2}}])

v=FullSimplify[Sum[(((((-1))^j)Cos[(2/3)π k])/((j^3)k)),{k,j-1}],j∈Integers]

FullSimplify[v,Mod[j,3]==0]

FullSimplify[v,Mod[j,3]==1]

FullSimplify[v,Mod[j,3]==2]

Piecewise[{FullSimplify[v,Mod[j,3]==#],Mod[j,3]==#}&/@Range[0,2]]

C==(∑_k^∞)((sin(((2 π k)/3)))/(k^2))==(((PolyGamma2[1, (1/3)])-(PolyGamma2[1, (2/3)]))/(6 (Sqrt[3])))

Sum[((Sin[(2/3)π k])/(k^2)),{k,∞}]

FullSimplify[%]