Beurling's Function

Author

Eric W. Weisstein
February 1, 2009

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

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

©2009 Wolfram Research, Inc. except for portions noted otherwise

Analytic Form

V5

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

FunctionExpand[%]

FullSimplify[%]

FullSimplify[Expand[%]]

V7

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

FullSimplify[%,z∈Reals]

FunctionExpand[%]

Expand[%]//Simplify//Apart

Plot

Plot[{1-((2 (-1-z+(z^2) PolyGamma[1,z]) ((Sin[π z])^2))/((π^2) (z^2))),
Sign[z]},{z,-3,3},PlotStyle->{Red,{Blue,Dashing[{.02}]}},
AxesLabel->TraditionalForm/@{x,HoldForm[B[x]]}]

Integral

V5?

Integrate[1-((2 (-1-z+(z^2) PolyGamma[1,z]) ((Sin[π z])^2))/((π^2) (z^2)))-Sign[z],{z,-∞,∞}]

Integrate[1-((2 (-1-z+(z^2) PolyGamma[1,z]) ((Sin[π z])^2))/((π^2) (z^2)))+1,{z,-∞,0}]

Integrate[1-((2 (-1-z+(z^2) PolyGamma[1,z]) ((Sin[π z])^2))/((π^2) (z^2)))+1,{z,0,∞}]

B[z_]:=1-((2 (-1-z+(z^2) PolyGamma[1,z]) ((Sin[π z])^2))/((π^2) (z^2)))

NIntegrate[B[x]-Sign[x],Evaluate[{x,Sequence@@Range[-500,-1],0,1,500}]]

V7

Integrate[1-((2 (-1-z+(z^2) PolyGamma[1,z]) ((Sin[π z])^2))/((π^2) (z^2)))-Sign[z],{z,-∞,∞}]

Integrate[1-((2 (-1-z+(z^2) PolyGamma[1,z]) ((Sin[π z])^2))/((π^2) (z^2)))-Sign[z],{z,-∞,0}]

Integrate[1-((2 (-1-z+(z^2) PolyGamma[1,z]) ((Sin[π z])^2))/((π^2) (z^2)))-Sign[z],{z,0,∞}]

NIntegrate[1-((2 (-1-z+(z^2) PolyGamma[1,z]) ((Sin[π z])^2))/((π^2) (z^2)))-Sign[z],{z,-∞,∞}]