K-Function

Author

Eric W. Weisstein
August 19, 2008

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

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

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

Definitions

Product

Table[Product[k^k,{k,n-1}],{n,5}]

V6

Product[k^k,{k,n-1}]

V7

Product[k^k,{k,n-1}]

MathWorld`SpecialFunctions

KFunction[z]

Table[KFunction[z],{z,5}]//FullSimplify

(((2 π))^(-(n/2))) exp((1/2) (n-1) n+(PolyGamma2[-2, n]))

FullSimplify[Exp[n(n-1)/2+PolyGamma[-2,n]](2Pi)^(-n/2)]

Table[Exp[n(n-1)/2+PolyGamma[-2,n]](2Pi)^(-n/2),{n,10}]//FunctionExpand//FullSimplify

Table[Hyperfactorial[n-1],{n,10}]

Hyperfactorial[π-1.]

With[{n=Pi-1.},Exp[n(n-1)/2+PolyGamma[-2,n]](2Pi)^(-n/2)]

With[{z=π-1.},(E^(-(1/12)+Derivative[1,0][Zeta][-1,z])) Glaisher]

FullSimplify[(E^(-(1/12)+Derivative[1,0][Zeta][-1,n]))-Exp[n(n-1)/2+PolyGamma[-2,n]](2Pi)^(-n/2)]

Plots

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

Real Axis

Plot[SpecialFunctions`Hyperfactorial[n-1],{n,-4,3}]

Plot[KFunction[x],{x,-4,3},
PlotStyle->Red,
AxesLabel->TraditionalForm/@{x,KFunction[x]}]

Complex Plane

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

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

Values

(∏_(k=1)^(n-1))(k^k)

Product[k^k,{k,n-1}]

Table[Product[k^k,{k,n-1}],{n,5}]