Whitney-Mikhlin Extension Constants

Author

Eric W. Weisstein
July 16, 2003

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Constants/Whitney-MikhlinExtensionConstants.nb.

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

©2005 Wolfram Research, Inc. except for portions noted otherwise

Recurrence

RSolve[{
a[k]==a[k-2]-(2(k-1)-1)a[k-1],
a[0]==E+1/E,a[1]==E-1/E},
a[k],{k}]//FullSimplify

RSolve[{
b[k]==b[k-2]+(2(k-1)-1)b[k-1],
b[0]==1/E,b[1]==1/E},
{b[k]},{k}]//FullSimplify

χ[kk_]:=Module[{k=(kk-1)/2},
Sqrt[1+a[k]b[k+1]/a[k+1]/b[k]]
]

a[k_]:=((2 I BesselI[-(1/2)+k,1] BesselK[(1/2),-1])/E)
b[k_]:=((2 I BesselK[(1/2),-1] BesselK[-(1/2)+k,1])/(E π))

FullSimplify[χ[k]]

FunctionExpand[Table[(((BesselI[(k/2),1] BesselK[(1/2) (-2+k),1]))^(-1/2)),{k,3,13,2}]]//FullSimplify