Functional Equation

Author

Eric W. Weisstein
June 27, 2004

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Calculus/FunctionalEquation.nb.

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

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

(tan^(-1))(1/x)

FullSimplify[f[x]-f[x+1]-f[x^2+x+1]/.f[x_]:>ArcTan[1/x]]

Table[%/.x->Random[Complex],{20}]//Chop//Union

ln(1+1/x)

FullSimplify[l[x]-l[2x+1]-l[2x]/.l[x_]:>Log[1+1/x]]

(l[x]-l[2x+1]-l[2x]/.l[x_]:>Sum[((a[n])/(x^n)),{n,∞}])

(tanh^(-1))(1/x)

FullSimplify[τ[x]-τ[x+1]-τ[x^2+x-1]/.τ[x_]:>ArcTanh[1/x]]

Table[%/.x->Random[Complex],{20}]//Chop//Union

(sinh^(-1))(1/x)

FullSimplify[σ[x]-σ[(Sqrt[(x^2)+1])]-σ[x(Sqrt[(x^2)+1])((Sqrt[(x^2)+1])+(Sqrt[(x^2)+2]))]/.σ[x_]:>ArcSinh[1/x]]

Table[%/.x->Random[Complex],{20}]//Chop//Union

(sin^(-1))(1/x)

FullSimplify[ρ[x]-ρ[(Sqrt[(x^2)+1])]-ρ[x(Sqrt[(x^2)+1])(x+(Sqrt[(x^2)-1]))]/.ρ[x_]:>ArcSin[1/x]]

Table[%/.x->Random[Complex],{20}]//Chop//Union

FullSimplify[ρ[x]-ρ[((x^2)/((x-1)(Sqrt[(x^2)-1])+(Sqrt[2x-1])))]+ρ[(x/(x-1))]/.ρ[x_]:>ArcSin[1/x]]

Table[%/.x->Random[Complex],{20}]//Chop//Union