Z-Transform

Author

Eric W. Weisstein
August 6, 2005

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Combinatorics/Z-Transform.nb.

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

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

Examples

f={KroneckerDelta[n],KroneckerDelta[n-m ],UnitStep[n-1],UnitStep[n-m ],n,n^2,n^3,n^k,b^n,n  b^n,n^2  b^n,n^k  b^ n,Cos[b n],Sin[b n]};

{#,ZTransform[#,n,z]}&/@f//TableForm

Special Cases

ExpToTrig[ExpandAll[ZTransform[#,n,z]]]&/@{Cos[α n],Sin[α n]}//FullSimplify

FullSimplify[Sum[UnitStep[n-m]/z^n,{n,0,∞}],m>0&&m∈Integers]

<<DiscreteMath`Combinatorica`

Table[ZTransform[n^k,n,z]-Sum[Eulerian[k,n] z^(n+1),{n,0,k}]/(z-1)^(k+1),{k,10}]//Simplify