Final value theorem

In mathematical analysis, the final value theorem (FVT) is one of several similar theorems used to relate frequency domain expressions to the time domain behavior as time approaches infinity. Mathematically, if f ( t ) {\displaystyle f(t)} in continuous time has (unilateral) Laplace transform F ( s ) {\displaystyle F(s)} , then a final value theorem establishes conditions under which lim t → ∞ f ( t ) = lim s → 0 s F ( s ) . {\displaystyle \lim _{t\,\to \,\infty }f(t)=\lim _{s\,\to \,0}{sF(s)}.} Likewise, if f [ k ] {\displaystyle f[k]} in discrete time has (unilateral) Z-transform F ( z ) {\displaystyle F(z)} , then a final value theorem establishes conditions under which lim k → ∞ f [ k ] = lim z → 1 ( z − 1 ) F ( z ) . {\displaystyle \lim _{k\,\to \,\infty }f[k]=\lim _{z\,\to \,1}{(z-1)F(z)}.} An Abelian final value theorem makes assumptions about the time-domain behavior of f ( t ) (or f [ k ] ) {\displaystyle f(t){\text{ (or }}f[k])} to calculate lim s → 0 s F ( s ) . {\textstyle \lim _{s\,\to \,0}{sF(s)}.} Conversely, a Tauberian final value theorem makes assumptions about the frequency-domain behaviour of F ( s ) {\displaystyle F(s)} to calculate lim t → ∞ f ( t ) {\displaystyle \lim _{t\to \infty }f(t)} (or lim k → ∞ f [ k ] ) {\displaystyle {\text{(or }}\lim _{k\to \infty }f[k])} (see Abelian and Tauberian theorems for integral transforms).