Final value theorem

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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).