Lesson plan
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) in continuous time has (unilateral) Laplace transform F(s) , then a final value theorem establishes conditions under which \lim_{t\,\to\,\infty}f(t) = \lim_{s\,\to\, 0}{sF(s)}. Likewise, if f[k] in discrete time has (unilateral) Z-transform F(z) , then a final value theorem establishes conditions under which \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) \text{ (or }f[k]) to calculate \lim_{s\,\to\, 0}{sF(s)}. Conversely, a Tauberian final value theorem makes assumptions about the frequency-domain behaviour of F(s) to calculate \lim_{t\to\infty}f(t) \text{(or }\lim_{k\to\infty}f[k]) (see Abelian and Tauberian theorems for integral transforms).
Article summary
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) in continuous time has (unilateral) Laplace transform F(s) , then a final value theorem establishes conditions under which \lim_{t\,\to\,\infty}f(t) = \lim_{s\,\to\, 0}{sF(s)}. Likewise, if f[k] in discrete time has (unilateral) Z-transform F(z) , then a final value theorem establishes conditions under which \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) \text{ (or }f[k]) to calculate \lim_{s\,\to\, 0}{sF(s)}. Conversely, a Tauberian final value theorem makes assumptions about the frequency-domain behaviour of F(s) to calculate \lim_{t\to\infty}f(t) \text{(or }\lim_{k\to\infty}f[k]) (see Abelian and Tauberian theorems for integral transforms).
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Discussion questions
- Summarise the article in your own words. Which single claim would you verify first, and why?
- Which claim in this article is most likely to be revised as the sources update? How would you check its current status?
- If two sources disagreed about one of these facts, what would you need in order to trust one over the other?
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