Final Value Theorem of Laplace Transform
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Theorem
Let $\laptrans {\map f t} = \map F s$ denote the Laplace transform of the real function $f$.
Then:
- $\ds \lim_{t \mathop \to \infty} \map f t = \lim_{s \mathop \to 0} s \, \map F s$
if those limits exist.
General Result
Let $\ds \lim_{t \mathop \to \infty} \dfrac {\map f t} {\map g t} = 1$.
Then:
- $\ds \lim_{s \mathop \to 0} \dfrac {\map F s} {\map G s} = 1$
if those limits exist.
Proof
From Laplace Transform of Derivative:
- $(1): \quad \laptrans {\map {f'} t} = s \, \map F s - \map f 0$
We have that:
\(\ds \lim_{s \mathop \to 0} \laptrans {\map {f'} t}\) | \(=\) | \(\ds \lim_{s \mathop \to 0} \int_0^\infty e^{-s t} \map {f'} t \rd t\) | Definition of Laplace Transform | |||||||||||
\(\ds \) | \(=\) | \(\ds \int_0^\infty \map {f'} t \rd t\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \lim_{L \mathop \to \infty} \int_0^L \map {f'} t \rd t\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \lim_{L \mathop \to \infty} \paren {\map f L - \map f 0}\) | Fundamental Theorem of Calculus | |||||||||||
\(\text {(2)}: \quad\) | \(\ds \) | \(=\) | \(\ds \lim_{t \mathop \to \infty} \map f t - \map f 0\) |
Hence:
\(\ds \lim_{s \mathop \to 0} \laptrans {\map {f'} t}\) | \(=\) | \(\ds \lim_{s \mathop \to 0} s \, \map F s - \map f 0\) | from $(1)$ | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \lim_{t \mathop \to \infty} \map f t - \map f 0\) | \(=\) | \(\ds \lim_{s \mathop \to 0} s \, \map F s - \map f 0\) | from $(2)$ | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \lim_{t \mathop \to \infty} \map f t\) | \(=\) | \(\ds \lim_{s \mathop \to 0} s \, \map F s\) |
$\Box$
Suppose that $f$ is not continuous at $t = 0$.
From Laplace Transform of Derivative with Discontinuity at Zero:
- $\laptrans {\map {f'} t} = s \, \map F s - \map f {0^+}$
which means:
- $(3): \quad \laptrans {\map {f'} t} = s \, \map F s - \ds \lim_{u \mathop \to 0} \map f u$
We have that:
\(\ds \lim_{s \mathop \to 0} \laptrans {\map {f'} t}\) | \(=\) | \(\ds \lim_{u \mathop \to 0} \paren {\lim_{s \mathop \to 0} \int_u^\infty e^{-s t} \map {f'} t \rd t}\) | Definition of Laplace Transform | |||||||||||
\(\ds \) | \(=\) | \(\ds \lim_{u \mathop \to 0} \paren {\int_u^\infty \map {f'} t \rd t}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \lim_{u \mathop \to 0} \paren {\lim_{L \mathop \to \infty} \int_u^L \map {f'} t \rd t}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \lim_{u \mathop \to 0} \paren {\lim_{L \mathop \to \infty} \paren {\map f L - \map f u} }\) | Fundamental Theorem of Calculus | |||||||||||
\(\ds \) | \(=\) | \(\ds \lim_{u \mathop \to 0} \paren {\lim_{t \mathop \to \infty} \map f t - \map f u}\) | ||||||||||||
\(\text {(4)}: \quad\) | \(\ds \) | \(=\) | \(\ds \lim_{t \mathop \to \infty} \map f t - \lim_{u \mathop \to 0} \map f u\) |
Hence:
\(\ds \lim_{s \mathop \to 0} \laptrans {\map {f'} t}\) | \(=\) | \(\ds \lim_{s \mathop \to 0} s \, \map F s - \lim_{u \mathop \to 0} \map f u\) | from $(3)$ | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \lim_{t \mathop \to \infty} \map f t - \lim_{u \mathop \to 0} \map f u\) | \(=\) | \(\ds \lim_{s \mathop \to 0} s \, \map F s - \lim_{u \mathop \to 0} \map f u\) | from $(4)$ | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \lim_{t \mathop \to \infty} \map f t\) | \(=\) | \(\ds \lim_{s \mathop \to 0} s \, \map F s\) |
$\blacksquare$
Examples
Example 1
Consider the real function $f: \R \to \R$ defined as:
- $\map f t = 3 e^{-2 t}$
From Laplace Transform of Exponential:
- $\laptrans {\map f t} = \dfrac 3 {s + 2}$
Then by the Initial Value Theorem of Laplace Transform:
\(\ds \lim_{t \mathop \to 0} 3 e^{-2 t}\) | \(=\) | \(\ds \lim_{s \mathop \to \infty} \dfrac 3 {s + 2}\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds 3\) | \(=\) | \(\ds 3\) |
Sources
- 1965: Murray R. Spiegel: Theory and Problems of Laplace Transforms ... (previous) ... (next): Chapter $1$: The Laplace Transform: Some Important Properties of Laplace Transforms: $12$. Final-value theorem: Theorem $1 \text{-} 17$
- 1965: Murray R. Spiegel: Theory and Problems of Laplace Transforms ... (previous) ... (next): Chapter $1$: The Laplace Transform: Solved Problems: Initial and Final Value Theorems: $26$