Definition:Inverse Laplace Transform
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Definition
Definition 1
Let $\map f s : S \to \R$ be a function, where $S \subset \R$.
Let $F$ be another function such that $F$ is the Laplace transform of $f$.
Then, $f$ is the inverse Laplace transform of $F$.
Definition 2
Let $\map f s: S \to \C$ be a complex function, where $S \subset \C$.
The inverse Laplace transform of $f$, denoted $\map F t: \R \to S$, is defined as:
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| \(\ds \map F t\) | \(=\) | \(\ds \dfrac 1 {2 \pi i} \PV_{c \mathop - i \, \infty}^{c \mathop + i \, \infty} e^{s t} \map f s \rd s\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \frac 1 {2 \pi i} \lim_{T \mathop \to \infty} \int_{c \mathop - i \, T}^{c \mathop + i \, T} e^{s t} \map f s \rd s\) |
where:
- $\PV$ is the Cauchy principal value of the integral
- $c$ is any real constant such that all the singular points of $\map f s$ lie to the left of the line $\map \Re s = c$ in the complex $s$ plane.
Also see
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- Results about inverse Laplace transforms can be found here.
Source of Name
This entry was named for Pierre-Simon de Laplace.