Integral transforms, operational calculus Definition

Inverse Laplace transform

Definition

Let $a$ be a real number and let $F$ be a complex function on the half-plane $\{s \in \mathbb{C} : \operatorname{Re} s > a\}$ . An inverse Laplace transform of $F$ is a piecewise continuous function $f : [0, \infty) \to \mathbb{C}$ with two properties. First, there are constants $M, b$ with $|f(t)| \le M e^{b t}$ for $t \ge 0$ . Second, the Laplace transform of $f$ equals $F(s)$ for each $s$ with $\operatorname{Re} s > \max(a, b)$ . The notation is $f = \mathcal{L}^{-1}\{F\}$ .

Remarks

By Lerch's theorem, two such functions with the same transform agree at each point where both are continuous. So $\mathcal{L}^{-1}\{F\}$ is unique in practice, and unique among continuous functions. Let $\gamma$ be a real number larger than the real parts of all singularities of $F$ . Under suitable conditions, the Bromwich integral recovers $f$ : $f(t) = \frac{1}{2 \pi i} \lim_{T \to \infty} \int_{\gamma - iT}^{\gamma + iT} e^{s t} F(s)\, ds$ . In practice, one splits $F$ into partial fractions and reads the inverse of each term from a table. For example, $\mathcal{L}^{-1}\{1/(s - a)\} = e^{a t}$ .

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Function concept
Laplace transform definition
Inverse Laplace transform definition

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Tag	`0JRW4X1MDC`
URI	`m2://0JRW4X1MDC`
Kind	definition
Writing	W1 stated
MSC	44A10
Defines	Inverse Laplace transform
Uses	Laplace transform

Inverse Laplace transform

Definition

Remarks

Reading path

Where this leads

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