Integral transforms, operational calculus Definition

Laplace transform

Definition

Let $f : [0, \infty) \to \mathbb{C}$ be a function that is integrable on each interval $[0, T]$ with $T > 0$ . The Laplace transform of $f$ is the function $F = \mathcal{L}\{f\}$ given by $F(s) = \int_0^\infty f(t)\, e^{-s t}\, dt = \lim_{T \to \infty} \int_0^T f(t)\, e^{-s t}\, dt,$ for each complex number $s$ for which the limit exists.

Remarks

Let $f$ be of exponential order: there are constants $M, a$ with $|f(t)| \le M e^{a t}$ for all $t \ge 0$ . Then the integral converges absolutely for $\operatorname{Re} s > a$ . Some transforms: $\mathcal{L}\{1\}(s) = 1/s$ and $\mathcal{L}\{e^{a t}\}(s) = 1/(s - a)$ for $\operatorname{Re} s > \operatorname{Re} a$ , and $\mathcal{L}\{\sin \omega t\}(s) = \omega / (s^2 + \omega^2)$ for $\operatorname{Re} s > 0$ and real $\omega$ . The transform is linear: $\mathcal{L}\{c f + g\} = c \mathcal{L}\{f\} + \mathcal{L}\{g\}$ where both sides converge. It turns derivatives into multiplication by $s$ . See the Laplace transform of a derivative.

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Function concept
Laplace transform definition

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Tag	`05XT5AVJBK`
URI	`m2://05XT5AVJBK`
Kind	definition
Writing	W1 stated
MSC	44A10
Defines	Laplace transform
Uses	Function

Laplace transform

Definition

Remarks

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