Integral transforms, operational calculus Laplace transform: operational rules Proposition

Scaling rule for the Laplace transform

Statement

Let $f \colon [0, \infty) \to \C$ be integrable on each interval $[0, T]$ , and let $F$ be its Laplace transform. Let $c > 0$ be real, and let $s$ be a complex number such that the Laplace integral of $f$ converges at $s / c$ . Then the Laplace integral of $t \mapsto f(c t)$ converges at $s$ , and $\mathcal{L}\{f(c t)\}(s) = \frac{1}{c} F\!\left(\frac{s}{c}\right).$

Remarks

The substitution $\tau = c t$ turns $\int_0^\infty f(c t) e^{-s t} \, dt$ into $\frac{1}{c} \int_0^\infty f(\tau) e^{-(s / c) \tau} \, d\tau$ . Example: from $\mathcal{L}\{\sin t\}(s) = 1 / (s^2 + 1)$ one gets $\mathcal{L}\{\sin \omega t\}(s) = \frac{1}{\omega} \cdot \frac{1}{(s / \omega)^2 + 1} = \frac{\omega}{s^2 + \omega^2}$ for $\omega > 0$ . The condition $c > 0$ is needed: for $c < 0$ the function $f(c t)$ uses values of $f$ at negative times.

Going deeper

For a student

The rule lets one derive a whole family of transforms from one transform, for example all $\sin \omega t$ from $\sin t$ . It also explains how the abscissa of convergence changes: it is multiplied by $c$ .

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Function concept
Laplace transform definition
Scaling rule for the Laplace transform statement

Tag	`QE2PT4TNDC`
URI	`m2://QE2PT4TNDC`
Kind	statement, proposition
Writing	W0 stub
MSC	44A10
Status	This copy of the site has no build data. Run `m2 status QE2PT4TNDC`.
About	Laplace transform
Uses	Laplace transform