Laplace Transform of Constant Multiple
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Theorem
Let $f$ be a function such that $\LL f$ exists.
Let $\laptrans {\map f t} = \map F s$ denote the Laplace transform of $f$.
Let $\map F s$ denote $\laptrans {\map f t}$.
Let $a \in \C$ or $\R$ be constant.
Then:
- $a \laptrans {\map f {a t} } = \map F {\dfrac s a}$
Proof
\(\ds a \laptrans {\map f {a t} }\) | \(=\) | \(\ds a \int_0^{\to + \infty} e^{-s t} \map f {a t} \rd t\) | Definition of Laplace Transform | |||||||||||
\(\ds \) | \(=\) | \(\ds a \paren {\frac 1 a} \int_0^{\to + \infty} e^{-s t} \map f {a t} \rd \paren {a t}\) | Primitive of Function of Constant Multiple | |||||||||||
\(\ds \) | \(=\) | \(\ds \int_0^{\to + \infty} e^{-u a t} \map f {a t} \rd \paren {a t}\) | where $u = \dfrac s a$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \int_0^{\to + \infty} e^{-u a t} \map f {a t} \rd \paren {a t}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \map F u\) | Definition of Laplace Transform | |||||||||||
\(\ds \) | \(=\) | \(\ds \map F {\dfrac s a}\) |
$\blacksquare$
Also presented as
This result can also be given as:
- $\laptrans {\map f {a t} } = \dfrac 1 a \map F {\dfrac s a}$
Also known as
This property of the Laplace transform operator is sometimes seen referred to as the change of scale property.
Examples
Example $1$
- $\laptrans {\sin 3 t} = \dfrac 3 {s^2 + 9}$
Sources
- 1965: Murray R. Spiegel: Theory and Problems of Laplace Transforms ... (previous) ... (next): Chapter $1$: The Laplace Transform: Some Important Properties of Laplace Transforms: $4$. Change of scale property: Theorem $1 \text{-} 5$
- 1965: Murray R. Spiegel: Theory and Problems of Laplace Transforms ... (previous) ... (next): Chapter $1$: The Laplace Transform: Solved Problems: Translation and Change of Scale Properties: $11$
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 32$: Table of General Properties of Laplace Transforms: $32.4$