Laplace Transform of Constant Multiple

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Theorem

Let $f$ be a function such that $\mathcal L f$ 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

\(\displaystyle a \laptrans {\map f {a t} }\) \(=\) \(\displaystyle a \int_0^{\to + \infty} e^{-s t} \map f {a t} \rd t\) Definition of Laplace Transform
\(\displaystyle \) \(=\) \(\displaystyle a \paren {\frac 1 a} \int_0^{\to + \infty} e^{-s t} \map f {a t} \rd \paren {a t}\) Primitive of Function of Constant Multiple
\(\displaystyle \) \(=\) \(\displaystyle \int_0^{\to + \infty} e^{-u a t} \map f {a t} \rd \paren {a t}\) where $u = \dfrac s a$
\(\displaystyle \) \(=\) \(\displaystyle \int_0^{\to + \infty} e^{-u a t} \map f {a t} \rd \paren {a t}\)
\(\displaystyle \) \(=\) \(\displaystyle \map F u\) Definition of Laplace Transform
\(\displaystyle \) \(=\) \(\displaystyle \map F {\dfrac s a}\)

$\blacksquare$


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