# Laplace Transform of Constant Multiple

## 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 $1$

$\laptrans {\sin 3 t} = \dfrac 3 {s^2 + 9}$