Laplace Transform of Function of Constant Multiple

From ProofWiki
Jump to navigation Jump to search

Theorem

Let $\laptrans {\map f t} = \map F s$ denote the Laplace transform of a real function $f$.


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
\(\ds \) \(=\) \(\ds \map F {\dfrac s a}\)

$\blacksquare$


Examples

Example $1$

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


Also presented as

The Laplace Transform of Function of Constant Multiple can also be given as:

$\laptrans {\map f {a t} } = \dfrac 1 a \map F {\dfrac s a}$


Also known as

The Laplace Transform of Function of Constant Multiple is sometimes seen referred to as the change of scale property.


Sources

Retrieved from "https://proofwiki.org/w/index.php?title=Laplace_Transform_of_Function_of_Constant_Multiple&oldid=743427"