Let
be a function on
. The Laplace transform of
is defined by the integral
![{\displaystyle F(s)={\mathcal {L}}\{f\}(s)=\int _{0}^{\infty }e^{-st}f(t)dt\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c4df44b634aee97a23c567ff60f219b8afb2ccee)
The domain of
is all values of
such that the integral exists.
Let
and
be functions whose Laplace transforms exist for
and let
and
be constants. Then, for
,
![{\displaystyle {\mathcal {L}}\{af+bg\}=a{\mathcal {L}}\{f\}+b{\mathcal {L}}\{g\}\,,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c43778872ce8183b1268f8ff5d090d8b14801580)
which can be proved using the properties of improper integrals.
If the Laplace transform
exists for
, then
![{\displaystyle {\mathcal {L}}\{e^{at}f(t)\}(s)=F(s-a)\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/62b269ff889e2a9c61765e09d8419dd44276a163)
for
.
Proof.
If
, then
- Proof:
![{\displaystyle {\mathcal {L}}\{f'(t)\}=\int _{0}^{\infty }f'(t)e^{-st}dt}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7c92685f4ec8e7fcc1058729b2d6c5261e785da7)
![{\displaystyle =\lim _{C\to \infty }\int _{0}^{C}f'(t)e^{-st}dt}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6375d066145a10af0c51cad4fab03ae7c6a9d0b1)
(integrating by parts)
![{\displaystyle =-f(0)+s\lim _{C\to \infty }\int _{0}^{C}f(t)e^{-st}dt}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eff9944afc70ffca64d9ccd409dd27a909d086a1)
![{\displaystyle =s{\mathcal {L}}\{f(t)\}-f(0)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9235ea8eaa4b930f8ad087199ed49ed1912da8ff)
![{\displaystyle =sF(s)-f(0)\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/53ebe2124575dc0620d2c04743746e7c1fb2b5d4)
Using the above and the linearity of Laplace Transforms, it is easy to prove that
If
, then
![{\displaystyle {\mathcal {L}}\{1\}={1 \over s}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f4c535c6497847806f150aed24fd7267144549d0)
![{\displaystyle {\mathcal {L}}\{e^{at}\}={1 \over s-a}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b0aea7565a5d8313ce84be8ec9dd8fed7b70416f)
![{\displaystyle {\mathcal {L}}\{\cos \omega t\}={s \over s^{2}+\omega ^{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/240a792675905a57956304664e5de4b1e7bbd3e6)
![{\displaystyle {\mathcal {L}}\{\sin \omega t\}={\omega \over s^{2}+\omega ^{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6f448b5b68d6f07f7e0cea00bd7e10b285dfe37b)
![{\displaystyle {\mathcal {L}}\{1\}={1 \over s}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f4c535c6497847806f150aed24fd7267144549d0)
![{\displaystyle {\mathcal {L}}\{t^{n}\}={n! \over s^{n+1}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1c004e46c95a557ecd40660049c864946285943d)