Laplace Transform

class: center, middle, inverse, title-slide

.title[
# Laplace Transform
]
.subtitle[
## Session 09
]
.author[
### Alejandro Ucan
]
.date[
### 2023-11-05
]

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# Goals:

* Motivate the Laplace transform. <br/><br/>
* Define the Laplace transform. <br/><br/>
* Compute the Laplace transform of some functions.

---
# Integral transforms

> The derivative and anti-derivative are transformations of functions, and can be thought of as linear transformations.

> Remember that through anti-derivatives we can define functions.

---
### Motivation:

Some problems (stated in some domain) are difficult to solve in their original presentation, but transformed can be simpler. The _integral transform_ maps an equation in its original "domain" into another "domain" in such a way that the manipulation of the equation is simpler than originally and the solution can be reconverted to the original domain.

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# Laplace Transform

> __Definition:__ the Laplace transform of a function `$f(t)$` with `$t\geq 0$` is the function `$F(s)$` defined by `$$F(s)=\int_0^\infty f(t)e^{-st}dt=\lim_{b\to\infty}\int_0^b f(t)e^{-st}$$` where `$s$` is a parameter. <br/><br/> The usual notation for the Laplace transform is `$\mathcal{L}\{f(t)\}=F(s).$`
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### Example:

> Compute the Laplace transforms of `$f(t)=1$` and `$f(t)=t.$`

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### Example:

> Compute the Laplace transform of: `$f(t)=-3t^2+2t+1,$` `$f(t)=\cos(3t)$` and `$f(t)=e^{2t}.$`
---
### The advantages

> The Laplace transform calculation is something that you do only once, because there are some formulaes that you can use directly.
<br/><br/>
> __Theorem:__ It follows that:
  * `$\mathcal{L}\{1\}=\frac{1}{s}.$`
  * `$\mathcal{L}\{t^n\}=\frac{n!}{s^{n+1}}.$` for an integer `$n>0$`
  * `$\mathcal{L}\{e^{at}\}=\frac{1}{s-a}.$`
  * `$\mathcal{L}\{\sin(kt)\}=\frac{k}{s^2+k^2}.$`
  * `$\mathcal{L}\{\cos(kt)\}=\frac{s}{s^2+k^2}.$`

---

## Laplace transform Linearity

> __Linearity:__ Let `$f$` and `$g$` be two functions defined on `$[0,\infty)$` such that their Laplace transforms exist, then `$$\mathcal{L}\{c_1 f+c_2 g\}=c_1\mathcal{L}\{f\}+c_2 \mathcal{L}\{g\}.$$`

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### Example

> Using Linearity in the Laplace transform, transform the following functions:
  * `$f(t)=-5+3t+4t^2$`
  * `$f(t)=5e^{t}-2e^{-3t}$`
  * `$f(t)=-3t^3+\cos(4t)+ 2\sin(4t).$`
  * `$f(t)=\frac{\sin(3t)-\cos(4t)}{5}.$`

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# Laplace Transforms for Piecewise Functions:

> Using the integral definition, compute the Laplace transform of the following functions: <br/>
  * `$f(t)=-1$` if `$0\leq t \leq 1$` and `$f(t)=1$` if `$t>1.$`
  * `$f(t)=0$` if `$0\leq t\leq a$` and `$f(t)=t$` if `$t>a.$`
  * `$f(t)=1$` if `$a\leq t\leq b$` and `$f(t)=0$` otherwise.