Laplace Transformations

[Downtrodden4]

I dont know if this is the right section for this, but I do know that Laplace Transforms have to do with integrals... So I kind of guessed. I dont have a specific math question with numbers... I was just wondering if anybody could actually explain to me what a Laplace transform is.
Like I said before, I know that it is a form of integral, and i know it moves the integrals (judging by the word Transform) but all the information I have read on it just gives a list of equations... Can anybody explain them???

Thanks in advance,
Downtrodden4

 

[tkhunny]

It is the long and celebrated history of mathematics. When no one else can solve a problem, you invent tools to solve it!

http://en.wikipedia.org/wiki/Pierre-Simon_Laplace

It's a start.

 

[galactus]

LaPlace transforms are wonderful things in the math world. They can be used to solve integrals which would otherwise be difficult using standard methods, series, DE's.

Say we wanted to evaluate \(\displaystyle \displaystyle \int_{0}^{\infty}\frac{cox(x)}{x^{2}+1}dx\)

This would be tough using elementary methods we learn in calc class. But, introduce a parameter t and transform it into a LaPlace.

\(\displaystyle \displaystyle f(t)=\int_{0}^{\infty}\frac{cos(tx)}{x^{2}+1}dx\)

Now, take the LaPLace, which can be looked up in a table.

\(\displaystyle \displaystyle F(s)=\int_{0}^{\infty}\frac{s}{(x^{2}+1)(x^{2}+s^{2})}dx\)

\(\displaystyle =\displaystyle \frac{s}{s^{2}-1}\int_{0}^{\infty}\left(\frac{1}{x^{2}+1}-\frac{1}{x^{2}+s^{2}}\right)dx\)

See?. Now it is in the form of two integrals we can easily integrate.

\(\displaystyle \displaystyle \frac{s}{s^{2}-1}\left(\frac{\pi}{2}-\frac{\pi}{2s}\right)\)

\(\displaystyle \displaystyle =\frac{\pi}{2(s+1)}\)

Now, take the inverse LaPlace and get:

\(\displaystyle \displaystyle \mathcal{L}^{-1}\{F(s);t\}=f(t)=\frac{\pi}{2e^{t}}, \;\ t>0\)

Since in our original integral we had cos(x), then t=1 and we have:

\(\displaystyle \displaystyle \frac{\pi}{2e}\)

LaPlace can also be used to solve various series, such as a series in the form:

\(\displaystyle \displaystyle \sum_{n=1}^{\infty}\frac{(-1)^{n}}{an+b}, \;\ a>0, \;\ b\geq 0\)

Say we wanted to find out what \(\displaystyle \displaystyle \sum_{n=1}^{\infty}\frac{(-1)^{n}}{3n+5}\) converges to. Of course, there are other ways, but I am just trying to state how versatile they are.

If you're into math. Research LaPLace. Lots of cool stuff to learn.