Laplace Transforms: Math's Elegant Solutions

[MostlyHarmless]

Recently in my DiffEq class, we learned how to use, and come up with, Laplace transforms. After doing my homework, I realized that Laplace Transforms are my new favorite concept in math(just beating out double/triple integrals and their applications)! The transforms just look so elegant on a white board!

The theorem we were taught said this: $L${$F(t)$}$=∫^∞_0e^{-st}F(t)dt$. My professor mentioned that there are more transforms out there, but we only had time for this one. Are the other transforms out there of this same form? Where can I find more information on them?

 

[MathematicalPhysicist]

It's not a theorem, it's its definition.

As for others, yes, there are plenty and as much as there are applications of it.

Radon transform, Mellin transform, etc...

In Functional Analysis we look on a kernel $$K(x,y)$$ and look on the next operator:

$$Kf(x) := \int f(y)K(x,y)dy$$

Obviously we can do this for a function f that has as many variables we wish, and then we look for possible convergence conditions of the integral, and other features as the theory of these kernels has progressed.

 

[SteamKing]

I don't know what you mean by 'the same form', but two other transforms are Fourier transforms and Hankel transforms:

http://en.wikipedia.org/wiki/Fourier_transform

http://en.wikipedia.org/wiki/Hankel_transform

 

[AlephZero]

Here's a list to get you started http://en.wikipedia.org/wiki/List_of_transforms

The "integral transforms" have similar forms to the Laplace transform. The most similar is the Fourier transform.

You might also look at the Z transform, which is analogous to the Laplace transform but for a series instead of a continuous function.

 

[JJacquelin]

Hi Jesse H. !

One of my favourite is the Riemann-Liouville transform which introduces to the derivatives and integrals of fractional orders.
http://mathworld.wolfram.com/Riemann-LiouvilleOperator.html
http://mathworld.wolfram.com/FractionalIntegral.html
A review for general public : "The fractional derivation. La dérivation fractionnaire"
http://www.scribd.com/JJacquelin/documents

 

[I like Serena]

Fourier Transform and Fourier Series, which is a whole family of transforms.
In particular, the Laplace Transform is the same as the Fourier Transform, except for a factor $i$.
The z-transform is also a (discrete) variant of these same transforms.

 

[MostlyHarmless]

Thank you all for your input, and I'm sorry I've not gotten a chance to reply. (Finals >.<)

Can't wait to finish up with finals so I can spare some time to look into those that you all have mentioned!

 

[ammarmechanica]

How to solve D.E by Laplace transform ??

 

[SteamKing]

How to solve D.E by Laplace transform ??

It's not cool to hijack an older thread at PF. Set up your own thread to ask a new question.

You can find out how to solve D.E.s using Laplace transforms by searching the net. You can google 'Laplace transform' to get you started.

 

[Jhenrique]

The biggest problem with these transforms is that not exist the rule of composition of function (chain rule, in derivative; integration for substituion, in integration). Thus, you need to calculate a mountain of possible cases.

 

[Jhenrique]

Fourier Transform and Fourier Series, which is a whole family of transforms.
In particular, the Laplace Transform is the same as the Fourier Transform, except for a factor $i$.
The z-transform is also a (discrete) variant of these same transforms.

Fourier Transform is useful for solve linear system too?