What are the properties of the Fancy L Transform and its inverse?

[Char. Limit]

Can someone give me a few properties of the Fancy L Transform Thingy (In case you can't tell, I can't remember the name) and its inverse? For example, is the sum of two transformed functions equal to the L transform of the sum of the functions, is the same true of products and division, and so on (I seem to remember a convolution popping up somewhere in there...).

 

[rock.freak667]

If you mean laplace transform

it is true that L[y₁(t)+y₂(t)]=L[y₁(t)]+L[y₂(t)]

but you will find that L[y_1(t)y_2(t)] \neq L[y_1(t)]L[y_2(t)]


But L[y_1(t)*y_2(t)]=L[y_1(t)]L[y_2(t)]

this site will help:

http://www.swarthmore.edu/NatSci/ec...onMethods/LaplaceZTable/LaplacePropTable.html

 

[Char. Limit]

I see... indeed, Laplace is what I was thinking of.

Is every Laplace transform and its inverse one-to-one?

 

[elibj123]

If you are working with differentiable function (in which case both the function and it's derivative are continuous) then yes, its one to one.
But notice that this is an integral transform, and if you take a function and change its value in a finite amount of points, then the integral remains unchanged, and in that case the transform is not one to one.

You should really look over the internet to find more accurate definitions, theorems and conditions.

 

[Char. Limit]

I prefer these forums because they have a certain... human quality that you just don't find in online lectures (khanacademy being the exception).

Diff EQs seem very abstract. Is this Laplace transform even useful in the real world?

 

[HallsofIvy]

Well, I, for one wouldn't consider "differential equations" very abstract! Certainly not compared with other mathematics. Differential equations are probably used in applications more than any other form of mathematics (well, except for arithmetic).

And Laplace transforms are used a lot "in the real world"- they give engineers a way to just "look up" solutions to differential equations.

 

[elibj123]

I prefer these forums because they have a certain... human quality that you just don't find in online lectures (khanacademy being the exception).

Diff EQs seem very abstract. Is this Laplace transform even useful in the real world?

Since Diff. Equations are very useful in the real world, I think it implies Laplace Transform useful as well.

You can also find a lot of Laplace (& Fourier) Transform in systems analysis, because it takes your point of view on the system from the time domain, in which you describe your system via Diff Eq. to the frequency (or more precisely, complex frequency) domain where your system is decribed by algebric eq.
Not only that many times it's easier to solve precisely (analytically or numerically) but also it gives you a more broad insight about your system behaviour.
For example, determining how your system responses to different input frequency, is important in filter designing (if you want for example, to cut off low frequency waves from your audio data).
In image processing, high frequency correspond to quick changes - edges, a property that may be used in filters that detect edges,smooth edges etc...

 

[Char. Limit]

See what I mean about the human element?

I guess I just haven't seeen Diff EQs at work enough to have known this.