Intuition about definition of laplace transform

[FOIWATER]

I always have made use of laplace transforms to integrate differential equations and solve them in less hastle. In terms of what it physically means, the relationship before the laplace transform is where the real intuition would lie, but then if we take this intuition - take the infinite sum of the function from zero to infinity, the intuition about the initial problem still exists. If we take the inverse laplace, the simplifications we make in that domain are perfectly valid. I agree with yungman (not to put words in his mouth, but,) it's best described as a "reality" in terms of mathematics.

 

[rbj]

I don't think we are talking about the same thing. I guess another way of looking at this is: Is there a derivation of

L(f(t))=\int_0^{\infty} e^{-st} f(t) dt

If this is derived from the existing theory, then I hope at least one of my 5 books would have mentioned where the equation comes from. In another word, the history of Laplace transform. We all know the application and the indefinite integral, the kernel etc. I am not particular familiar with Laplace transform, all I can based on is the 5 books I have. When you talk about LTI, is Laplace transform derive base on that? If yes, that's good enough for me, you prove your point. If Laplace transform fit into the LTI, that does not prove anything.

You don't need to repeat all the theory to characterize Laplace transform, just the original history of Laplace transform, I think that's what the OP was asking since he is not interested in the definition and all. I assume he know enough about the application of it.

the OP was asking how this laplace transform actually works?

you are asking if

\mathcal{L}\{f(t)\}=\int_0^{\infty} e^{-st} f(t) dt

is derived, and i am saying it is an extension of the Fourier Transform which is an extension of Fourier series and you'll see the first integral that evolves to be the F.T. and L.T. from the derived Fourier coefficients in the F.T.

and the reason why sinusoidal and exponential functions are used as the basis functions for F.T. and L.T. are because they are eigenfunctions for Linear Time-Invariant systems.

it's not magic, and the definition of L.T. did not appear by magic. there is a rhyme and reason to it and a decent modern course in Signals and Systems (what we used to call Linear System Theory) would spell this out.

 

[jim hardy]

i'm not mathematically inclined, let alone gifted. ODE was as far as i went.

Might Euler's equation be part of the intuitive explanation OP inquired about??

when we multiply some arbitrary function by e^st
where s includes a real term σ and a jω and has correct dimension (t^-1),

it's not difficult for me to imagine that operation multipies our function by sine/cosine and exponential functions , analagous to a frequency sweep of a circuit plus ringing it with pulses ,

and integrating over 0 to ∞ collects the results and makes it somehow represent what a previous poster called a transform into a new plane,,, frequency dependent behavior being its ordinate and exponential behavior its abcissa?

Please excuse this musing of a math ignoramus, its just i woke up in middle of night with that question.
My alleged brain chews on concepts like this for months and ususlly discards them
maybe somebody can accelerate that rejection process for me, or say it's worthy of more thought.
Right now it's the best lead I've run across.
It ties the transform to something i can conceive of doing with hardware.Thanks guys,
old jim (a chid of the lesser gods)

 

[Gauss M.D.]

Bumping because I find it mind boggling that such a neat concept is so poorly understood. There isn't much to fuss about regarding the intuition behind the Laplace transform. You guys should check out OCW:s differential equations lecture series, prof. Mattuck gives a fantastic explanation. The Laplace transform is simply the continuous analogue of a power series expansion, where instead of summing an infinite amount of c_n x^n, with the terms separated by 1, you're integrating an infinite amount of e^x terms, each separated by dx.

I remember when they first went over power series expansions in out Calc 1 class, I thought to myself "hmm, I wonder if you can use integrals instead of sums to represent functions". Turns out you can, and the result is the Laplace transform!

 

[dawin]

Bumping because I find it mind boggling that such a neat concept is so poorly understood. There isn't much to fuss about regarding the intuition behind the Laplace transform. You guys should check out OCW:s differential equations lecture series, prof. Mattuck gives a fantastic explanation. The Laplace transform is simply the continuous analogue of a power series expansion, where instead of summing an infinite amount of c_n x^n, with the terms separated by 1, you're integrating an infinite amount of e^x terms, each separated by dx.

I remember when they first went over power series expansions in out Calc 1 class, I thought to myself "hmm, I wonder if you can use integrals instead of sums to represent functions". Turns out you can, and the result is the Laplace transform!

Speaking just from my experience in engineering school, the Laplace transform is, more or less, as a "gift from the gods". (borrowing the phrase from http://www.codee.org/library/articles/the-laplace-transform-motivating-the-definition).

Understanding that e^t is an eigenfunction of d/dt can help understand that the Laplace transform works by design and not accident.