Where can I find a rigorous derivation of the Laplace transform?

[sponsoredwalk]

I wonder does anyone have a source for the derivation of the Laplace transform in print
that is analogous to the derivation given in this video:

https://www.youtube.com/watch?v=zvbdoSeGAgI

https://www.youtube.com/watch?v=hqOboV2jgVo

I've browsed booksgoogle but have really been unable to find it. I understand it but would
just love to read a derivation from more than one source, I really mean it it would be really
helpful!

 

[╔(σ_σ)╝]

I don't think the laplace transform was "derieved" I think it was just defined; I might be wrong though.

For what I understand there are things called integral transforms and laplace is just one of them. http://en.wikipedia.org/wiki/Integral_transform

 

[sponsoredwalk]

Well in the video he says that this is where it comes from so I assumed it's a derivation.
In any case it's extremely interesting & I'd love to read a derivation in a book that
builds the theory up with that clarity of explanation. Maybe I'll write to Mattuck.

 

[╔(σ_σ)╝]

Well in the video he says that this is where it comes from so I assumed it's a derivation.
In any case it's extremely interesting & I'd love to read a derivation in a book that
builds the theory up with that clarity of explanation. Maybe I'll write to Mattuck.

Get a book on Integral transforms.

.

 

[sponsoredwalk]

Get a book on Integral transforms.

.

I've searched books on Integral transforms, on mathematical methods & on differential
equations on amazon/googlebooks trying to find a derivation that looks anything like
the way Mattuck lays it out to no avail & done it more than once over the past month.
I've started to think I'm just not recognising it or haven't searched hard enough hence
posting this question If I find anything I'll post it up on this page

 

[╔(σ_σ)╝]

I've searched books on Integral transforms, on mathematical methods & on differential
equations on amazon/googlebooks trying to find a derivation that looks anything like
the way Mattuck lays it out to no avail & done it more than once over the past month.
I've started to think I'm just not recognising it or haven't searched hard enough hence
posting this question If I find anything I'll post it up on this page

I still don't believe you will find a "proof". It's just like asking for a proof of linear transformations or integrals.
I think you will find things like when can we find a laplace transform, is it unique blah blah but not a proof.

 

[sponsoredwalk]

I'll give a partial quote from his response:

"Thanks. I can't give a reference, I just pulled it out of my head one day
since students seemed unhappy about being handed a machine for solving
differential equations with no idea of where it came from. I think
Laplace introduced the integral in his treatise on Probability theory,
where it came up naturally, but without motivation."



This man is a genius

 

[gomunkul51]

I'll give a partial quote from his response:

"Thanks. I can't give a reference, I just pulled it out of my head one day
since students seemed unhappy about being handed a machine for solving
differential equations with no idea of where it came from. I think
Laplace introduced the integral in his treatise on Probability theory,
where it came up naturally, but without motivation."



This man is a genius

Are you quoting Prof. Arthur Mattuck? If yes, where did you got it and if no, who is it? :)

 

[sponsoredwalk]

Are you quoting Prof. Arthur Mattuck? If yes, where did you got it and if no, who is it? :)

I e-mailed him nearly 2 hours ago asking him about the derivation and what I quoted
is what he said in his e-mail response to me just over a half hour ago, I'm absolutely
stunned by this, it's just pure genius. Here is some more from him:



"If you want to fool around with it a little more, see if you can use the
same ideas to show that the formula for the convolution f(t)*g(t) of two
functions is just the continuous analog for functions of the discrete
formula for finding the coefficient c_n of the power x^n when you multiply
two power series: sum a_k x^k times sum b_j x^j . (Which also explains
the Laplace transform formula L(f*g) = L(f) L(g)."

This should be in [STRIKE]a[/STRIKE] every textbook!

 

[gomunkul51]

Thank you for your response !

P.S. I have always believed that series' are very important :)

 

[RoyLB]

The book Advanced Calculus by David Widder has a power series derivation of the Laplace Transform very much like Prof Mattuck's.

I like the idea of the LT as an (infinite dimensional) dot product with exp(-st). The Fourier transform is similar - just use exp(j omega t). This approach can be found in Signals and Systems by Girod, Rabenstein, and Stenger

- Roy

 

[sponsoredwalk]

Fantastic, thanks a lot

 

[hunt_mat]

I believe it comes from the more general inverse scattering transform, it's when you define the Lax pair I think.

 

[rppearso]

That is definately going in my notes, is the derivation for Fourier transforms similar. I will have to check my advanced engineering text but I am pretty sure it was just stated and not nessicarily derived. I understand Fourier series ok but got twisted up when it came to the transforms.

 

[gomunkul51]

...Remembered this post when I was using Z-Transforms and was told that they are just discrete Laplace Transforms :)

 

[QuarkCharmer]

Old thread, and I really have nothing more to add than a "Thanks for posting this". This question has been bothering me for the better part of the day.

 

[rppearso]

I believe the Fourier transform is recognizing that any sort of a shape can be modeled by a series of sines and cosines. So instead of a power series its a series of sines and cosines. Many texts lack the basics of these sorts of things but understanding it is extremely powerful. Its not like someone dug up books that had the answers in them one day someone had to reason this stuff out.

 

[lurflurf]

Here is another thread
https://www.physicsforums.com/showthread.php?t=86897
I suppose these type of things are not in most books because they are obvious.
Someone should make a book of obvious things.

 

[chollapete]

Thanks for posting this, OP. I really enjoyed the video clip, and Prof. Mattuck's reply. You're right about it being not rigorously proved, but that is because it is intended as "motivation"--it motivates the definition of a continuous analogue to a discrete definition. My idea of a "rigorous proof" is taking everything back to Postulates 1-12 out of Ch 1 of Spivak's Calculus. But, if you do that, you'll be drowning in details and lose sight of the "motivation". Both are necessary, but they serve different purposes, and "motivation" was Prof. Mattuck's purpose, here.

P.S. I think I'll start watching OCW videos.