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Math Help - Laplace transformation?

  1. #1
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    Laplace transformation?

    I Have four question about Laplace transformation.

    1. Can anyone show me full but easy example of Laplace transformation. I don't quite understand how transforming some basic functions for example:
    2x or x/2 or sqrt(x+1)

    2. And what mean parameter S in laplace transformation?

    3. Let say I Have very trivial function: 2x and i wanna laplace transform that, is there any integration/derivation type rules how I do it or do i have to memorize transformations or look from the transformation table?

    4. Finally, Am I miss something, I understand that with Laplace transformation you can for example digitalize sin / cosine curve and any analogical curve. However if I draw Laplace tranformed function I cant see any digitialization in chart. What Laplace function represents actually?
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    Quote Originally Posted by tabularasa View Post
    I Have four question about Laplace transformation.

    1. Can anyone show me full but easy example of Laplace transformation. I don't quite understand how transforming some basic functions for example:
    2x or x/2 or sqrt(x+1)
    memorize the basic Laplace transformation rules.

    there is a rule that says: \mathcal{L} \left[t^n \right](s) = \frac {n!}{s^{n + 1}} for n \in \mathbb{Z}, n \ge 0 and s is the Laplace transform parameter

    thus: \mathcal{L} [2x](s) = 2 \mathcal{L}[x](s)= 2 \cdot \frac 1{s^2} = \frac 2{s^2} ......(the Laplace transform is like a linear operator in this sense, we can pull constants out when convenient)

    and: \mathcal{L} \left[ \frac x2 \right] = \frac 12 \mathcal{L}[x](s)= \frac 1{2s^2}

    write \sqrt{x + 1} as (x + 1)^{1/2}. now the power is not an integer. the Laplace transform is a bit more complicated. we would use the Gamma function here. the rule is: \mathcal{L} \left[ t^a \right](s) = \frac {\Gamma (a + 1)}{s^{a + 1}} for \mathcal{R}[a] > -1

    what do you think the Laplace transform is?


    2. And what mean parameter S in laplace transformation?
    think of it as a change of variable. you are transforming a function in some variable, say x or t, into another variable s. this variable is brought about by the Laplace transform of the function in the original variable. and I suppose you know the definition of the Laplace transform.


    3. Let say I Have very trivial function: 2x and i wanna laplace transform that,
    i answered this already

    is there any integration/derivation type rules how I do it or do i have to memorize transformations or look from the transformation table?
    the rule to find the Laplace transform (which will not always be easy) for any function is:

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

    but it is really a good idea to memorize the basic functions. i can give you a good list to know if you want.

    4. Finally, Am I miss something, I understand that with Laplace transformation you can for example digitalize sin / cosine curve and any analogical curve. However if I draw Laplace tranformed function I cant see any digitialization in chart. What Laplace function represents actually?
    not sure what you are talking about. we never did much with graphing Laplace transforms when i did differential equations. not sure exactly how such a graph would relate to the problem per say. maybe you could elaborate
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    Quote Originally Posted by tabularasa View Post
    4. Finally, Am I miss something, I understand that with Laplace transformation you can for example digitalize sin / cosine curve and any analogical curve. However if I draw Laplace tranformed function I cant see any digitialization in chart. What Laplace function represents actually?
    I will try to answer the last part

    Mostly you are a part electrical engineer(like me).
    All our linear time invariant systems are modeled by constant coefficient differential equations and hence the mathematical way of looking at Laplace transform holds for us too. What you are talking about is fourier transform of periodic signals I suppose. That will map all periodic "analogical"(or just analog) signals to sum of sinusoids. And you will need only a countable number of sinusoids. >That< is discretization (or equivalently digitalization). However drawing Laplace transforms in one go is not easy. Imagine how many dimensions are actually needed?(how many?)
    The function helps in learning a system with initial conditions and can be generally applied to a huge family of signals. Fourier is a special case of Laplace and in general many signals do not follow Dirichlet's conditions. So Laplace Transform is necessary.
    So I told you why we use it, but "what is it" is a mathematical "complex"(pun intended ) question.

    Probably The great CaptainBlack or TPH will answer it and help us out
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    Well, I'm just a boy who trying to understand Calculus better.
    I realise that S is kind of change of variable. So it is really same x than before Laplace transformation. Is it true that S=X when:
    <br />
\mathcal{L} [2x](s) = 2 \mathcal{L}[x](s)= 2 \cdot \frac 1{s^2} = \frac 2{s^2}<br />

    ?

    I have some fundamental hole in my understanding about Laplace. So questions are still very basic ones. Why to use Laplace transformation? What I can achieve with Laplace transformation? And last but not the least: Could you show me some practical example from real life how to use Laplace transform?

    I'm enthusiastic hooked to learn more.
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    Quote Originally Posted by tabularasa View Post
    I have some fundamental hole in my understanding about Laplace. So questions are still very basic ones. Why to use Laplace transformation? What I can achieve with Laplace transformation? And last but not the least: Could you show me some practical example from real life how to use Laplace transform?
    Laplace transforms are used to transform a differencial equation into simply an algebraic equation. Solving this transformed equation would be a lot easier. The final step is to invert the Laplace transform that will give you the function, which would be the solution to the differencial equation.
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    Quote Originally Posted by tabularasa View Post
    Well, I'm just a boy who trying to understand Calculus better.
    I realise that S is kind of change of variable. So it is really same x than before Laplace transformation. Is it true that S=X when:
    <br />
\mathcal{L} [2x](s) = 2 \mathcal{L}[x](s)= 2 \cdot \frac 1{s^2} = \frac 2{s^2}<br />

    ?

    I have some fundamental hole in my understanding about Laplace. So questions are still very basic ones. Why to use Laplace transformation? What I can achieve with Laplace transformation? And last but not the least: Could you show me some practical example from real life how to use Laplace transform?

    I'm enthusiastic hooked to learn more.
    see post #5 here as an example of how we use Laplace transforms to solve an initial value ODE problem. in this example we had to use partial fractions, but they're not always that difficult (or easy )
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    Quote Originally Posted by tabularasa View Post
    Well, I'm just a boy who trying to understand Calculus better.
    I realise that S is kind of change of variable. So it is really same x than before Laplace transformation. Is it true that S=X when:
    <br />
\mathcal{L} [2x](s) = 2 \mathcal{L}[x](s)= 2 \cdot \frac 1{s^2} = \frac 2{s^2}<br />

    ?

    I have some fundamental hole in my understanding about Laplace. So questions are still very basic ones. Why to use Laplace transformation? What I can achieve with Laplace transformation? And last but not the least: Could you show me some practical example from real life how to use Laplace transform?

    I'm enthusiastic hooked to learn more.
    Oh, I answered that in a general sense, because you talked about digitalisation of sin and cosine. If you want practical applications there, the entire branch of Digital Signal processing (which does audio processing, video processing etc) deals with Laplace transform and design of filters using it.

    However as TPH explained, it is a cunning way to solve differential equations
    (if you find algebra easy).

    However I would like to add one point. For engineers(or physicist) every ODE has an initial conditions specified. Laplace takes care of that too. It is thus a good enough reason to learn Laplace transforms and apply them.
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  8. #8
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    Quote Originally Posted by Isomorphism View Post
    For engineers(or physicist) every ODE has an initial conditions specified.
    In enegineering every ODE always has constant coefficients.
    Thus, when enginerring students take a course in differencial equations the only thing they should know how to solve are equations with constant coefficients.
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    Quote Originally Posted by Isomorphism View Post
    Oh, I answered that in a general sense, because you talked about digitalisation of sin and cosine. If you want practical applications there, the entire branch of Digital Signal processing (which does audio processing, video processing etc) deals with Laplace transform and design of filters using it.

    However as TPH explained, it is a cunning way to solve differential equations
    (if you find algebra easy).

    However I would like to add one point. For engineers(or physicist) every ODE has an initial conditions specified. Laplace takes care of that too. It is thus a good enough reason to learn Laplace transforms and apply them.

    Thank you. I'm looking practical example of Laplace. I think that I don't quite understand Laplace without some real life concrete example. Is this right thread/forum to looking for that? Algebra is not a problem. But still this S-constant is and also what Laplace transformed function actually represents.

    If I have that very basic linear function 2x and I Laplace transform that:
    <br />
\mathcal{L} [2x](s) = 2 \mathcal{L}[x](s)= 2 \cdot \frac 1{s^2} = \frac 2{s^2}<br />

    Ok, now if x goes to -10 to 10 and I want to know what that Laplace function gives answer for instance when x=5. What will be the result and what that result represents?
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    Quote Originally Posted by Isomorphism View Post
    Oh, I answered that in a general sense, because you talked about digitalisation of sin and cosine. If you want practical applications there, the entire branch of Digital Signal processing (which does audio processing, video processing etc) deals with Laplace transform and design of filters using it.
    The Laplace transform has nothing to do with DSP. The Laplace transform is a transform of functions of a continuous variable. For discrete systems you need the z-transform, which is the discrete analogue of the Laplace Transform.


    Now with a little jiggerey pokery under which eveything changes it's meaning we can end up working with distributions and the Laplace transform and the z-transform are seen to be two aspects of the same thing. But you don't really want to know that.

    RonL
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    Quote Originally Posted by CaptainBlack View Post
    The Laplace transform has nothing to do with DSP. The Laplace transform is a transform of functions of a continuous variable. For discrete systems you need the z-transform, which is the discrete analogue of the Laplace Transform.
    Yes, agreed! Not DSP, but the basic ideas about signals and systems.
    I wanted to tell the poster where it is used.I believed the transforms are important since it simplifies our analysis of LTI systems. Considering the fact that this allows us to just learn the behaviour for an input of e^{st}.Since every other signal, by the transform can be written using e^{st}. Am I wrong?

    Now with a little jiggerey pokery under which eveything changes it's meaning we can end up working with distributions and the Laplace transform and the z-transform are seen to be two aspects of the same thing. But you don't really want to know that.
    RonL
    I do want to know. Tell me more about this "jiggerey pokery". Or at least point me in some suitable direction
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  12. #12
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    Quote Originally Posted by Isomorphism View Post
    Yes, agreed! Not DSP, but the basic ideas about signals and systems.
    I wanted to tell the poster where it is used.I believed the transforms are important since it simplifies our analysis of LTI systems. Considering the fact that this allows us to just learn the behaviour for an input of e^{st}.Since every other signal, by the transform can be written using e^{st}. Am I wrong?


    I do want to know. Tell me more about this "jiggerey pokery". Or at least point me in some suitable direction

    The Wikipedia page on the Laplace transform is a good place to start.

    RonL
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