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Laplace Transform

  1. Apr 18, 2009 #1
    1) If Laplace transform...Cos(3t) = s/(s^2+9)
    but how about laplace transform ....Cos(3) = ???

    2) Cos(0) =1
    but how about Cosh(0) =? ; Cosh(1) =?
    Sinh(0) = ?; Sinh(1) = ?

    3) How to do the partial fractions for ...

    (s+13)/(s^2 +2s+10) = ??


    Thx for help =)
     
  2. jcsd
  3. Apr 18, 2009 #2

    Tom Mattson

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    First, what do you think about these questions?
     
  4. Apr 18, 2009 #3
    Some clues:

    Definition of the Laplace Transform:

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

    Plug in cos(3) or cosh(0) or whatever else in here for f(t) and then integrate.



    Alternatively:

    Well-known Result (i.e. derivation is trivial):

    [tex]L(1) = 1/s[/tex]

    A consequence of the definition of the Laplace transform is Linearity:

    [tex]L(c f1 + d f2) = c L(f1) + d L(f2) [/tex]

    In other words, the Laplace transform of a constant times a function is the constant times the Laplace transform of the function. (That makes sense too, since the Laplace transform is just an integral.)

    Definitions of Hyperbolic functions:

    [tex]sinh(x) = \frac{e^x - e^{-x}}{2}[/tex]

    [tex]cosh(x) = \frac{e^x + e^{-x}}{2}[/tex]

    Does that make sense? The hyperbolic functions will give you some constant and you know how to get the LT from there.
     
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