Prove that a function could be a Laplace transform of another function

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I'm preparing for some exams and this is a question that is usually asked.



Given a function $F(s)$ prove that it can be a Laplace transform of a function $f(t)$



While we've been given the conditions for the Laplace transform of a function to exist, we haven't been given anything on the inverse part.
My guess is we'll have to show that the Inverse transform integral converges. Will that be enough?



However, we rarely even use the inverse transform integral. We usually try to bring the function $F(s)$ to a form that is familiar and we just figure out by memory the function $f(t)$ that has a Laplace transform $F(s)$. I assume that this is not a proof in this case...










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    up vote
    0
    down vote

    favorite












    I'm preparing for some exams and this is a question that is usually asked.



    Given a function $F(s)$ prove that it can be a Laplace transform of a function $f(t)$



    While we've been given the conditions for the Laplace transform of a function to exist, we haven't been given anything on the inverse part.
    My guess is we'll have to show that the Inverse transform integral converges. Will that be enough?



    However, we rarely even use the inverse transform integral. We usually try to bring the function $F(s)$ to a form that is familiar and we just figure out by memory the function $f(t)$ that has a Laplace transform $F(s)$. I assume that this is not a proof in this case...










    share|cite|improve this question























      up vote
      0
      down vote

      favorite









      up vote
      0
      down vote

      favorite











      I'm preparing for some exams and this is a question that is usually asked.



      Given a function $F(s)$ prove that it can be a Laplace transform of a function $f(t)$



      While we've been given the conditions for the Laplace transform of a function to exist, we haven't been given anything on the inverse part.
      My guess is we'll have to show that the Inverse transform integral converges. Will that be enough?



      However, we rarely even use the inverse transform integral. We usually try to bring the function $F(s)$ to a form that is familiar and we just figure out by memory the function $f(t)$ that has a Laplace transform $F(s)$. I assume that this is not a proof in this case...










      share|cite|improve this question













      I'm preparing for some exams and this is a question that is usually asked.



      Given a function $F(s)$ prove that it can be a Laplace transform of a function $f(t)$



      While we've been given the conditions for the Laplace transform of a function to exist, we haven't been given anything on the inverse part.
      My guess is we'll have to show that the Inverse transform integral converges. Will that be enough?



      However, we rarely even use the inverse transform integral. We usually try to bring the function $F(s)$ to a form that is familiar and we just figure out by memory the function $f(t)$ that has a Laplace transform $F(s)$. I assume that this is not a proof in this case...







      laplace-transform






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      asked Sep 5 at 10:07









      John Katsantas

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