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Math Help - Laplace Transform Gamma Function

  1. #1
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    Laplace Transform Gamma Function

    I am just looking for a proof or help to solve the Laplace Transform that:

    f(t^α) = [ Γ(α+1) ] / [ s^(α+1) ]

    I dont even know where to begin working on this and any help would be greatly appreciated.
    Thanks.
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  2. #2
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    You need the definition of the Laplace transform and make a substitution as follows:
    \int_0^{+\infty}e^{-st}t^a dt
    Now set:
    st=u \qquad s>0
    You can rewrite this after the substitution as:
    \frac{1}{s^{a+1}}\int_0^{+\infty}e^{-u}u^a du
    Which can be written using the definition of the Gamma function to the required result.
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  3. #3
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    I'm not seeing how substituting in u for st is turning the t into a u.
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  4. #4
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    Quote Originally Posted by eeverett1 View Post
    I'm not seeing how substituting in u for st is turning the t into a u.
    \int_0^{\infty} e^{-st}t^a dt = \int_0^{\infty} e^{-st} \left( \frac{st}{s} \right)^a dt = \frac{1}{s^a}\int_0^{\infty} e^{-st}(st)^a dt.
    Now let \mu = st so then \frac{1}{s^{a+1}} \int_0^{\infty} e^{-\mu} \mu^a d\mu.
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