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Math Help - Gamma Function -Substituting Variables-

  1. #1
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    Gamma Function -Substituting Variables-

    The expression for the Gamma function is given as:

    \Gamma(z)=\int^\infty_0t^{z-1}e^{-t}dt

    for Re(z)>0, and we change the variable t into t=ns

    for n\in\mathbb N.

    It then says that by this, we get:

    \Gamma(z)=n^z\int^\infty_0s^{z-1}e^{-ns}dt

    Which I am not sure how they got.

    Here is what I tried:
    Given t=ns
    Then I assume  dt=nds and \frac{1}{n}dt=ds

    Substituting variables, I get:

    \Gamma(z)=\frac{1}{n}\int^\infty_0(ns)^{z-1}e^{-ns}ds

     \Gamma(z)=\frac{1}{n}\int^\infty_0(n^{z-1})s^{z-1}e^{-ns}ds

    \Gamma(z)=\frac{n^{z-1}}{n}\int^\infty_0s^{z-1}e^{-ns}ds<br />

    Which is not the same as what I'm suppose to get.

    If someone could show how it should be done, that would be greatly appreciated.

    Thank you.
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  2. #2
    Senior Member
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    You're making a mistake when changing dt to ds.

    dt=n \, ds

    should give you:

     \Gamma(z)=n \int_0^{\infty} (ns)^{z-1} e^{-ns} \, ds
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  3. #3
    Junior Member
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    Quote Originally Posted by drumist View Post
    You're making a mistake when changing dt to ds.

    dt=n \, ds

    should give you:

     \Gamma(z)=n \int_0^{\infty} (ns)^{z-1} e^{-ns} \, ds

    Thank you!
    I completely overlooked that simple mistake.
    Thanks again.
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