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Derivative of x!

  1. Feb 7, 2010 #1
    Hi,
    I want to find the derivative of factorial function f(x)=x! and i found this integral,

    [tex]f(x)=x!=\int_{0}^{\infty}e^{-t}t^xdt[/tex] when i take derivative of this

    [tex]\frac{d}{dx}f(x)=\frac{d}{dx}\int_{0}^{\infty}e^{-t}t^xdt=\int_{0}^{\infty}e^{-t}t^xlnxdx[/tex]

    How do i find this integral? Please help me. Thanks.
     
  2. jcsd
  3. Feb 7, 2010 #2

    arildno

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    Incorrect differentiation!

    You should get:
    [tex]\frac{df}{dx}=\int_{0}^{\infty}\ln(t)e^{-t}t^{x}dt[/tex]

    There is no nice expression for the anti-derivative with respect to "t" here, so it must be evaluated numerically.
     
  4. Feb 7, 2010 #3
    Okey if i want to find

    [tex]f'(2)=\int_{0}^{\infty}\ln(t)e^{-t}t^{2}dt[/tex]

    can i integrate it?
     
  5. Feb 7, 2010 #4

    Borek

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    Have you actually read arildno's post?
     
  6. Feb 7, 2010 #5
  7. Feb 7, 2010 #6

    arildno

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    To anything simple?

    Nope.
     
  8. Feb 7, 2010 #7
    Yes, http://mathworld.wolfram.com/Euler-MascheroniConstant.html" [Broken].
     
    Last edited by a moderator: May 4, 2017
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