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Functions that integrate to a gamma function?

  1. Mar 23, 2008 #1
    maple syntax:

    int(theta^y * exp(-theta*(1-alpha) ) , theta)

    I have a distribution that I need to integrate, and I know the result should have a gamma function in it.

    The only thing I have found helpful is:
    http://en.wikipedia.org/wiki/Gamma_function

    My function is kind of in that form (theta^something * exp(something) ), but the "somethings" dont seem to be able to be manipulated into that form. Any hints on how to go about this?

    Thanks!

    David
     
  2. jcsd
  3. Mar 23, 2008 #2
    You could match it with this generalized form of the gamma function (sometimes called the Plica Function):

    [tex]\Gamma (a,z)=\int_z^{\infty } t^{a-1} e^{-t} dt[/tex]

    Or you could also match it with the exponential integral function (n is an integer):

    [tex]E_n (z)=\int_1^{\infty } \frac{e^{-z t}}{t^n} dt[/tex]

    If you must match it to a non-generalized gamma expression, show us the final form you are aiming for.
     
  4. Mar 23, 2008 #3
    I dont understand how I would match it with those? The problem is I have

    t^alpha

    and I need
    t^(alpha - 1)

    The only way I would know to "mold" mine into the correct one is to multiply by t^(-1) and t, then take t outside the integral and combine t^alpha with t^(-1) to get t^(alpha-1). However, t is the integration variable, so I can't do that!!

    any other thoughts?

    Thanks,

    David
     
  5. Mar 23, 2008 #4

    Mute

    User Avatar
    Homework Helper

    Use the recursive property of the gamma function:

    [tex]\Gamma(\alpha+1) = \alpha \Gamma(\alpha)[/tex]

    This does require, however, that your limits of integration are 0 to infinity.
     
  6. Mar 23, 2008 #5
    Use integration by parts, differentiate t^alpha.
     
  7. Mar 26, 2008 #6
    If the integral you are looking for is the following (assuming the limits are correct):

    [tex]\int_{0}^{\infty}\theta^y \cdot e^{-\theta(1-\alpha)} d\theta[/tex]

    You can transform it using:

    [tex]\theta(1-\alpha)=u[/tex]

    This gives you something related to the Gamma function.
     
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