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Gamma Function

  1. Jan 27, 2007 #1

    JPC

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    Hey , i am in grade 11
    not yet studied gamma function, and not sure if it will be in the program

    but i have studied it a bit on my own

    f(x) = gamma(x)

    x = 1 : y = 1
    x = 2 : y = 1
    x = 3 : y = 2
    x = 4 : y = 6
    x = 5 : y = 24
    x = 6 : y = 120

    and i found a pattern :

    1 * 1 = 1
    2 * 1 = 2
    3 * 2 = 6
    4 * 6 = 24
    5 * 24 = 120

    so if we consider the gamma sequence , is it ?

    U(n+1) = Un * (n-1)
    with n0 = 1

    but now , how do u find Gamma of any x (x a any real number) ?
     
  2. jcsd
  3. Jan 27, 2007 #2

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    You say you have studied the gamma function on your own- didn't you start with the definition? How did you get those numbers? Yes, it is true that for n any positive integer
    [tex]\Gamma(n)= (n-1)![/itex]
    but that is a consequence of the definition.

    The gamma function is defined by
    [tex]\Gamma(x)= \int_0^\infty t^{x-1}e^{-t}dt[/itex]

    Using that formula, you can show by induction on n (and integration by parts for the induction step) that
    [tex]\Gamma(n)= \int_0^\infty t^{n-1}e^{-t}dt= (n-1)![/tex]
    for n any positive integer.

    The gamma function is defined for all real numbers except negative integers.

    Perhaps you are asking how you do that integral for x not a positive integer- the answer is that, in general, you don't! Generally numerical integration is used.

    However, it is an interesting exercise for simple values like x= 1/2.
    [tex]\Gamma(\frac{1}{2})= \int_0^\infty t^{-\frac{1}{2}}e^{-t} dt[/tex]
    Let u= t1/2 so that t= u2 and dt= 2udu. Then the integral is
    [tex]\int_0^\infty \frac{1}{u} e^{-u^2}(2 udu)= \int_0^\infty e^{-u^2}du[/itex]

    Do you know how to show that that is [itex]\sqrt{\pi}[/itex]?
    [tex]\Gamma(\frac{1}{2})= \sqrt{\pi}[/tex]
     
  4. Jan 27, 2007 #3

    JPC

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    hum

    sorry havent done calculus yet
    didnt know the gamma function was that complex

    but is there a link between the gamma function , and the gamma level adjustments when you edit pictures ?
     
  5. Jan 27, 2007 #4
  6. Jan 27, 2007 #5

    uart

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    Science Advisor

    Hi JPC, what you have been studying is (almost) the factorial function, not the gamma function. For postive whole numbers the gamma function is almost the same thing as the gamma function.
     
  7. Jan 27, 2007 #6

    JPC

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    ok

    but why have a gamma function ?
    whats its use ?
     
  8. Jan 27, 2007 #7
    It's a generalisation of the factorial. The factorial is only defined for integers, and the gamma function is the only function that satisfies some nice "smoothness" criteria (it's continuous, strictly increasing, convex, as well as some more things) which goes through all the points the factorial does. So you might as well define the factorial for non-integers and negative numbers by the gamma function. Then you can calculate the "factorial" of numbers like -1/2.
     
  9. Jan 27, 2007 #8

    HallsofIvy

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    Actually, the gama function is a lot more that just a generalization of the factorial. With simple substitutions you can convert to the "Beta" function which is important in the Beta probability distribution.
     
  10. Jan 29, 2007 #9

    Gib Z

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    Homework Helper

    Gamma function also lets us compute things that are normally thought to be done decretely like derivatives.
     
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