Results 1 to 4 of 4

Math Help - Gamma function related to fractions

  1. #1
    Senior Member
    Joined
    Sep 2013
    From
    Portland
    Posts
    481
    Thanks
    70

    Gamma function related to fractions

    I am working on on understanding $S_n=\frac{2\pi^\frac{n}{2}}{\Gamma(\frac{n}{2})}$ which is the hyper-surface area of a unit n-sphere

    this post is specifically about the gamma function. I just need some direction

    I know it can be thought of as $\Gamma(n)=(n-1)!$ but if I am doing n=1/2,3/2,.... I have a fraction so that doesn't work. I want to understand how the gamma function works when n is a fraction

    I found the Euler reflection formula $\Gamma(x)\Gamma(1-x) = \frac{\pi}{\sin \pi x}$ and I thought maybe this was what I needed but then trying some rationals in it

    i.e. let $x=\frac{1}{2}$

    $\Gamma(\frac{1}{2}) = \sqrt{\pi}$ whereas $\Gamma(\frac{1}{2})\Gamma(1-\frac{1}{2}) = \pi$ and in this case one is just the square of the other but that wasn't the case for other rationals like $\frac{1}{4}$

    so this is where I am now
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Forum Admin topsquark's Avatar
    Joined
    Jan 2006
    From
    Wellsville, NY
    Posts
    9,966
    Thanks
    350
    Awards
    1

    Re: Gamma function related to fractions

    Quote Originally Posted by Jonroberts74 View Post
    I am working on on understanding $S_n=\frac{2\pi^\frac{n}{2}}{\Gamma(\frac{n}{2})}$ which is the hyper-surface area of a unit n-sphere

    this post is specifically about the gamma function. I just need some direction

    I know it can be thought of as $\Gamma(n)=(n-1)!$ but if I am doing n=1/2,3/2,.... I have a fraction so that doesn't work. I want to understand how the gamma function works when n is a fraction

    I found the Euler reflection formula $\Gamma(x)\Gamma(1-x) = \frac{\pi}{\sin \pi x}$ and I thought maybe this was what I needed but then trying some rationals in it

    i.e. let $x=\frac{1}{2}$

    $\Gamma(\frac{1}{2}) = \sqrt{\pi}$ whereas $\Gamma(\frac{1}{2})\Gamma(1-\frac{1}{2}) = \pi$ and in this case one is just the square of the other but that wasn't the case for other rationals like $\frac{1}{4}$

    so this is where I am now
    The Gamma function is not a nice function. There might be other "nice" solutions but, for example, Gamma(1/4) is transcendental. (According to W|A anyway.)

    -Dan
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Aug 2011
    Posts
    246
    Thanks
    58

    Re: Gamma function related to fractions

    You should consider the integral definition of the Gamma function : Eq.(3) in : Gamma Function -- from Wolfram MathWorld
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Senior Member
    Joined
    Sep 2013
    From
    Portland
    Posts
    481
    Thanks
    70

    Re: Gamma function related to fractions

    Quote Originally Posted by topsquark View Post
    The Gamma function is not a nice function. There might be other "nice" solutions but, for example, Gamma(1/4) is transcendental. (According to W|A anyway.)

    -Dan
    yeah both Gamma(1/4) and [Gamma(1-1/4)Gamma(1/4)] are transcendental. whats also interesting is $\Gamma(1-x)\Gamma(x)\sin \pi x = \pi$ for all non-integer numbers

    it's an interesting function

    Quote Originally Posted by JJacquelin View Post
    You should consider the integral definition of the Gamma function : Eq.(3) in : Gamma Function -- from Wolfram MathWorld
    I have played with the integral a bit but I'm not getting what I want, I'm going to keep investigating.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. gamma function
    Posted in the Discrete Math Forum
    Replies: 0
    Last Post: February 8th 2013, 08:19 AM
  2. Replies: 5
    Last Post: May 27th 2012, 01:33 PM
  3. Gamma Function
    Posted in the Differential Geometry Forum
    Replies: 7
    Last Post: March 24th 2011, 09:40 PM
  4. Gamma function
    Posted in the Calculus Forum
    Replies: 1
    Last Post: September 21st 2010, 05:50 AM
  5. [SOLVED] The error function and incomplete gamma function
    Posted in the Advanced Applied Math Forum
    Replies: 0
    Last Post: January 31st 2010, 03:24 PM

Search Tags


/mathhelpforum @mathhelpforum