Results 1 to 5 of 5

Math Help - Wonderful Limit

  1. #1
    Newbie
    Joined
    Feb 2011
    Posts
    17

    Wonderful Limit

    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Apr 2005
    Posts
    15,326
    Thanks
    1298

    Re: Wonderful Limit

    So \Gamma'(t)= (t-1)\int_0^\infty y^{t- 2}e^{-y}dy
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor chisigma's Avatar
    Joined
    Mar 2009
    From
    near Piacenza (Italy)
    Posts
    2,162
    Thanks
    5

    Re: Wonderful Limit

    We start with the well known 'infinite product'...

    \Gamma(x)= \frac{e^{-\gamma x}}{x}\ \prod_{n=1}^{\infty} \frac{e^{\frac{x}{n}}}{1+\frac{x}{n}} (1)

    ... where \gamma is the so called Euler's constant, then from (1)...

    \ln \Gamma(x)= -\ln x - \gamma x + \sum_{n=1}^{\infty} \{\frac{x}{n}-\ln (1+\frac{x}{n}) \} (2)

    ... and then from (2)...

    \frac{d}{dx} \ln \Gamma (x) = -\frac{1}{x} - \gamma + \sum_{n=1}^{\infty} \frac{x}{n\ (n+x)} (3)

    Now is...

    \frac{d}{dx} \ln \Gamma (x) = \frac{\Gamma^{'}(x)}{\Gamma(x)} (4)

    ... so that is...

    \Gamma^{'}(x)= \Gamma(x)\ \{-\frac{1}{x} - \gamma + \sum_{n=1}^{\infty} \frac{x}{n\ (n+x)}\} (5)

    ... and, taking into account (1) and (5), we can conclude that...

    \lim_{x \rightarrow 0+} x^{2}\ \Gamma^{'}(x) = -1 (6)

    Kind regards

    \chi \sigma
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Newbie
    Joined
    Feb 2011
    Posts
    17

    Re: Wonderful Limit

    Quote Originally Posted by HallsofIvy View Post
    So \Gamma'(t)= (t-1)\int_0^\infty y^{t- 2}e^{-y}dy
    This wrong
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Newbie
    Joined
    Feb 2011
    Posts
    17

    Re: Wonderful Limit

    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 1
    Last Post: November 26th 2010, 06:43 PM
  2. wonderful integral
    Posted in the Calculus Forum
    Replies: 2
    Last Post: July 2nd 2010, 10:20 PM
  3. Limit, Limit Superior, and Limit Inferior of a function
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: September 3rd 2009, 05:05 PM
  4. Wonderful Calculus Quiz problem
    Posted in the Calculus Forum
    Replies: 1
    Last Post: November 3rd 2008, 09:10 AM
  5. Replies: 1
    Last Post: January 24th 2006, 04:21 AM

Search Tags


/mathhelpforum @mathhelpforum