Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Trouble understanding the Gamma function

  1. Feb 17, 2008 #1
    The http://en.wikipedia.org/wiki/Gamma_function" [Broken] is the integral [tex]\Gamma(z)=\int_{0}^{\infty}{dt\, t^{z-1}e^{-t}}[/tex] . It has poles for integers of z less than 1 and is finite everywhere else. But to me it seems like it should be infinite for non integer values of z less than 0.

    My reasoning: when t is close to 0, the function t[tex]^{z-1}[/tex]e[tex]^{-t}[/tex] is approximately equal to t[tex]^{z-1}[/tex].

    The integral [tex]\int_{t_1}^{t_2}{dt\, t^{z-1}}[/tex] is equal to ([tex]t_2^{z}[/tex] - [tex]t_1^{z}[/tex])/z. When z<0 and [tex]t_1 = 0[/tex], [tex]t_1^{z}[/tex] is infinite. Thus it seems the integral [tex]\int_{0}^{\infty}{dt\, t^{z-1}e^{-t}}[/tex] has an infinite contribution right after t=0 for z<0.
    Last edited by a moderator: May 3, 2017
  2. jcsd
  3. Feb 18, 2008 #2


    User Avatar
    Homework Helper

    properties of the integrand may be seen by expanding it into a Laurent series
  4. Feb 18, 2008 #3


    User Avatar

    Only when z has positive real part. It is extended to the rest of the complex plane by analytic continuation or, equivalently, by enforcing that [itex]\Gamma(z+1)=z\Gamma(z)[/itex] holds everywhere.
    Last edited by a moderator: May 3, 2017
  5. Feb 18, 2008 #4
    Ok thanks
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook