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

Expanding Gamma function around poles

  1. Jun 15, 2012 #1
    Can someone help me to expand the following gamma functions around the pole ε, at fisrt order in ε

    [itex]\Gamma[(1/2) \pm (ε/2)][/itex]

    where ε= d-4
     
  2. jcsd
  3. Jun 15, 2012 #2

    Bill_K

    User Avatar
    Science Advisor

    Γ(½ ± ε/2) ≈ Γ(½) ± ε/2 Γ'(½)

    No, seriously.. :smile:

    Well, you also need to use the digamma function, ψ(x) = Γ'(x)/Γ(x). And the values Γ(½) = √π and ψ(½) = - γ - 2 ln 2 where γ is Euler's constant.
     
  4. Jun 15, 2012 #3

    Hepth

    User Avatar
    Gold Member

    [tex]
    \Gamma(\frac{1}{2} - \frac{\epsilon}{2}) = \sqrt{\pi }+\frac{1}{2} \sqrt{\pi } \epsilon (\gamma_E +\log (4))+O\left(\epsilon ^2\right)
    [/tex]

    [tex]
    \Gamma(\frac{1}{2} + \frac{\epsilon}{2}) = \sqrt{\pi }+\frac{\sqrt{\pi } \epsilon \psi ^{(0)}\left(\frac{1}{2}\right)}{2}+O\left(\epsilon ^2\right)
    [/tex]
     
  5. Jun 15, 2012 #4
    Bill_K and Hepth, I am so grateful for your help

    I am new in this subject

    :smile:
     
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook