Expanding Gamma function around poles

  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. Bill_K

    Bill_K 4,159
    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. Hepth

    Hepth 508
    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. Bill_K and Hepth, I am so grateful for your help

    I am new in this subject

    :smile:
     
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