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
Γ(½ ± ε/2) ≈ Γ(½) ± ε/2 Γ'(½) No, seriously.. 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.
[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]