Expanding Gamma function around poles

[DMESONS]

Can someone help me to expand the following gamma functions around the pole ε, at fisrt order in ε

$\Gamma[(1/2) \pm (ε/2)]$

where ε= d-4

 

[Bill_K]

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.

 

[Hepth]

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

$$\Gamma(\frac{1}{2} + \frac{\epsilon}{2}) = \sqrt{\pi }+\frac{\sqrt{\pi } \epsilon \psi ^{(0)}\left(\frac{1}{2}\right)}{2}+O\left(\epsilon ^2\right)$$

 

[DMESONS]

Bill_K and Hepth, I am so grateful for your help

I am new in this subject