# Expanding Gamma function around poles

by DMESONS
Tags: expanding, function, gamma, poles
 P: 27 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
 Sci Advisor Thanks P: 3,860 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.
 PF Gold P: 445 $$\Gamma(\frac{1}{2} - \frac{\epsilon}{2}) = \sqrt{\pi }+\frac{1}{2} \sqrt{\pi } \epsilon (\gamma_E +\log (4))+O\left(\epsilon ^2\right)$$ $$\Gamma(\frac{1}{2} + \frac{\epsilon}{2}) = \sqrt{\pi }+\frac{\sqrt{\pi } \epsilon \psi ^{(0)}\left(\frac{1}{2}\right)}{2}+O\left(\epsilon ^2\right)$$
P: 27

## Expanding Gamma function around poles

Bill_K and Hepth, I am so grateful for your help

I am new in this subject

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