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Expanding Gamma function around poles 
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#1
Jun1512, 03:17 PM

P: 27

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= d4 


#2
Jun1512, 04:43 PM

Sci Advisor
Thanks
P: 4,160

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. 


#3
Jun1512, 05:15 PM

PF Gold
P: 466

[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] 


#4
Jun1512, 06:15 PM

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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