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Expanding Gamma function around poles

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Jun15-12, 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= d-4
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Jun15-12, 04:43 PM
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Bill_K's Avatar
P: 4,160
Γ( ε/2) ≈ Γ() ε/2 Γ'()

No, seriously..

Well, you also need to use the digamma function, ψ(x) = Γ'(x)/Γ(x). And the values Γ() = √π and ψ() = - γ - 2 ln 2 where γ is Euler's constant.
Jun15-12, 05:15 PM
PF Gold
Hepth's Avatar
P: 471
\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)

Jun15-12, 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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