# analytic continuation of gamma function

The last of the parent entry (http://planetmath.org/GammaFunction) may be expressed as

 $\displaystyle\Gamma(z)\;=\;\frac{\Gamma(z\!+\!n)}{z(z\!+\!1)(z\!+\!2)\cdots(z% \!+\!n\!-\!1)}.$ (1)

According to the standard definition

 $\Gamma(z)\;:=\;\int_{0}^{\infty}\!e^{-t}t^{z-1}\,dt,$

the left hand side of (1) is defined only in the right half-plane$\Re{z}>0$,  whereas the expression $\Gamma(z+n)$ is defined and holomorphic for  $\Re{z}>-n$  and thus the right hand side of (1) is holomorphic in the half-plane  $\Re{z}>-n$  except the points

 $0,\,-1,\,-2,\,\ldots,\,-(n\!-\!1)$

where it has the poles of order 1.  Because the both sides of (1) are equal for  $\Re{z}>0$,  the left side of (1) is the analytic continuation of $\Gamma(z)$ to the half-plane  $\Re{z}>-n$.  And since the positive integer $n$ can be chosen arbitrarily, the Euler’s $\Gamma$-function has been defined analytically to the whole complex plane.

Accordingly, the gamma function is unambiguous and holomorphic everywhere in $\mathbb{C}$ except in the points

 $\displaystyle 0,\,-1,\,-2,\,-3,\,\ldots$ (2)

which are poles of order 1 of the function.  Hence, $\Gamma(z)$ is a meromorphic function.

For determining the residue of the function in the points (2), we rewrite the equation (1) as

 $\Gamma(z)\;=\;\frac{\Gamma(z\!+\!n\!+\!1)}{z(z\!+\!1)(z\!+\!2)\cdots(z\!+\!n)}.$

In the point  $z=-n$  we have

 $\Gamma(z\!+\!n\!+\!1)\;=\;\Gamma(1)\;=\;0!\;=\;1,$

which implies (see the rule in the entry coefficients of Laurent series) that

 $\operatorname{Res}(\Gamma;\,-n)\;=\;\frac{(-1)^{n}}{n!}.$

## References

• 1 R. Nevanlinna & V. Paatero: Funktioteoria.  Kustannusosakeyhtiö Otava. Helsinki (1963).
 Title analytic continuation of gamma function Canonical name AnalyticContinuationOfGammaFunction Date of creation 2013-03-22 17:03:07 Last modified on 2013-03-22 17:03:07 Owner pahio (2872) Last modified by pahio (2872) Numerical id 10 Author pahio (2872) Entry type Derivation Classification msc 30D30 Classification msc 30B40 Classification msc 33B15 Synonym residues of gamma function Related topic AnalyticContinuation Related topic EmptyProduct Related topic ResiduesOfTangentAndCotangent Related topic RolfNevanlinna