analytic continuation of gamma function

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

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

According to the standard definition

$$\mathrm{\Gamma }(z):={\int }_0^{\mathrm{\infty }}{e}^{-t}{t}^{z-1}𝑑t,$$

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

$$0,-1,-2,\mathrm{\dots },-(n-1)$$

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

Accordingly, the gamma function is unambiguous and holomorphic everywhere in $ℂ$ except in the points

$0,-1,-2,-3,\mathrm{\dots }$ (2)

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

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

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

In the point  $z=-n$  we have

$$\mathrm{\Gamma }(z+n+1)=\mathrm{\Gamma }(1)=\mathrm{\hspace{0.33em}0}!=\mathrm{\hspace{0.33em}1},$$

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

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

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