analytic continuation of gamma function

The last of the parent entry ( may be expressed as

Γ(z)=Γ(z+n)z(z+1)(z+2)(z+n-1). (1)

According to the standard definition


the left hand side of (1) is defined only in the right half-planez>0,  whereas the expression Γ(z+n) is defined and holomorphic for  z>-n  and thus the right hand side of (1) is holomorphic in the half-plane  z>-n  except the points


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

Accordingly, the gamma functionDlmfDlmfMathworldPlanetmath is unambiguous and holomorphic everywhere in except in the points

0,-1,-2,-3, (2)

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

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


In the point  z=-n  we have

Γ(z+n+1)=Γ(1)= 0!= 1,

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



  • 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