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analytic continuation of gamma function
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(Derivation)
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The last formula of the parent entry may be expressed as
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(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
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(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 $$\Res(\Gamma;\,-n) \;=\; \frac{(-1)^n}{n!}.$$
- 1
- R. NEVANLINNA & V. PAATERO: Funktioteoria. Kustannusosakeyhtiö Otava. Helsinki (1963).
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"analytic continuation of gamma function" is owned by pahio.
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Cross-references: coefficients of Laurent series, implies, equation, residue, meromorphic, function, unambiguous, gamma function, complex plane, Euler's, integer, positive, analytic continuation, order, poles, points, right hand side, holomorphic, expression, right, left hand side
There is 1 reference to this entry.
This is version 7 of analytic continuation of gamma function, born on 2007-05-06, modified 2009-11-12.
Object id is 9342, canonical name is AnalyticContinuationOfGammaFunction.
Accessed 1964 times total.
Classification:
| AMS MSC: | 33B15 (Special functions :: Elementary classical functions :: Gamma, beta and polygamma functions) | | | 30B40 (Functions of a complex variable :: Series expansions :: Analytic continuation) | | | 30D30 (Functions of a complex variable :: Entire and meromorphic functions, and related topics :: Meromorphic functions, general theory) |
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Pending Errata and Addenda
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