TheoremStatement: Literature-sourcedProof: AI-generatedprecheck passaudited 2026-08-30

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Meromorphic continuation of Gamma

Statement

Gamma extends to a meromorphic function on $C$ with simple poles at the nonpositive integers, and

$Res (Γ, - n) = \frac{(- 1)^{n}}{n!} (n = 0, 1, 2, \dots) .$

More generally, for every integer $m \geq 0$ and every $z \notin {0, - 1, \dots, - m}$ ,

$Γ (z) = \frac{Γ (z + m + 1)}{z (z + 1) \dots (z + m)} .$

Facts & Assumptions

Given: The holomorphic Gamma function on $Re z > 0$ .

[L1]

The functional equation $Γ (z + 1) = z Γ (z)$ holds on the right half-plane (The Gamma functional equation).

[L2]

$Γ (1) = 1$ and $Γ (n + 1) = n!$ for integers $n \geq 0$ (Gamma at the positive integers).

Proof

technique · direct

1.1

For each integer $m \geq 0$ , define $G_{m} (z) : = \frac{Γ (z + m + 1)}{z (z + 1) \dots (z + m)}$ on the half-plane $Re z > - m - 1$ with the nonpositive integers $0, - 1, \dots, - m$ removed. By repeated use of [L1], $G_{m} (z) = Γ (z)$ whenever $Re z > 0$ .

givenL1construct

2.1

The functions $G_{m}$ and $G_{m + 1}$ agree on their common domain because both equal $Γ (z)$ on the nonempty open half-plane $Re z > 0$ . Hence the $G_{m}$ glue to a meromorphic continuation of Gamma to $C$ , and step 1.1 is exactly the displayed continuation formula on the domain of $G_{m}$ . The denominator in step 1.1 shows that the only possible poles are the nonpositive integers, and each is simple.

step 1.1L1algebra

3.1

Near $z = - n$ , take $m = n$ . Then $Γ (z) = \frac{Γ (z + n + 1)}{z (z + 1) \dots (z + n - 1) (z + n)} .$ Using [L2], the numerator tends to $Γ (1) = 1$ and the product excluding $z + n$ tends to $(- n) (- n + 1) \dots (- 1) = (- 1)^{n} n!$ . Therefore $Res (Γ, - n) = \frac{1}{(- 1)^{n} n!} = \frac{(- 1)^{n}}{n!} .$

step 2.1L2algebra∎

Depends on

Used by

The residue of Gamma at z=-2 Example
FALSE: the Gamma function is entire False statement
Euler's limit formula for Gamma Theorem
Hankel's representation for reciprocal Gamma Theorem
The Weierstrass product for reciprocal Gamma Theorem

Dependency tree · two levels

5 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

K. Chandrasekharan, Lectures on the Riemann Zeta-Function, Lecture 6 §2 (standard reference, not scraped)
M. Weber, Complex Analysis, §3.7 (standard reference, not scraped)