Residue of Gamma Function

Theorem

Let $\Gamma$ be the Definition:Gamma Function.

Let $n$ be a non-negative integer.

Then:

$\Res \Gamma {-n} = \dfrac {\paren {-1}^n} {n!}$

Proof

By Poles of Gamma Function, $\Gamma$ has simple poles at the non-positive integers, so $-n$ is a simple pole of $\Gamma$.

Then:

\(\ds \Res \Gamma {-n}\)

\(=\)

\(\ds \lim_{z \mathop \to -n} \paren {z - \paren {-n} } \map \Gamma z\)

Residue at Simple Pole

\(\ds \)

\(=\)

\(\ds \lim_{z \mathop \to -n} \paren {z + n} \paren {\frac {z \paren {z + 1} \ldots \paren {z + n} } {z \paren {z + 1} \ldots \paren {z + n} } } \map \Gamma z\)

multiplying by $1$

\(\ds \)

\(=\)

\(\ds \lim_{z \mathop \to -n} \paren {z + n} \frac {\map \Gamma {z + n + 1} } {z \paren {z + 1} \ldots \paren {z + n} }\)

Gamma Difference Equation

\(\ds \)

\(=\)

\(\ds \lim_{z \mathop \to -n} \frac {\map \Gamma {z + n + 1} } {z \paren {z + 1} \ldots \paren {z + n - 1} }\)

cancelling $z + n$

\(\ds \)

\(=\)

\(\ds \frac {\map \Gamma 1} {-n \paren {-n + 1} \ldots \paren {-1} }\)

\(\ds \)

\(=\)

\(\ds \frac 1 {\paren {-1}^n \paren {n \paren {n - 1} \ldots \paren 1} }\)

$\map \Gamma 1 = 1$

\(\ds \)

\(=\)

\(\ds \frac 1 {\paren {-1}^n n!}\)

Definition of Factorial

\(\ds \)

\(=\)

\(\ds \frac {\paren {-1}^n} {n!}\)

$\blacksquare$