Theorem: Gauss Multiplication Formula

Let $\Gamma$ denote the Gamma Function.
Let $n \in \N_{>0}$ where $\N_{>0}$ denotes the non-zero natural numbers.
Then:
:$\ds \forall z \notin \set {-\frac m n: m \in \N}: \prod_{k \mathop = 0}^{n - 1} \map \Gamma {z + \frac k n} = \paren {2 \pi}^{\paren {n - 1} / 2} n^{1/2 - n z} \map \Gamma {n z}$

Proof:
{{begin-eqn}}
{{eqn | l = \map \Gamma {z + \frac k n}
      | r = \paren {z + \frac k n - 1} \map \Gamma {z + \frac k n - 1}
      | c = Gamma Difference Equation
}}
{{eqn | r = \lim_{m \mathop \to \infty} \paren {z + \frac k n - 1} \frac {m! m^{z + k / n - 1} } {\paren {z + \frac k n - 1} \paren {z + \frac k n} \paren {z + \frac k n + 1} \paren {z + \frac k n + 2} \cdots \paren {z + \frac k n - 1 + m} } 
      | c = {{Defof|Euler Form of Gamma Function}}
}}
{{eqn | r = \lim_{m \mathop \to \infty} \frac {\sqrt {2 \pi m} \paren {\frac m e}^m m^{z + k / n - 1} } {\paren {z + \frac k n} \paren {z + \frac k n + 1} \paren {z + \frac k n + 2} \cdots \paren {z + \frac k n - 1 + m} }
      | c = Stirling's Formula and canceling $\paren {z + \frac k n - 1}$
}}
{{eqn | r = \lim_{m \mathop \to \infty} \frac {\sqrt {2 \pi m} \paren {\frac m e}^m m^{z + k / n - 1} } {\paren {z + \frac k n} \paren {z + \frac k n + 1} \paren {z + \frac k n + 2} \cdots \paren {z + \frac k n - 1 + m} } \times \frac {n^m} {n^m}
      | c = multiplying top and bottom by $n^m$
}}
{{eqn | r = \lim_{m \mathop \to \infty} \frac {\sqrt {2 \pi} \paren {\frac {m n} e}^m m^{z + k / n - 1/2} } {\paren {n z + k} \paren {n z + k + n} \paren {n z + k + 2n} \cdots \paren {n z + k - n + m n} }
      | c = 
}}
{{end-eqn}}
Taking the product for $k = 0$ to $n - 1$, we have:
{{begin-eqn}}
{{eqn | l = \prod_{k \mathop = 0}^{n - 1} \map \Gamma {z + \frac k n}
      | r = \lim_{m \mathop \to \infty} \frac {\paren {\sqrt {2 \pi} }^n \paren {\frac {m n} e}^{m n} m^{n z  - n / 2} m^{\sum_{k \mathop = 0}^{n - 1} k / n} } {\paren {n z} \paren {n z + 1} \cdots \paren {n z - 1 + m n} }
      | c = 
}}
{{eqn | r = \lim_{m \mathop \to \infty} \frac {\paren {\sqrt {2 \pi} }^n \paren {\frac {m n} e}^{m n} m^{n z  - n / 2 + n / 2 - 1 / 2 } } {\paren {n z} \paren {n z + 1} \cdots \paren {n z - 1 + m n} }
      | c = Sum of Arithmetic Sequence
}}
{{eqn | r = \lim_{m \mathop \to \infty} \frac {\paren {\sqrt {2 \pi} }^n \paren {\frac m e}^m m^{n z  - 1 / 2} n^{1 / 2 - n z} } {\paren {n z} \paren {n z + 1} \cdots \paren {n z - 1 + m} }
      | c = substituting $m n \mapsto m$
}}
{{eqn | r = \lim_{m \mathop \to \infty} \frac {\paren {\sqrt {2 \pi} }^{n - 1} m! m^{n z  - 1} n^{1 / 2 - n z} } {\paren {n z} \paren {n z + 1} \cdots \paren {n z - 1 + m} }
      | c = Stirling's Formula
}}
{{eqn | r = \lim_{m \mathop \to \infty} \frac {\paren {n z - 1} } {\paren {n z - 1} } \times \frac {\paren {\sqrt {2 \pi} }^{n - 1} m! m^{n z  - 1} n^{1 / 2 - n z} } {\paren {n z} \paren {n z + 1} \cdots \paren {n z - 1 + m} }
      | c = multiplying top and bottom by $n z - 1$
}}
{{eqn | r = \paren {2 \pi}^{\paren {n - 1} / 2} n^{1 / 2 - n z} \paren {n z - 1} \lim_{m \mathop \to \infty} \frac {m! m^{n z  - 1} } {\paren {n z - 1} \paren {n z} \paren {n z + 1} \cdots \paren {n z - 1 + m} }
      | c = bringing $\paren {2 \pi}^{\paren {n - 1} / 2} n^{1 / 2 - n z} \paren {n z - 1}$ outside the limit
}}
{{eqn | r = \paren {2 \pi}^{\paren {n - 1} / 2} n^{1 / 2 - n z} \paren {n z - 1} \map \Gamma {n z - 1}
      | c = {{Defof|Euler Form of Gamma Function}}
}}
{{eqn | r = \paren {2 \pi}^{\paren {n - 1} / 2} n^{1 / 2 - n z} \map \Gamma {n z}
      | c = Gamma Difference Equation
}}
{{end-eqn}}
{{qed}}