Gamma Difference Equation/Proof 2

Theorem

$\map \Gamma {z + 1} = z \, \map \Gamma z$

Proof

$\ds \frac {\map \Gamma {z + 1} } {\map \Gamma z}$

$=$

$\ds \paren {\frac 1 {z + 1} \lim_{m \mathop \to \infty} \prod_{n \mathop = 1}^m \frac {\paren {1 + \frac 1 n}^{z + 1} } {1 + \frac {z + 1} n} } \div \paren {\frac 1 z \lim_{m \mathop \to \infty} \prod_{n \mathop = 1}^m \frac {\paren {1 + \frac 1 n}^z} {1 + \frac z n} }$

Definition of Euler Form of Gamma Function

$\ds $

$=$

$\ds \frac z {z + 1} \lim_{m \mathop \to \infty} \prod_{n \mathop = 1}^m \paren {\frac {\paren {1 + \frac 1 n}^{z + 1} \paren {1 + \frac z n} } {\paren {1 + \frac 1 n}^z \paren {1 + \frac {z + 1} n} } }$

Product Rule for Complex Sequences

$\ds $

$=$

$\ds \frac z {z + 1} \lim_{m \mathop \to \infty} \prod_{n \mathop = 1}^m \paren {\frac {n + 1} n} \paren {\frac {z + n} {z + n + 1} }$

$\ds $

$=$

$\ds \frac z {z + 1} \lim_{m \mathop \to \infty} \paren {\frac {m + 1} 1} \paren {\frac {z + 1} {z + m + 1} }$

Telescoping Product

$\ds $

$=$

$\ds z$

Limit to Infinity of Complex Rational Function

$\blacksquare$