# Bound on Complex Values of Gamma Function

## Theorem

Let $\map \Gamma z$ denote the Gamma function.

Then for any complex number $z = s + i t$, we have for $\size b \le \size t$:

$\size {\map \Gamma {s + i t} } \le \dfrac {\size {s + i b} } {\size {s + i t} } \size {\map \Gamma {s + i b} }$

## Proof

From the Euler Form of the Gamma Function:

 $\ds \size {\map \Gamma {s + i t} }$ $=$ $\ds \lim_{M \mathop \to \infty} \size {\dfrac 1 {s + i t} \prod_{n \mathop = 1}^M \dfrac {\paren {1 + \frac 1 n}^{s + i t} } {1 + \frac {s + i t} n} }$ $\ds$ $=$ $\ds \lim_{M \mathop \to \infty} \dfrac 1 {\size {s + i t} } \prod_{n \mathop = 1}^M \size {\dfrac {\paren {1 + \frac 1 n}^{s + i t} } {1 + \frac {s + i t} n} }$ $\ds$ $=$ $\ds \lim_{M \mathop \to \infty} \dfrac 1 {\size {s + i t} } \prod_{n \mathop = 1}^M \dfrac {\size {\paren {1 + \frac 1 n}^{s + i t} } } {\size {1 + \frac {s + i t} n} }$ $\ds$ $=$ $\ds \lim_{M \mathop \to \infty} \dfrac 1 {\size {s + i t} } \prod_{n \mathop = 1}^M \dfrac {\size {\paren {1 + \frac 1 n}^s} } {\size {1 + \frac {s + i t} n} }$ Modulus of Exponential of Imaginary Number is One

Because $\size b \le \size t$, we have that:

 $\ds b^2$ $\le$ $\ds t^2$ $\ds \leadsto \ \$ $\ds \paren {\dfrac b n}^2$ $\le$ $\ds \paren {\dfrac t n}^2$ $\ds \leadsto \ \$ $\ds \paren {1 + \dfrac s n}^2 + \paren {\dfrac b n}^2$ $\le$ $\ds \paren {1 + \dfrac s n}^2 + \paren {\dfrac t n}^2$ $\ds \leadsto \ \$ $\ds \size {1 + \frac {s + i b} n}$ $\le$ $\ds \size {1 + \frac {s + i t} n}$ Definition of Complex Modulus $\ds \leadsto \ \$ $\ds \dfrac 1 {\size {1 + \frac {s + i b} n} }$ $\ge$ $\ds \dfrac 1 {\size {1 + \frac {s + i t} n} }$

Using this we obtain:

 $\ds \size {\map \Gamma {s + i t} }$ $=$ $\ds \lim_{M \mathop \to \infty} \dfrac 1 {\size {s + i t} } \prod_{n \mathop = 1}^M \dfrac {\size {\paren {1 + \frac 1 n}^s} } {\size {1 + \frac {s + i t} n} }$ $\ds$ $\le$ $\ds \lim_{M \mathop \to \infty} \dfrac 1 {\size {s + i t} } \prod_{n \mathop = 1}^M \dfrac {\size {\paren {1 + \frac 1 n}^s} } {\size {1 + \frac {s + i b} n} }$ $\ds$ $=$ $\ds \dfrac {\size {s + i b} } {\size {s + i t} } \lim_{M \mathop \to \infty} \dfrac 1 {\size {s + i b} } \prod_{n \mathop = 1}^M \dfrac {\size {\paren {1 + \frac 1 n}^s} } {\size {1 + \frac{s + i b} n} }$ $\ds$ $=$ $\ds \dfrac {\size {s + i b} } {\size {s + i t} } \lim_{M \mathop \to \infty} \dfrac 1 {\size {s + i b} } \size {\prod_{n \mathop = 1}^M \dfrac {\paren {1 + \frac 1 n}^{s + i b} } {1 + \frac{s + i b} n} }$ $\ds$ $=$ $\ds \dfrac {\size {s + i b} } {\size {s + i t} } \size {\map \Gamma {s + i b} }$

$\blacksquare$