# Thread: Uniqueness of Gamma Function

1. ## Uniqueness of Gamma Function

Is the $\Gamma$ function uniquely determined (as a meromorphic function on $\mathbb{C}$) by the functional equation $\Gamma(s + 1) = s\Gamma(s)$ and a single value, say $\Gamma(1) = 1$?

And/or is there any other characterization of the Gamma function that uniquely determines it on $\mathbb{C}$?

2. Originally Posted by EinStone
Is the $\Gamma$ function uniquely determined (as a meromorphic function on $\mathbb{C}$) by the functional equation $\Gamma(s + 1) = s\Gamma(s)$ and a single value, say $\Gamma(1) = 1$?

And/or is there any other characterization of the Gamma function that uniquely determines it on $\mathbb{C}$?
Here are the criterion that uniquely determine $\Gamma(z)$ in $\mathbb{C}$:

1.) $\Gamma(1)=1$

2.) $\Gamma(s+1)=s\Gamma(s)$

3.) $\lim_{n\to\infty}\frac{\Gamma(s+n)}{n^s\Gamma(n)}= 1$

Ask if you'd like to see why this is true.

This is a nice theorem, because if you want to show something is equal to the Gamma function, all you need to do is show it satisfies these three properties!

3. Can you give a counter example why my first definition does not work?

4. Originally Posted by EinStone
Can you give a counter example why my first definition does not work?
Let $F(s) = \Gamma(s)\cos(2\pi s)$

1.) $F(1) = \Gamma(1)\cos(2\pi) = 1$

2.) $F(s+1) = \Gamma(s+1)\cos(2\pi(s+1)) = s\Gamma(s)\cos(2\pi s) = sF(s)$

But $F(s)\neq\Gamma(s)$

5. Originally Posted by chiph588@
Let $F(s) = \Gamma(s)\cos(2\pi s)$

1.) $F(1) = \Gamma(1)\cos(2\pi) = 1$

2.) $F(s+1) = \Gamma(s+1)\cos(2\pi(s+1)) = s\Gamma(s)\cos(2\pi s) = sF(s)$

But $F(s)\neq\Gamma(s)$
To support my uniqueness claim of $\Gamma(s)$, let's compute $\lim_{n\to\infty}\frac{F(s+n)}{n^sF(n)}$.

$\lim_{n\to\infty}\frac{F(s+n)}{n^sF(n)} = \lim_{n\to\infty}\frac{\Gamma(s+n)\cos(2\pi(s+n))} {n^s\Gamma(n)\cos(2\pi n)} = 1\cdot\lim_{n\to\infty}\frac{\cos(2\pi(s+n))}{\cos (2\pi n)} \not\equiv 1$

6. Another characterization (on $\mathbb{R}$) : $\Gamma$ is the only function on $(0,+\infty)$ that is log-convex (i.e. $\log\Gamma$ is convex) and satisfies $\Gamma(1)=1$ and $\Gamma(s+1)=s\Gamma(s)$ for all $s>0$. This is called Artin's theorem I think. (and the proof is really nice; probably you can find it somewhere online)

7. Originally Posted by Laurent
Another characterization (on $\mathbb{R}$) : $\Gamma$ is the only function on $(0,+\infty)$ that is log-convex (i.e. $\log\Gamma$ is convex) and satisfies $\Gamma(1)=1$ and $\Gamma(s+1)=s\Gamma(s)$ for all $s>0$. This is called Artin's theorem I think. (and the proof is really nice; probably you can find it somewhere online)
You are referring to the Bohr–Mollerup theorem.

Here's the proof.

8. Thank you guys. Was looking for this...