Equivalence of Definitions of Gamma Function

Theorem
The following definitions of the Gamma function are equivalent:

Weierstrass Form equivalent to Euler Form
First it is shown that the Weierstrass form is equivalent to the Euler form.

Combining the limits:

But:
 * $(1): \quad \displaystyle m = \frac{m!}{\left({m - 1}\right)!} = \frac 2 1 \cdot \frac 3 2 \dots \frac{x+1} x \dots \frac m {m-1}$

Each term in $(1)$ is just $\dfrac{x+1} x = 1 + \dfrac 1 x$, so:
 * $\displaystyle m = \prod_{n \mathop = 1}^{m-1} \left({1 + \frac 1 n}\right)$

Thus the expression for $\dfrac 1 {\Gamma \left({z}\right)}$ becomes:

Hence:
 * $\displaystyle \Gamma \left({z}\right) = \frac 1 z \prod_{n \mathop = 1}^{\infty} \left({1 + \frac 1 n}\right)^z \left({1 + \frac z n}\right)^{-1}$

which is the Euler form of the Gamma function.

Integral Form equivalent to Euler Form
It is taken for granted that the Gamma function increases monotonically for real numbers greater than or equal to $1$.

We begin with an inequality that can easily be verified using cross multiplication.

$x$ is a real number between $0$ and $1$ whereas $n$ is a positive integer:


 * $\displaystyle\frac{\log\Gamma(n-1)-\log\Gamma(n)}{(n-1)-n}\leq\frac{\log\Gamma(x+n)-\log\Gamma(n)}{(x+n)-n}\leq\frac{\log\Gamma(n+1)-\log\Gamma(n)}{(n+1)-n}$

Since n is a positive integer, we can make use of the identity $\displaystyle \Gamma(n)=(n-1)!$. Simplifying, we get


 * $\displaystyle\log(n-1)\leq\frac{\log\Gamma(x+n)-\log((n-1)!)}{x}\leq\log(n)$

We now make use of the identity $\displaystyle \Gamma(x+n)=\prod_{k=1}^{n}(x+n-k)\Gamma(x)$, along with the fact that the Gamma function is log-convex, to simplify the inequality:


 * $\displaystyle(n-1)^x(n-1)!\prod_{k=1}^{n}(x+n-k)^{-1}\leq\Gamma(x)\leq n^x(n-1)!\prod_{k=1}^{n}(x+n-k)^{-1}$

Taking the limit as n goes to infinity and using the Squeeze Theorem, we get


 * $\displaystyle \Gamma(x)=\lim_{n\to\infty}n^{x}n!\prod_{k=0}^{n}(x+n-k)^{-1}$

which is another representation of Euler's form. This product can also be shown to be equivalent to the Weierstrass form.

We proved the theorem for x between 0 and 1. If this satisfies the Gamma functional equation, then our proof is valid for all reals but the simple poles of the Gamma function. It turns out this indeed satisfies the functional equation. The proof of this is trivial and hence not presented.