Weierstrass product

Here we shall deduce the product expansion

$$\frac{1}{\Gamma (z)} = z \exp (\gamma z) \prod_{n=1}^{\infty}(1+z/n)\exp (-z/n)$$

from Wielandt's theorem. First, however, we shall show that the infinite product defines an entire function. Then we prove that it is equal to $1/\Gamma (z)$.