# What is an elementary proof of the Weierstrass Factorization Theorem?

I was researching the gamma function and then I stumbled into this: $$\Gamma(x)=\frac{e^{-\gamma x}}{x}\prod_{i\geq1}{\frac{e^{x/i}}{1+x/i}}$$ Turns out it's related to Weierstrass' Factorization Theorem. I've seen this before in: $$\sin(x)=x(1-\frac{x}{\pi})(1+\frac{x}{\pi})(1-\frac{x}{2\pi})(1+\frac{x}{2\pi})\dotsb$$ Figured now would be a good time to understand this theorem. Someone please help with a derivation of the theorem as well as how it connects to the product for the Gamma function for someone who is an undergraduate freshman (currently doing multivariable). Thanks

• See this. It implies $\phi(s) = \frac{\Gamma'(s)}{\Gamma(s)}+ \sum_{n=0}^\infty \frac{1}{s+n}-\frac{1}{n}$ is an entire and bounded function, thus constant. – reuns Sep 13 '17 at 6:00
• Impossible to explain if you have not taken complex analysis. – timur Sep 14 '17 at 2:01
• Yeah that's what I thought. What do you think of Needham's book "Visual Complex Analysis"? – Matthew K Sep 16 '17 at 2:29