Can someone show me how u can derive the infinite product representation of digamma(z) /gamma(z)

Question

Here is the picture from Wikipedia which shows the infinite product. I am confused about how to derive this infinite product below. Infinite product picture

Answer 1

According to the problem book "Problems in the Theory of Functions of a Complex Variable", L. Volkovysky, G. Lunts and L. Aramanovich, Problem 951, such a function, with simple zeros $x_1, x_2, x_0, x_3, x_4, x_5...$ has a Weierstrass product expansion, $$f(z)=f(0)e^{\frac{f'(0)}{f(0)}z}\prod_{k=0}^{\infty}(1-\frac{z}{x_k})e^{\frac{z}{x_k}}.$$ Honsetly, I do not know the proof. They skipped in the solution set very much. The rest of the problem is easier.

Here, $f(z)=\frac{\psi(z)}{\Gamma(z)}=\frac{\Gamma'(z)}{\Gamma^2(z)}$. We know the Laurent expansion: How to obtain the Laurent expansion of gamma function around $z=0$? Then, by limit, we can show that $f(0)=-1$ and $f'(0)=-2\gamma$.