What are the two definitions of gamma function and how are they related?

[LHS1]

I spend some time studying special functions recently. I found two definitions of gamma function, one in form of integral and the other in form of infinite products, and I cannot prove of their equivalence. I found the definition in infinite product form important in proofing many properties of gamma function such as gamma reflection formula, therefore I eager to learn it. Could anyone help me, please !

 

[Santa1]

See;

http://en.wikipedia.org/wiki/Bohr–Mollerup_theorem

If the product has those properties it is the Gamma function.

 

[LHS1]

Thank you very much!