Weierstrass infinite product for the Gamma function
Weierstrass infinite product for the Gamma function I want to derive in this post the Weierstrass infinite product gamma . It is a very useful expression to solve many problems involving the gamma function. 1.Weierstrass infinite product 1. The Weierstrass infinite product is given by Recall Gauss infinite product for the Gamma function derived in this post (1.1) Now, and finally taking the limit! (1.2) or And recalling the functional equation of the Gamma Function we get (1.3) 2. Series Expansion Digamma function If we take logs of (1.3) and differentiate with respect to x we get the Series representation of the digamma function , extremely useful (2.1) 3. Series expansion Trigamma Function and the evaluation of an Infinite series If we differentiate (2.1) we get (3.1) if we let and we get that putting all together (3.2) On the other hand we have the reflection formula for the Gamma function taking log on both sides and differenti...