## Infinite Product Representation of the Gamma Function

**Point of post: **In this post we prove that for every we have that

where is the Euler-Maschernoi constant.

**Theorem: ***Let . Then,*

**Proof: **We recall the well-known Euler definition of the Gamma function, namely for we have that

From this the rest is just computation:

where the only part lacking possible rigor is the splitting of the limit over the product. But, this is justified since trivially

converges to a non-zero value and thus

must converge since the whole limit converges. Or, if this doesn’t quite meet your standards recall that for a sequence of positive constants one has

converges. So, taking large enough so that we may apply this and note that

either way, we’re done.

**Corollary: **Let , then

**References:**

1. Apostol, Tom M. *Mathematical Analysis.* London: Addison Wesley Longman, 1974. Print.

2. Edwards, Harold M. *Riemann’s Zeta Function*. Mineola, NY: Dover Publications, 2001. Print.

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