This post is intended as a continuation of my last post, even if the chosen title does not suggests so. I want to prove some more results about the gamma function and in doing so, obtain an infinite product formula for , without using complex analysis. I’ll finish by using this infinite product (due to Euler) to show that
as opposed to the usual clever proof by polar integration.
Lemma. (de Moivre’s formula) Let . Then
Proof. We prove de Movire’s formula by induction on . The base case is obvious. Suppose the formula is true for some . Then
where the last equality follows from an application of the sine and cosine angle sum formulas.
We can use de Moivre’s formula to show that , for is a polynomial in . Indeed,
Making the substitution completes the proof of the claim. In fact, this polynomial is of degree . Indeed, has distinct roots (modulo ) at , for . Hence, the polynomial has roots at , for . So we can write as the product
We now compute the constant . If we divide both sides by and compute the limit as , we obtain
which implies that
Lemma. (Euler’s Sine Product)
Proof. Fix and positive integers and such that . The identity for allows us to write
Set . By Jordan’s inequality, for and therefore
Without loss of generality, we may assume that is sufficiently large so that .
where . Hence,
We obtain the estimate
Since , we obtain that . Since the series converges, we see that the infinite product converges absolutely and therefore we can evaluate the limit of the product by computing the limit of each factor. By the squeeze theorem,
Letting , we conclude from another application of the squeeze theorem that
Using the identity , for , we can write
Since and , we conclude that