**The Gamma Function Part 2: Weierstrass form and Stirling’s formulas**

(Part 1 is here)

Let us now define a sequence of functions . We note first that has simple poles at 0, -1, -2, …, -n. Next,

,

and

.

We also look at the logarithm of and the derivatives thereof:

Thus:

So is convex.

Now, let us consider the limit of this sequence, which can be shown to exist via complex analysis:

We thus see from the above that has poles at 0, -1, -2, -3,…; and the properties of above tell us:

,

,

And is convex. As we noted at the end of the previous part, the only logarithmically convex function , x>0, for which and is the gamma function. Thus

.

Next, I will remind readers of the Euler-Mascheroni constant, *γ*, which is defined by the limit:

giving *γ*=0.57721566…

We note:

Now, suppose we multiply each term in the product by , multiplying out front with the reciprocal terms:

.

Now, we take the limit as , and see that the limit of the bracketed terms in the first exponent is –*γ*, obtaining:

.

This in known as the Weierstrass form of the gamma function.

Also useful are Stirling’s approximations, which allow one to approximate for large *x*, and *n*! for large *n*:

A derivation of the second approximation using Laplace’s method can be found here.

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Tags: Factorial, Gamma Function, Math, Monday Math, Stirling's Approximation, Stirling's Formulas, Weierstrass

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