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The Gamma Function: Leonhard Euler's Generalization of the Factorial Function

Leonhard Euler is considered one of the top ten mathematicians in human history. He
was an extremely prolific mathematician and a very ingenious one at that. In 1729 Euler
proposed a generalization of the factorial function n!=n(n-1)(n-2)··3·2·1.
from integers to any real number. His generalization is called the gamma function
Γ(x), which is defined as

Since lim_{m→∞} [(x+1)/m] = 0 this means the first term is
equal to zero. Thus

xΓ(x) = Γ(x+1)·0 + Γ(x+1) = Γ(x+1)

The proof of course could have been streamlined to end up with

xΓ(x) = Γ(x+1){lim_{m→∞} [1 + (x+1)/m]
= Γ(x+1)

Since Γ(x+1)=xΓ(x) and Γ(1)=1, Γ(x+1)=x! for x equal to a
non-negative integer. For negative integers Γ(x+1) = ∞.

The evaluation of Γ(x) for x non-integer using Euler's formula is not easy to
evaluate analytically and
now an equivalent integral formula is used in its place.