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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
With this definition Γ(n+1)=n!.
The formula may be used to evaluate Γ(1); i.e.,
Since Γ(n)=(n-1)!, Γ(1)=0! and hence 0!=1.
The crucial relationship for the gamma function is Γ(x+1)=xΓ(x).
Proof:
Consider
This can be evaluated as the sum of limit of two terms, providing that the limit of each term exists. The first term is
The second term is
This second term is none other than Γ(x+1).
The first term may be expressed as
The limit of this expression is the product of the limits, providing both limits exist; i.e.,
Since lim_{m→∞} [(x+1)/m] = 0 this means the first term is equal to zero. Thus
The proof of course could have been streamlined to end up with
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.
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