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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

#### Γ(x) = limm→∞ {mxm!/[x(x+1)(x+2)···(x+m)]}

With this definition Γ(n+1)=n!.

The formula may be used to evaluate Γ(1); i.e.,

#### Γ(1) = limm→∞ {m·m!/(m+1)!} = limm→∞ {m/(m+1)} = 1

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

#### xΓ(x) = limm→∞ {mxm!/[(x+1)(x+2)···(x+m)]} = limm→∞ {mxm!(x+1+m)/[(x+1)(x+2)···(x+m)(x+1+m)]}

This can be evaluated as the sum of limit of two terms, providing that the limit of each term exists. The first term is

#### limm→∞ {mxm!(x+1)/[(x+1)(x+2)···(x+m)(x+1+m)]}

The second term is

#### limm→∞ {mxm!(m)/[(x+1)(x+2)···(x+m)(x+1+m)]} which reduces to limm→∞ {mx+1m!/[(x+1)(x+2)···(x+m)(x+1+m)]}

This second term is none other than Γ(x+1).

The first term may be expressed as

#### limm→∞ {mxm![(x+1)/m]/[(x+1)(x+2)···(x+m)(x+1+m)]}

The limit of this expression is the product of the limits, providing both limits exist; i.e.,

#### {limm→∞ {mx+1m!/[(x+1)(x+2)···(x+m)(x+1+m)]}·{limm→∞ [(x+1)/m]

Since limm→∞ [(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){limm→∞ [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.