Sometime around 1729 Euler defined the Gamma function in the form
Γ(x) = limn→∞(n−1)!nx
x(1 + x)(2 + x)(3 + x)···(n−1 + x)= limn→∞nx
x(1 + x 1 )(1 + x 2 )···(1 + nx− 1 )
(12.106)Karl Weierstrass modified Euler’s form for the Gamma function and represented
it in the form
1
Γ(z)=z eγz∏∞n=1[(
1 + z
n)
e−z/n]
(12.107)where γ is Euler’s constant from equation (12.104). Here the infinite product
∏∞n=1
[(
1 + z
n)
e−z/n]is convergent for all values of zpositive, negative, real or complex.
Other forms of the Gamma function can be found in the mathematical literature.
The representation of the Gamma function in the complex plane provides new incites
into properties of the Gamma function. As an interesting exercise check out some
textbooks on the Gamma function to see how all of the above forms of the Gamma
function are equivalent. This type of exercise is one example illustrating the concept
that functions and ideas, which occur in mathematical studies, can be presented in
a variety of ways.
Euler formula for the Gamma function
Having a variety of forms for representing the Gamma function provides one the
opportunity to seek out and discover other properties of the Gamma function. For
example, employ the Weirstrass representation
1
Γ(x)=x e γx∏∞n=1[(
1 + x
n)
e−x/n]
(12.108)and show
1
Γ(x)1
Γ(−x)=−x^2 eγxe−γx∏∞n=1(
1 + x
n)(
1 −x
n)
e−x/nex/n (12.109)This equation simplifies using the property Γ(1 −x) = −xΓ(−x). One can verify
equation (12.109) simplifies to
1
Γ(x)1
Γ(1 −x)=x∏∞n=1(
1 −x^2
n^2)or
1
Γ(x)1
Γ(1 −x)=x(
1 −x^2
12)(
1 −x^2
22)(
1 −x^2
32)
··· (12.110)