The Gamma Function

The Gamma Function The gamma function is a generalization of the factorial function for any positive integer n: Γ( n) = ( n −1)! For example, Γ(6) = (5)! =120

Gamma is defined by the improper integral ∞

Γ(n) = ∫ e −t t n −1 dt 0

Integration by parts readily reveals that Γ( n) = ( n −1)Γ( n −1) . We may write the previous result as Γ( x +1) = xΓ( x )

where x is any real number. Let’s consider the following case: 1 2

∞

For x = ½: Γ  = ∫ e −t t

1 −1 2

dt

0

∞

= ∫ e −t t

−1

2

dt

0

∞

1

(Let u = t 2 , then du =

1 − 12 t dt ) 2

= 2 ∫ e −u du 2

0

= π

(The value of the previous integral requires multivariable calculus.) We may use this value to evaluate the following values of gamma: π 3  1 1  Γ  = Γ  = 2 2 2 2     5  3 3 3 π Γ  = Γ  = 4 2 2 2  7  5  5  15 π Γ  = Γ  = . 8 2 2 2