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Gamma Function

Let h(x)=xα-1exp(-x) for 0<x<. The Gamma function determines how the integral of this function over the range (0,) varies with α.

Γ(α)=0xα-1exp(-x)dx,

for α>0. The Gamma function can be evaluated in R by the command gamma(a):

> gamma(0.5)        # Calculate the gamma function at 0.5
[1] 1.772454
> x_seq(1,10)       # Let x = (1,2,3,4,5,6,7,8,9,10)
> x
 [1]  1  2  3  4  5  6  7  8  9 10
> gamma(x)          # Calculate the gamma function at these values
 [1]  1    1    2    6    24   120   720   5040  40320 36288

Properties:

  1. 1.

    Recurrence relation (see below): Γ(α+1)=αΓ(α) for α>0.

  2. 2.

    Positive integers (see notes): Γ(1)=1, Γ(α)=(α-1)! for α a positive integer.

  3. 3.

    Special (see below): Γ(1/2)=π.

  4. 4.

    Limits (see below): Γ(α) as α0 or α.

Proofs:

Proof that Γ(α+1)=(α)Γ(α) for α>1.

Γ(α+1) =0tαexp(-t)dt
=[tα(-1)exp(-t)]0+0αtα-1exp(-t)dt
=0-0+αΓ(α)

for α>0.

Proof that Γ(1/2)=π.

In Math113 you saw that

I=-e-x2dx=π.

Substituting x=t1/2, so dx=t-1/2/2dt, we have, by symmetry,

I=20e-x2dx=20e-tt-1/2/2dt=0t1/2-1e-tdt=Γ(1/2).

Proof that Γ(α) as α0.

Γ(α)>01xα-1e-xdx>01xα-1e-1dx=e-1[xαα]01=e-1α-1

as α0.

Proof that Γ(α) as α.

Γ(α)>2xα-1e-xdx>22α-1e-xdx=2α-1[-e-x]2=2α-1e-2

as α.