Gamma Function

Let $h(x)=x^{\alpha -1}\exp (-x)$ for . The Gamma function determines how the integral of this function over the range $(0,\mathrm{\infty })$ varies with $\alpha$.

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

Properties:

1. 1.

   Recurrence relation (see below): $\mathrm{\Gamma }(\alpha +1)=\alpha \mathrm{\Gamma }(\alpha )$ for $\alpha >0$.

2. 2.

   Positive integers (see notes): $\mathrm{\Gamma }(1)=1$, $\mathrm{\Gamma }(\alpha )=(\alpha -1)!$ for $\alpha$ a positive integer.

3. 3.

   Special (see below): $\mathrm{\Gamma }(1/2)=\sqrt{\pi }$.

4. 4.

   Limits (see below): $\mathrm{\Gamma }(\alpha )\to \mathrm{\infty }$ as $\alpha \to 0$ or $\alpha \to \mathrm{\infty }$.

Proofs:

Proof that $\mathrm{\Gamma }(\alpha +1)=(\alpha )\mathrm{\Gamma }(\alpha )$ for $\alpha >1$.

for $\alpha >0$.

Proof that $\mathrm{\Gamma }(1/2)=\sqrt{\pi }$.

In Math113 you saw that

Substituting $x=t^{1/2}$, so $\mathrm{d}x=t^{-1/2}/2\mathrm{d}t$, we have, by symmetry,

Proof that $\mathrm{\Gamma }(\alpha )\to \mathrm{\infty }$ as $\alpha \to 0$.

as $\alpha \to 0$.

Proof that $\mathrm{\Gamma }(\alpha )\to \mathrm{\infty }$ as $\alpha \to \mathrm{\infty }$.

as $\alpha \to \mathrm{\infty }$.