10.3: C- Clinic on the Gamma Function

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The gamma function Γ(s) is defined, for s > 0, by

\[ \Gamma(s)=\int_{0}^{\infty} x^{s-1} e^{-x} d x.\]

Upon seeing any integral, your first thought is to evaluate it. Stay calm. . . first make sure that the integral exists. A quick check shows that the integral above converges when s > 0.

There is no simple formula for the gamma function for arbitrary s. But for s = 1,

\[ \Gamma(1)=\int_{0}^{\infty} e^{-x} d x=-\left.e^{-x}\right|_{0} ^{\infty}=1.\]

For s > 1 we may integrate by parts,

\[ \int_{0}^{\infty} x^{s-1} e^{-x} d x=-x^{s-1}\left.e^{-x}\right|_{0} ^{\infty}+(s-1) \int_{0}^{\infty} x^{s-2} e^{-x} d x,\]

giving

\[ \Gamma(s)=(s-1) \Gamma(s-1) \quad \text { for } s>1.\]

Apply equation (C.4) repeatedly for n a positive integer,

\[ \Gamma(n)=(n-1) \Gamma(n-1)=(n-1)(n-2) \Gamma(n-2)=(n-1)(n-2) \cdots 2 \cdot 1 \cdot \Gamma(1)=(n-1) !,\]

to find a relation between the gamma function and the factorial function. Thus the gamma function generalizes the factorial function to non-integer values, and can be used to define the factorial function through

\[ x !=\Gamma(x+1) \quad \text { for any } x>-1.\]

In particular,

\[ 0 !=\Gamma(1)=1.\]

(It is a deep and non-obvious result that the gamma function is in fact the simplest generalization of the factorial function.)

The gamma function can be simplified for half-integral arguments. For example

\[ \Gamma\left(\frac{1}{2}\right)=\int_{0}^{\infty} x^{-1 / 2} e^{-x} d x=\int_{0}^{\infty} y^{-1} e^{-y^{2}}(2 y d y)=\int_{-\infty}^{\infty} e^{-y^{2}} d y=\sqrt{\pi}\]

where we used the substitution \(y=\sqrt{x}\). Thus

\[ \Gamma\left(\frac{3}{2}\right)=\frac{1}{2} \Gamma\left(\frac{1}{2}\right)=\frac{\sqrt{\pi}}{2}=\left(\frac{1}{2}\right) !,\]

\[ \Gamma\left(\frac{5}{2}\right)=\frac{3}{2} \Gamma\left(\frac{3}{2}\right)=\frac{3}{4} \sqrt{\pi},\]

and so forth.