26
Dec 21

Gamma function

Gamma function

The gamma function and gamma distribution are two different things. This post is about the former and is a preparatory step to study the latter.

Definition. The gamma function is defined by

\Gamma \left( t\right) =\int_{0}^{\infty }x^{t-1}e^{-x}dx,\ t> 0.

The integrand f(t)=x^{t-1}e^{-x} is smooth on \left( 0,\infty \right) , so its integrability is determined by its behavior at \infty and 0. Because of the exponent, it is integrable in the neighborhood of \infty . The singularity at 0 is integrable if t>0. In all calculations involving the gamma function one should remember that its argument should be positive.

Properties

1) Factorial-like property. Integration by parts shows that

\Gamma \left( t\right) =-\int_{0}^{\infty }x^{t-1}\left( e^{-x}\right)^{\prime }dx=-x^{t-1}e^{-x}|_{0}^{\infty }+\left( t-1\right)\int_{0}^{\infty }x^{t-2}e^{-x}dx

=\left( t-1\right) \Gamma \left( t-1\right) if t>1.

2) \Gamma \left( 1\right) =1 because \int_{0}^{\infty }e^{-x}dx=1.

3) Combining the first two properties we see that for a natural n

\Gamma \left( n+1\right) =n\Gamma ( n) =...=n\times \left(n-1 \right) ... \times 1 \times \Gamma\left( 1 \right) =n!

Thus the gamma function extends the factorial to non-integer t>0.

4) \Gamma \left( 1/2\right) =\sqrt{\pi }.

Indeed, using the density f_{z} of the standard normal z we see that

\Gamma \left( 1/2\right) =\int_{0}^{\infty }x^{-1/2}e^{-x}dx=

(replacing x^{1/2}=u)

=\int_{0}^{\infty }\frac{1}{u}e^{-u^{2}}2udu=2\int_{0}^{\infty}e^{-u^{2}}du=\int_{-\infty }^{\infty }e^{-u^{2}}du=

(replacing u=z/\sqrt{2})

=\frac{\sqrt{\pi }}{\sqrt{2\pi }}\int_{-\infty }^{\infty }e^{-z^{2}/2}dz=\sqrt{\pi }\int_{R}f_{z}\left( t\right) dt=\sqrt{\pi }.

Many other properties are not required in this course.

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