B.4 The Gamma function

The gamma function is a useful special function that extends the domain of the factorial function to non-integer values. It is defined by the integral²² 2 I like to write $\mathrm{\Gamma }(z)$ like this $$\mathrm{\Gamma }(z)={\int }_0^{\mathrm{\infty }}{x}^z{e}^{-x}\frac{\mathrm{d}x}{x}$$ (B.16) since the measure $\mathrm{d}x/x$ is invariant under a rescaling $x^{\prime }=\lambda x$, i.e. $\mathrm{d}x/x=\mathrm{d}{x}^{\prime }/x^{\prime }$. This is one reason why clever math folk defined $\mathrm{\Gamma }(z)$ with the power $z-1$ in Eq. (B.17) instead of just $z$.

and has the familiar recursive relationship

starting from $\mathrm{\Gamma }(1)=1$. We have

While the gamma function is defined for all complex numbers except the non-positive integers, analytical expressions are only known where $n$ is an integer or half-integer. In particular,

So for instance, using the recursion, $\mathrm{\Gamma }(3/2)=\frac{1}{2}\mathrm{\Gamma }(\frac{1}{2})=\sqrt{\pi }/2$.

In statistical mechanics, the gamma function occurs frequently in integrals involving the Maxwell–Boltzmann distribution. In addition, the area of a sphere in $d$ dimensions is

which comes up a lot later in the course.