Chapter 11 - Special Functions

Reference Mathematical Methods in the Physical Sciences (3e) by Mary L. Boas.

Factorial functions $n!$

Factorial functions $n!$ are defined as $$n!={\int }_0^{\infty }{x}^n{e}^{-x}dx$$ More memorably, factorials are the product of their sums: $$5!=5\ast 4\ast 3\ast 2\ast 1=120$$ A table of the initial ones is below.

Gamma functions $\Gamma (n)$

For any $n>0$, $$\Gamma (n)={\int }_0^{\infty }{x}^{n-1}{e}^{-x}$$ The $\Gamma (n)$ is further easily defined in terms of factorials: $$\Gamma (n)=(n-1)!\Gamma (n+1)=n!$$ Useful is also the recursion relation: $$\Gamma (n+1)=n\Gamma (n)=n(n-1)!$$

The gamma function seems useful for working with both fractions and complex numbers.

For negative numbers where $n\le 0$, $$\Gamma (n)=\frac{1}{n}\Gamma (n+1)$$ Also, some special formulae: $$\begin{array}{r}\Gamma (1/2)=\sqrt{\pi }\\ \Gamma (n)\Gamma (1-n)=\frac{\pi }{\sin (\pi n)}\end{array}$$ Note: this last equation is undefined for integer $n=1,2,3\dots$

Beta functions $\beta (p,q)$

Beta functions, I'm not sure what they're useful for. But Boas gives several representations of it, which I'll include below.

For $p,q>0$: $$\begin{array}{rl}\beta (p,q)& ={\int }_0^1{x}^{p-1}(1-x{)}^{q-1}dx\\ & =\frac{1}{a^{p+q-1}}{\int }_0^a{y}^{p-1}(a-y{)}^{q-1}dy\\ & =2{\int }_0^{2\pi }(\sin \theta {)}^{2p-1}(\cos \theta {)}^{2q-1}d\theta \\ & ={\int }_0^{\infty }\frac{y^{p-1}}{(1+y{)}^{p+q}}dy\end{array}$$

Also, note that $\beta (p,q)=\beta (q,p)$.

In terms of gamma functions, $$\beta (p,q)=\frac{\Gamma (p)\Gamma (q)}{\Gamma (p+q)}$$