what does the ${_2F_1}\left[\cdot, \cdot, \cdot, \cdot\right]$ function mean?

Question

I was reading the integral tables, where it says:

\begin{equation} \int \cos^p ax dx = -\frac{1}{a(1+p)}{\cos^{1+p} ax} \times {_2F_1}\left[ \frac{1+p}{2}, \frac{1}{2}, \frac{3+p}{2}, \cos^2 ax \right] \end{equation}

May I ask what does the ${_2F_1}\left[\cdot, \cdot, \cdot, \cdot\right]$ function mean?

Answer 1

It is hypergeometric function. We usually write it as $_2F_1[a,b;c;z]$, not $_2F_1[a,b,c,z]$.

For $|z|\lt 1$, the hypergeometric function is defined as $$_2F_1[a,b;c;z]=\sum_{n=0}^{\infty}\frac{(a)_n(b)_n}{(c)_n}\cdot\frac{z^n}{n!}$$ where $$(q)_n=\begin{cases}q(q+1)\cdots (q+n-1)&\text{if}\ n\gt 0\\1&\text{if}\ n=0\end{cases}$$ Here, $(q)_n$ is called as Pochhammer symbol.