I was trying to solve this: $\int_\;\frac{1}{(1-x^2)^{3/4}}\;dx$
but I couldn't come up with any good idea so I used "Wolfram Alpha". After then, the trouble began...
The following is the solution according to "Wolfram":
I haven't seen 'Hypergeometric Function' in my life never again.So my questions are:
- How is the above integral equal to this?
- What exactly is a Hypergeometric Function and how can someone use it?
- Are there any books that somebody would suggest in order to learn more about this function?
I'm pretty sure this is quite elementary but I have no idea how to proceed. Any help would be valuable.
Thanks in advance!!!
We can use https://web.archive.org/web/20120127210623/http://www.dsi.unifi.it/~resp/GouldBK.pdf : BC1,BS
i.e. and apply combinatorial and Pochhammer identities
To get
$\frac{1}{\left(1-x^{2}\right)^{\frac{3}{4}}}={\displaystyle \sum_{k=0}^{\infty}}\left(\begin{array}{c} \left(-\frac{3}{4}\right)\\ k \end{array}\right)\left(-x^{2}\right)^{k}={\displaystyle \sum_{k=0}^{\infty}\left(-1\right)^{k}}\left(\begin{array}{c} \left(k+\frac{3}{4}-1\right)\\ k \end{array}\right)\left(-x^{2}\right)^{k}$
$={\displaystyle \sum_{k=0}^{\infty}}\left(\begin{array}{c} \left(k-\frac{1}{4}\right)\\ k \end{array}\right)\left(x^{2}\right)^{k}=\sum_{k=0}^{\infty}\frac{\Gamma\left(\left(k-\frac{1}{4}\right)+1\right)}{\left(\Gamma\left(k-\left(k-\frac{1}{4}\right)\right)\right)k!}x^{2k}=\sum_{k=0}^{\infty}\frac{\Gamma\left(k+\frac{3}{4}\right)}{\left(\Gamma\left(\frac{3}{4}\right)\right)k!}x^{2k} $
Integrating
$x\cdot{\displaystyle \sum_{k=0}^{\infty}}\frac{\left(\frac{3}{4}\right)_{k}}{\left(2k+1\right)}\frac{\left(x^{2}\right)^{k}}{k!}=x\cdot{\displaystyle \sum_{k=0}^{\infty}}\frac{\left(\frac{3}{4}\right)_{k}\left(\frac{1}{2}\right)_{k}}{\left(\frac{3}{2}\right)_{k}}\frac{\left(x^{2}\right)^{k}}{k!} $
$=x\cdot_{2}F_{1}\left(\frac{1}{2},\frac{3}{4};\frac{3}{2};x^{2}\right) $
The secret to Hypergeometric functions (Generalized or not) is that successive coefficients of the Taylor type expansion ( $ x^{n} /n! $ ) are rational polynomials in n and the coefficients. These might be considered as things like combinatorial's $\left(\begin{array}{c} n\\ k \end{array}\right) $ on steroids. There are various introductions; of greater and lesser prices. A short one is the reduce-algebra Zeilberger module or you might try Wikipedia. There is real substance in these functions; and mysteries still remain (at least to me).