The wikipedia page on [Srinivasa Ramanujan](http://en.wikipedia.org/wiki/Srinivasa_Ramanujan) gives a very strange formula: 



> 
> **Ramanujan:** If $0 < a < b + \frac{1}{2}$ then, $$\int\limits\_{0}^{\infty} \frac{ 1 + x^{2}/(b+1)^{2}}{ 1 + x^{2}/a^{2}} \times \frac{ 1 + x^{2}/(b+2)^{2}}{ 1 + x^{2}/(a+1)^{2}} \times \cdots \ \textrm{dx} = \frac{\sqrt{\pi}}{2} \small{\frac{ \Gamma(a+\frac{1}{2}) \cdot \Gamma(b+1)\: \Gamma(b-a+\frac{1}{2})}{\Gamma(a) \cdot \Gamma(b+\frac{1}{2}) \cdot \Gamma(b-a+1)}}$$
> 
> 
> 


* Question I would like to pose to this community is: What could be the Intuition behind discovering this formula.
* Next, I see that Ramanujan has discovered a lot of formulas for expressing $\pi$ as series. May I know what is the advantage of having a same number expressed as a series in a different way. Is it useful at all?
* From what I know Ramanujan basically worked on *Infinite series, Continued fractions,* $\cdots$ etc. I have never seen applications of *continued fractions*, in the real world. I would also like to know if continued fractions has any applications.


Hope I haven't asked too many questions. As I was posting this question the last question on *application of continued fractions* popped up and I thought it would be a good idea to pose it here, instead of posing it as a new question. 



Ramanujan's eccentric Integral formula
This is one of those precious cases when Ramanujan himself provided (a sketch of) a proof. The identity was published in his paper ["Some definite integrals"](http://www.imsc.res.in/~rao/ramanujan/CamUnivCpapers/Cpaper11/page1.htm) (*Mess. Math.* 44 (1915), pp. 10-18) together with several related formulae. 


It might be instructive to look first at the simpler identity (i.e. the limiting case when $b\to\infty$; the identity mentioned in the original question can be obtained by a similar approach):
$$\int\limits\_{0}^{\infty} \prod\_{k=0}^{\infty}\frac{1}{ 1 + x^{2}/(a+k)^{2}}dx = \frac{\sqrt{\pi}}{2} \frac{ \Gamma(a+\frac{1}{2})}{\Gamma(a)},\quad a>0.\qquad\qquad\qquad(1)$$
Ramanujan derives (1) by using a partial fraction decomposition of the product $\prod\_{k=0}^{n}\frac{1}{ 1 + x^{2}/(a+k)^{2}}$, integrating term-wise, and passing to the limit $n\to\infty$. He also indicates that alternatively (1) is implied by the factorization 
$$\prod\_{k=0}^{\infty}\left[1+\frac{x^2}{(a+k)^2}\right] = \frac{ [\Gamma(a)]^2}{\Gamma(a+ix)\Gamma(a-ix)},$$
which follows readily from Euler's product formula for the gamma function. Thus (1) is equivalent to the formula 
$$\int\limits\_{0}^{\infty}\Gamma(a+ix)\Gamma(a-ix)dx=\frac{\sqrt{\pi}}{2} \Gamma(a)\Gamma\left(a+\frac{1}{2}\right).$$




---


There is a nice paper ["Wallis-Ramanujan-Schur-Feynman"](http://arminstraub.com/files/publications/ws.pdf) by Amdeberhan et al (*American Mathematical Monthly* 117 (2010), pp. 618-632) that discusses interesting combinatorial aspects of formula (1) and its generalizations. 


