Consider Euler's Function defined as (and not to be confused with the totient!)
$$ \phi(x) = (1-x)(1 - x^2) (1 - x^3 ) .... = \prod_{i=1}^{\infty} \left[(1 - x)^i \right] = (1;x)_{\infty}$$
Above are 3 different ways to notate it. Now the article: http://en.wikipedia.org/wiki/Euler_function describes special values of the given function found by ramanujan such as:
$$ \phi(e^{-\pi}) = \frac{e^{\frac{\pi}{24}}\Gamma(\frac{1}{4})}{2^{\frac{7}{8}}\pi^{\frac{3}{4}}} $$
And there are a couple more. But no link is given to their motivation/derivation. I googled around and couldn't find anything. Why are these true?
Where does even begin to retrieve the values?