Consider the following polynomial series:


$S(x) = \sum\_{i=1}^{\infty}(-1)^{i+1}x^{i^{2}}$


Between 0 and 1, this looks like a well-behaved function - is there any way to write this function in this interval without using a series?


Given $0 < S(x) < 1$, I need to solve the equation for $x$ (in the 0 to 1 interval), but an analytic solution would be much nicer than a numerical one...



Polynomial series
I don't know what you mean by "polynomial series" since your function doesn't seem to have much to do with polynomials (perhaps you could elaborate?).


$S(x) = -\frac12 \theta(\frac12, \frac{\log x}{\pi i }) - 1$, where $\theta$ is [Jacobi's theta function](http://en.wikipedia.org/wiki/Theta_function). I don't know of any nice algebraic methods to take inverses of modular forms (even if you restrict to real $x$, i.e., $\tau$ purely imaginary), so I'm not sure if this is the sort of answer you want.


