Abstract
This entry defines the Ramanujan theta function
\[f(a,b) = \sum_{n=-\infty}^\infty a^{\frac{n(n+1)}{2}} b^{\frac{n(n-1)}{2}}\] and derives from it the more commonly known Jacobi theta function on the unit disc \[\vartheta_{00}(w,q) = \sum_{n=-\infty}^\infty w^{2n} q^{n^2},\ \] its version in the complex plane \[\vartheta_{00}(z;\tau) = \sum_{n=-\infty}^\infty \exp(i\pi (2nz + n^2\tau))\] as well as its half-period variants $\vartheta_{01}$, $\vartheta_{10}$, and $\vartheta_{11}$.The most notable single result in this work is the proof of Jacobi's triple product \[\prod_{n=1}^\infty (1-q^{2m})(1+q^{2m-1}w^2)(1+q^{2m-1}w^{-2}) = \sum_{k=-\infty}^\infty q^{k^2}w^{2k}\] and its corollary, Euler's famous pentagonal number theorem: \[\prod_{n=1}^\infty (1-q^n) = \sum_{k=-\infty}^\infty (-1)^k q^{k(3k-1)/2}\]