Theta Functions

This entry defines the Ramanujan theta function

$$f(a,b)=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{a}^{\frac{n(n+1)}{2}}{b}^{\frac{n(n-1)}{2}}$$ and derives from it the more commonly known Jacobi theta function on the unit disc $${\vartheta }_{00}(w,q)=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{w}^{2n}{q}^{n^2},$$ its version in the complex plane $${\vartheta }_{00}(z;\tau )=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\exp (i\pi (2nz+n^2\tau ))$$ as well as its half-period variants ${\vartheta }_{01}$, ${\vartheta }_{10}$, and ${\vartheta }_{11}$.

Notable results formalised include the fact that ${\vartheta }_{00}$ is a solution to the one-dimensional heat equation $\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{z}^2}f(z,t)=4i\pi \frac{\mathrm{\partial }}{\mathrm{\partial }t}f(z,t)$, and Jacobi's triple product $$\prod _{n=1}^{\mathrm{\infty }}(1-q^{2m})(1+q^{2m-1}{w}^2)(1+q^{2m-1}{w}^{-2})=\sum _{k=-\mathrm{\infty }}^{\mathrm{\infty }}{q}^{k^2}{w}^{2k}$$ as well as its corollary, Euler's famous pentagonal number theorem: $$\prod _{n=1}^{\mathrm{\infty }}(1-q^n)=\sum _{k=-\mathrm{\infty }}^{\mathrm{\infty }}(-1{)}^k{q}^{k(3k-1)/2}$$