= Jacobi theta duplication identity
{c}
{title2=$\vartheta_{00}(z)\vartheta_{01}(z)\vartheta_{10}(z)\vartheta_{11}(z)=\tfrac12\vartheta_{00}(0)\vartheta_{01}(0)\vartheta_{10}(0)\vartheta_{11}(2z)$}
For the binary characteristic convention, each <theta function with characteristics> has one simple zero per lattice cell. The four zeros of the numerator, at the four half-periods, cancel those of $\vartheta_{11}(2z)$. Their quotient is an entire <elliptic function>, hence constant. Taking the limit at zero determines the factor one half. The product identity remains valid at the zeros, where the initial quotient requires removable extension.
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