ヤコビの二平方定理
テータ関数の比は楕円関数(二重周期を持つ有理型関数)になり、楕円関数の導関数も楕円関数になるから、
F ( v ) = ∂ ∂ v ( θ 1 ( v , τ ) θ 2 ( v , τ ) ) = θ 1 ′ ( v , τ ) θ 2 ( v , τ ) − θ 1 ( v , τ ) θ 2 ′ ( v , τ ) θ 2 ( v , τ ) 2 G ( v ) = ( θ 3 ( v , τ ) θ 2 ( v , τ ) ) ( θ 4 ( v , τ ) θ 2 ( v , τ ) ) = θ 3 ( v , τ ) θ 4 ( v , τ ) θ 2 ( v , τ ) 2 {\displaystyle {\begin{aligned}F(v)&={\frac {\partial }{\partial {v}}}\left({\frac {\vartheta _{1}(v,\tau )}{\vartheta _{2}(v,\tau )}}\right)={\frac {\vartheta _{1}'(v,\tau )\vartheta _{2}(v,\tau )-\vartheta _{1}(v,\tau )\vartheta _{2}'(v,\tau )}{\vartheta _{2}(v,\tau )^{2}}}\\G(v)&=\left({\frac {\vartheta _{3}(v,\tau )}{\vartheta _{2}(v,\tau )}}\right)\left({\frac {\vartheta _{4}(v,\tau )}{\vartheta _{2}(v,\tau )}}\right)={\frac {\vartheta _{3}(v,\tau )\vartheta _{4}(v,\tau )}{\vartheta _{2}(v,\tau )^{2}}}\\\end{aligned}}}
となり、ヤコビの三重積の公式により