ヤコビの三重積
θ ( v + τ ; τ ) = ∑ n = − ∞ ∞ e π i τ n 2 + 2 π i n ( v + τ ) = ∑ n = − ∞ ∞ e π i τ ( n + 1 ) 2 − π i τ + 2 π i ( n + 1 ) v − 2 π i v = ∑ n = − ∞ ∞ e π i τ n 2 − π i τ + 2 π i n v − 2 π i v = e − π i τ e − 2 π i v θ ( v ; τ ) {\displaystyle {\begin{aligned}\vartheta (v+\tau ;\tau )&=\sum _{n=-\infty }^{\infty }{e^{{\pi }i{\tau }n^{2}+2{\pi }in(v+\tau )}}\\&=\sum _{n=-\infty }^{\infty }{e^{{\pi }i{\tau }(n+1)^{2}-{\pi }i{\tau }+2{\pi }i(n+1)v-2{\pi }iv}}\\&=\sum _{n=-\infty }^{\infty }{e^{{\pi }i{\tau }n^{2}-{\pi }i{\tau }+2{\pi }inv-2{\pi }iv}}\\&=e^{-{\pi }i\tau }e^{-2{\pi }i{v}}\vartheta (v;\tau )\\\end{aligned}}}