ヤコビの虚数変換式
「岩波数学公式集III」p.48.では誤った式が書かれているので注意せよ。
{\displaystyle {\begin{aligned}\theta _{0}(z,\tau )&:=\theta _{01}(z,\tau )\\&:=\sum _{n=-\infty }^{\infty }e^{\pi i\tau n^{2}+2\pi in\left(z+{\frac {1}{2}}\right)}\\&=1+2\sum _{n=1}^{\infty }(-1)^{n}e^{\pi i\tau n^{2}}\cos 2n\pi z,\\\theta _{1}(z,\tau )&:=-\theta _{11}(z,\tau )\\&:=-\sum _{n=-\infty }^{\infty }{e^{\pi i\tau \left(n+{\frac {1}{2}}\right)^{2}+2\pi i\left(n+{\frac {1}{2}}\right)\left(z+{\frac {1}{2}}\right)}}\\&=2\sum _{n=0}^{\infty }(-1)^{n}e^{\pi i\tau \left(n+{\frac {1}{2}}\right)^{2}}\sin(2n+1)\pi z,\\\theta _{2}(z,\tau )&:=\theta _{10}(z,\tau )\\&:=\sum _{n=-\infty }^{\infty }e^{\pi i\tau \left(n+{\frac {1}{2}}\right)^{2}+2\pi i\left(n+{\frac {1}{2}}\right)z}\\&=2\sum _{n=0}^{\infty }e^{\pi i\tau \left(n+{\frac {1}{2}}\right)^{2}}\cos(2n+1)\pi z,\\\theta _{3}(z,\tau )&:=\theta _{00}(z,\tau )\\&:=\sum _{n=-\infty }^{\infty }e^{\pi i\tau n^{2}+2\pi inz}\\&=1+2\sum _{n=1}^{\infty }e^{{\pi }i{\tau }n^{2}}\cos 2n\pi z.\end{aligned}}}