자코비 형식
정의
- 2변수 함수로서의 정의
\begin{align*} % \theta_1(z;\tau) \theta_{11}(z;\tau) & = \sum_{n \in \mathbb{Z}} q^{\frac{1}{2} \left( n+ \frac{1}{2} \right)^2} \, \E^{2 \pi i \left(n+\frac{1}{2} \right) \, \left( z+\frac{1}{2} \right) } = - i \, \theta_1(z; \tau) , % \\ % & = % i \, q^{\frac{1}{8}} \, \E^{\pi \I z} \, % \left( % q, \E^{-2 \pi \I z}, \E^{2 \pi \I z} \, q % \right)_\infty \\[2mm] % % \theta_{2}(z;\tau) \theta_{10}(z;\tau) & = \sum_{n \in \mathbb{Z}} q^{\frac{1}{2} \left( n + \frac{1}{2} \right)^2} \, \E^{2 \pi i \left( n+\frac{1}{2} \right) z} = \theta_2(z;\tau) , \\[2mm] % % \theta_3 (z;\tau) \theta_{00} (z;\tau) & = \sum_{n \in \mathbb{Z}} q^{\frac{1}{2} n^2} \, \E^{2 \pi i n z} = \theta_3 (z;\tau) , \\[2mm] % % \theta_0 (z;\tau) \theta_{01} (z;\tau) & = \sum_{n \in \mathbb{Z}} q^{\frac{1}{2} n^2} \, \E^{2 \pi i n \left( z+\frac{1}{2} \right) } = \theta_4(z;\tau) , \end{align*}
모듈라 성질
\begin{equation} \begin{pmatrix} \theta_{11}(z;\tau) \\ \theta_{10}(z;\tau) \\ \theta_{00}(z;\tau) \\ \theta_{01}(z;\tau) \end{pmatrix}
=
\sqrt{ \frac{i}{\tau} } \, \E^{- \pi i \frac{z^2}{\tau}} \,
\begin{pmatrix}
i & & & \\
& & & 1 \\
& & 1 & \\
& 1 & &
\end{pmatrix} \,
\begin{pmatrix}
\theta_{11}\left( \frac{z}{\tau}; -\frac{1}{\tau} \right) \\
\theta_{10}\left( \frac{z}{\tau}; -\frac{1}{\tau} \right) \\
\theta_{00}\left( \frac{z}{\tau}; -\frac{1}{\tau} \right) \\
\theta_{01}\left( \frac{z}{\tau}; -\frac{1}{\tau} \right)
\end{pmatrix} .
\end{equation}