ヤコビの虚数変換式
f ( 1 2 , τ ) = − i τ e π i / 4 τ θ 3 ( 1 2 , τ ) θ 3 ( 1 2 τ , − 1 τ ) = − i τ ∑ n = − ∞ ∞ e n 2 π i τ + n π i ∑ n = − ∞ ∞ e n 2 − π i / τ + n π i / τ e − π i / 4 τ = − i τ ∑ n = − ∞ ∞ ( − 1 ) n e n 2 π i τ ∑ n = − ∞ ∞ e − ( 2 n − 1 ) 2 π i / 4 τ = − i τ ∑ n = − ∞ ∞ ( − 1 ) n e n 2 π i τ 2 ∑ n = 1 ∞ e − ( 2 n − 1 ) 2 π i / 4 τ {\displaystyle {\begin{aligned}f\left({\frac {1}{2}},\tau \right)&={\frac {{\sqrt {-i\tau }}e^{{\pi }i/4\tau }\vartheta _{3}\left({\frac {1}{2}},\tau \right)}{\vartheta _{3}\left({\frac {1}{2\tau }},-{\frac {1}{\tau }}\right)}}={\frac {{\sqrt {-i\tau }}\sum _{n=-\infty }^{\infty }{e^{n^{2}{\pi }i\tau +n{\pi }i}}}{\sum _{n=-\infty }^{\infty }{e^{n^{2}{-\pi }i/\tau +n{\pi }i/\tau }e^{-{\pi }i/4\tau }}}}\\&={\frac {{\sqrt {-i\tau }}\sum _{n=-\infty }^{\infty }{(-1)^{n}e^{n^{2}{\pi }i{\tau }}}}{\sum _{n=-\infty }^{\infty }{e^{-(2n-1)^{2}{\pi }i/4\tau }}}}\\&={\frac {{\sqrt {-i\tau }}\sum _{n=-\infty }^{\infty }{(-1)^{n}e^{n^{2}{\pi }i{\tau }}}}{2\sum _{n=1}^{\infty }{e^{-(2n-1)^{2}{\pi }i/4\tau }}}}\\\end{aligned}}}
ヤコビの虚数変換式(Jacobi's imaginary transformation)は、楕円テータ関数に関する以下のような恒等式である。
θ 3 ( v τ , − 1 τ ) = e − π i / 4 τ 1 / 2 e π i v 2 / τ θ 3 ( v , τ ) {\displaystyle \vartheta _{3}\left({\frac {v}{\tau }},-{\frac {1}{\tau }}\right)=e^{-\pi i/4}\tau ^{1/2}e^{{\pi }iv^{2}/\tau }\vartheta _{3}\left(v,\tau \right)}
θ 1 ( v τ , − 1 τ ) = − i e − π i / 4 τ 1 / 2 e π i v 2 / τ θ 1 ( v , τ ) {\displaystyle \vartheta _{1}\left({\frac {v}{\tau }},-{\frac {1}{\tau }}\right)=-ie^{-\pi i/4}\tau ^{1/2}e^{{\pi }iv^{2}/\tau }\vartheta _{1}\left(v,\tau \right)}